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# Polytope of Type {4,6,4}

Atlas Canonical Name : {4,6,4}*768c
if this polytope has a name.
Group : SmallGroup(768,1088921)
Rank : 4
Schlafli Type : {4,6,4}
Number of vertices, edges, etc : 4, 48, 48, 16
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,6,4}*384a, {2,6,4}*384b
4-fold quotients : {4,6,4}*192a, {4,6,4}*192b, {2,6,4}*192
8-fold quotients : {2,6,4}*96a, {4,6,2}*96a, {2,3,4}*96, {2,6,4}*96b, {2,6,4}*96c
12-fold quotients : {4,2,4}*64
16-fold quotients : {2,3,4}*48, {2,6,2}*48
24-fold quotients : {2,2,4}*32, {4,2,2}*32
32-fold quotients : {2,3,2}*24
48-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (  1,193)(  2,194)(  3,195)(  4,196)(  5,197)(  6,198)(  7,199)(  8,200)
(  9,201)( 10,202)( 11,203)( 12,204)( 13,205)( 14,206)( 15,207)( 16,208)
( 17,209)( 18,210)( 19,211)( 20,212)( 21,213)( 22,214)( 23,215)( 24,216)
( 25,217)( 26,218)( 27,219)( 28,220)( 29,221)( 30,222)( 31,223)( 32,224)
( 33,225)( 34,226)( 35,227)( 36,228)( 37,229)( 38,230)( 39,231)( 40,232)
( 41,233)( 42,234)( 43,235)( 44,236)( 45,237)( 46,238)( 47,239)( 48,240)
( 49,253)( 50,254)( 51,255)( 52,256)( 53,257)( 54,258)( 55,259)( 56,260)
( 57,261)( 58,262)( 59,263)( 60,264)( 61,241)( 62,242)( 63,243)( 64,244)
( 65,245)( 66,246)( 67,247)( 68,248)( 69,249)( 70,250)( 71,251)( 72,252)
( 73,277)( 74,278)( 75,279)( 76,280)( 77,281)( 78,282)( 79,283)( 80,284)
( 81,285)( 82,286)( 83,287)( 84,288)( 85,265)( 86,266)( 87,267)( 88,268)
( 89,269)( 90,270)( 91,271)( 92,272)( 93,273)( 94,274)( 95,275)( 96,276)
( 97,289)( 98,290)( 99,291)(100,292)(101,293)(102,294)(103,295)(104,296)
(105,297)(106,298)(107,299)(108,300)(109,301)(110,302)(111,303)(112,304)
(113,305)(114,306)(115,307)(116,308)(117,309)(118,310)(119,311)(120,312)
(121,313)(122,314)(123,315)(124,316)(125,317)(126,318)(127,319)(128,320)
(129,321)(130,322)(131,323)(132,324)(133,325)(134,326)(135,327)(136,328)
(137,329)(138,330)(139,331)(140,332)(141,333)(142,334)(143,335)(144,336)
(145,349)(146,350)(147,351)(148,352)(149,353)(150,354)(151,355)(152,356)
(153,357)(154,358)(155,359)(156,360)(157,337)(158,338)(159,339)(160,340)
(161,341)(162,342)(163,343)(164,344)(165,345)(166,346)(167,347)(168,348)
(169,373)(170,374)(171,375)(172,376)(173,377)(174,378)(175,379)(176,380)
(177,381)(178,382)(179,383)(180,384)(181,361)(182,362)(183,363)(184,364)
(185,365)(186,366)(187,367)(188,368)(189,369)(190,370)(191,371)(192,372);;
s1 := (  3,  4)(  5,  9)(  6, 10)(  7, 12)(  8, 11)( 15, 16)( 17, 21)( 18, 22)
( 19, 24)( 20, 23)( 27, 28)( 29, 33)( 30, 34)( 31, 36)( 32, 35)( 39, 40)
( 41, 45)( 42, 46)( 43, 48)( 44, 47)( 51, 52)( 53, 57)( 54, 58)( 55, 60)
( 56, 59)( 63, 64)( 65, 69)( 66, 70)( 67, 72)( 68, 71)( 75, 76)( 77, 81)
( 78, 82)( 79, 84)( 80, 83)( 87, 88)( 89, 93)( 90, 94)( 91, 96)( 92, 95)
( 99,100)(101,105)(102,106)(103,108)(104,107)(111,112)(113,117)(114,118)
(115,120)(116,119)(123,124)(125,129)(126,130)(127,132)(128,131)(135,136)
(137,141)(138,142)(139,144)(140,143)(147,148)(149,153)(150,154)(151,156)
(152,155)(159,160)(161,165)(162,166)(163,168)(164,167)(171,172)(173,177)
(174,178)(175,180)(176,179)(183,184)(185,189)(186,190)(187,192)(188,191)
(193,205)(194,206)(195,208)(196,207)(197,213)(198,214)(199,216)(200,215)
(201,209)(202,210)(203,212)(204,211)(217,229)(218,230)(219,232)(220,231)
(221,237)(222,238)(223,240)(224,239)(225,233)(226,234)(227,236)(228,235)
(241,253)(242,254)(243,256)(244,255)(245,261)(246,262)(247,264)(248,263)
(249,257)(250,258)(251,260)(252,259)(265,277)(266,278)(267,280)(268,279)
(269,285)(270,286)(271,288)(272,287)(273,281)(274,282)(275,284)(276,283)
(289,301)(290,302)(291,304)(292,303)(293,309)(294,310)(295,312)(296,311)
(297,305)(298,306)(299,308)(300,307)(313,325)(314,326)(315,328)(316,327)
(317,333)(318,334)(319,336)(320,335)(321,329)(322,330)(323,332)(324,331)
(337,349)(338,350)(339,352)(340,351)(341,357)(342,358)(343,360)(344,359)
(345,353)(346,354)(347,356)(348,355)(361,373)(362,374)(363,376)(364,375)
(365,381)(366,382)(367,384)(368,383)(369,377)(370,378)(371,380)(372,379);;
s2 := (  1, 57)(  2, 59)(  3, 58)(  4, 60)(  5, 53)(  6, 55)(  7, 54)(  8, 56)
(  9, 49)( 10, 51)( 11, 50)( 12, 52)( 13, 69)( 14, 71)( 15, 70)( 16, 72)
( 17, 65)( 18, 67)( 19, 66)( 20, 68)( 21, 61)( 22, 63)( 23, 62)( 24, 64)
( 25, 81)( 26, 83)( 27, 82)( 28, 84)( 29, 77)( 30, 79)( 31, 78)( 32, 80)
( 33, 73)( 34, 75)( 35, 74)( 36, 76)( 37, 93)( 38, 95)( 39, 94)( 40, 96)
( 41, 89)( 42, 91)( 43, 90)( 44, 92)( 45, 85)( 46, 87)( 47, 86)( 48, 88)
( 97,153)( 98,155)( 99,154)(100,156)(101,149)(102,151)(103,150)(104,152)
(105,145)(106,147)(107,146)(108,148)(109,165)(110,167)(111,166)(112,168)
(113,161)(114,163)(115,162)(116,164)(117,157)(118,159)(119,158)(120,160)
(121,177)(122,179)(123,178)(124,180)(125,173)(126,175)(127,174)(128,176)
(129,169)(130,171)(131,170)(132,172)(133,189)(134,191)(135,190)(136,192)
(137,185)(138,187)(139,186)(140,188)(141,181)(142,183)(143,182)(144,184)
(193,261)(194,263)(195,262)(196,264)(197,257)(198,259)(199,258)(200,260)
(201,253)(202,255)(203,254)(204,256)(205,249)(206,251)(207,250)(208,252)
(209,245)(210,247)(211,246)(212,248)(213,241)(214,243)(215,242)(216,244)
(217,285)(218,287)(219,286)(220,288)(221,281)(222,283)(223,282)(224,284)
(225,277)(226,279)(227,278)(228,280)(229,273)(230,275)(231,274)(232,276)
(233,269)(234,271)(235,270)(236,272)(237,265)(238,267)(239,266)(240,268)
(289,357)(290,359)(291,358)(292,360)(293,353)(294,355)(295,354)(296,356)
(297,349)(298,351)(299,350)(300,352)(301,345)(302,347)(303,346)(304,348)
(305,341)(306,343)(307,342)(308,344)(309,337)(310,339)(311,338)(312,340)
(313,381)(314,383)(315,382)(316,384)(317,377)(318,379)(319,378)(320,380)
(321,373)(322,375)(323,374)(324,376)(325,369)(326,371)(327,370)(328,372)
(329,365)(330,367)(331,366)(332,368)(333,361)(334,363)(335,362)(336,364);;
s3 := (  1, 98)(  2, 97)(  3,100)(  4, 99)(  5,102)(  6,101)(  7,104)(  8,103)
(  9,106)( 10,105)( 11,108)( 12,107)( 13,110)( 14,109)( 15,112)( 16,111)
( 17,114)( 18,113)( 19,116)( 20,115)( 21,118)( 22,117)( 23,120)( 24,119)
( 25,122)( 26,121)( 27,124)( 28,123)( 29,126)( 30,125)( 31,128)( 32,127)
( 33,130)( 34,129)( 35,132)( 36,131)( 37,134)( 38,133)( 39,136)( 40,135)
( 41,138)( 42,137)( 43,140)( 44,139)( 45,142)( 46,141)( 47,144)( 48,143)
( 49,170)( 50,169)( 51,172)( 52,171)( 53,174)( 54,173)( 55,176)( 56,175)
( 57,178)( 58,177)( 59,180)( 60,179)( 61,182)( 62,181)( 63,184)( 64,183)
( 65,186)( 66,185)( 67,188)( 68,187)( 69,190)( 70,189)( 71,192)( 72,191)
( 73,146)( 74,145)( 75,148)( 76,147)( 77,150)( 78,149)( 79,152)( 80,151)
( 81,154)( 82,153)( 83,156)( 84,155)( 85,158)( 86,157)( 87,160)( 88,159)
( 89,162)( 90,161)( 91,164)( 92,163)( 93,166)( 94,165)( 95,168)( 96,167)
(193,290)(194,289)(195,292)(196,291)(197,294)(198,293)(199,296)(200,295)
(201,298)(202,297)(203,300)(204,299)(205,302)(206,301)(207,304)(208,303)
(209,306)(210,305)(211,308)(212,307)(213,310)(214,309)(215,312)(216,311)
(217,314)(218,313)(219,316)(220,315)(221,318)(222,317)(223,320)(224,319)
(225,322)(226,321)(227,324)(228,323)(229,326)(230,325)(231,328)(232,327)
(233,330)(234,329)(235,332)(236,331)(237,334)(238,333)(239,336)(240,335)
(241,362)(242,361)(243,364)(244,363)(245,366)(246,365)(247,368)(248,367)
(249,370)(250,369)(251,372)(252,371)(253,374)(254,373)(255,376)(256,375)
(257,378)(258,377)(259,380)(260,379)(261,382)(262,381)(263,384)(264,383)
(265,338)(266,337)(267,340)(268,339)(269,342)(270,341)(271,344)(272,343)
(273,346)(274,345)(275,348)(276,347)(277,350)(278,349)(279,352)(280,351)
(281,354)(282,353)(283,356)(284,355)(285,358)(286,357)(287,360)(288,359);;
poly := Group([s0,s1,s2,s3]);;

```
Finitely Presented Group Representation (GAP) :
```F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1, s2*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s3*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s1*s2 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(384)!(  1,193)(  2,194)(  3,195)(  4,196)(  5,197)(  6,198)(  7,199)
(  8,200)(  9,201)( 10,202)( 11,203)( 12,204)( 13,205)( 14,206)( 15,207)
( 16,208)( 17,209)( 18,210)( 19,211)( 20,212)( 21,213)( 22,214)( 23,215)
( 24,216)( 25,217)( 26,218)( 27,219)( 28,220)( 29,221)( 30,222)( 31,223)
( 32,224)( 33,225)( 34,226)( 35,227)( 36,228)( 37,229)( 38,230)( 39,231)
( 40,232)( 41,233)( 42,234)( 43,235)( 44,236)( 45,237)( 46,238)( 47,239)
( 48,240)( 49,253)( 50,254)( 51,255)( 52,256)( 53,257)( 54,258)( 55,259)
( 56,260)( 57,261)( 58,262)( 59,263)( 60,264)( 61,241)( 62,242)( 63,243)
( 64,244)( 65,245)( 66,246)( 67,247)( 68,248)( 69,249)( 70,250)( 71,251)
( 72,252)( 73,277)( 74,278)( 75,279)( 76,280)( 77,281)( 78,282)( 79,283)
( 80,284)( 81,285)( 82,286)( 83,287)( 84,288)( 85,265)( 86,266)( 87,267)
( 88,268)( 89,269)( 90,270)( 91,271)( 92,272)( 93,273)( 94,274)( 95,275)
( 96,276)( 97,289)( 98,290)( 99,291)(100,292)(101,293)(102,294)(103,295)
(104,296)(105,297)(106,298)(107,299)(108,300)(109,301)(110,302)(111,303)
(112,304)(113,305)(114,306)(115,307)(116,308)(117,309)(118,310)(119,311)
(120,312)(121,313)(122,314)(123,315)(124,316)(125,317)(126,318)(127,319)
(128,320)(129,321)(130,322)(131,323)(132,324)(133,325)(134,326)(135,327)
(136,328)(137,329)(138,330)(139,331)(140,332)(141,333)(142,334)(143,335)
(144,336)(145,349)(146,350)(147,351)(148,352)(149,353)(150,354)(151,355)
(152,356)(153,357)(154,358)(155,359)(156,360)(157,337)(158,338)(159,339)
(160,340)(161,341)(162,342)(163,343)(164,344)(165,345)(166,346)(167,347)
(168,348)(169,373)(170,374)(171,375)(172,376)(173,377)(174,378)(175,379)
(176,380)(177,381)(178,382)(179,383)(180,384)(181,361)(182,362)(183,363)
(184,364)(185,365)(186,366)(187,367)(188,368)(189,369)(190,370)(191,371)
(192,372);
s1 := Sym(384)!(  3,  4)(  5,  9)(  6, 10)(  7, 12)(  8, 11)( 15, 16)( 17, 21)
( 18, 22)( 19, 24)( 20, 23)( 27, 28)( 29, 33)( 30, 34)( 31, 36)( 32, 35)
( 39, 40)( 41, 45)( 42, 46)( 43, 48)( 44, 47)( 51, 52)( 53, 57)( 54, 58)
( 55, 60)( 56, 59)( 63, 64)( 65, 69)( 66, 70)( 67, 72)( 68, 71)( 75, 76)
( 77, 81)( 78, 82)( 79, 84)( 80, 83)( 87, 88)( 89, 93)( 90, 94)( 91, 96)
( 92, 95)( 99,100)(101,105)(102,106)(103,108)(104,107)(111,112)(113,117)
(114,118)(115,120)(116,119)(123,124)(125,129)(126,130)(127,132)(128,131)
(135,136)(137,141)(138,142)(139,144)(140,143)(147,148)(149,153)(150,154)
(151,156)(152,155)(159,160)(161,165)(162,166)(163,168)(164,167)(171,172)
(173,177)(174,178)(175,180)(176,179)(183,184)(185,189)(186,190)(187,192)
(188,191)(193,205)(194,206)(195,208)(196,207)(197,213)(198,214)(199,216)
(200,215)(201,209)(202,210)(203,212)(204,211)(217,229)(218,230)(219,232)
(220,231)(221,237)(222,238)(223,240)(224,239)(225,233)(226,234)(227,236)
(228,235)(241,253)(242,254)(243,256)(244,255)(245,261)(246,262)(247,264)
(248,263)(249,257)(250,258)(251,260)(252,259)(265,277)(266,278)(267,280)
(268,279)(269,285)(270,286)(271,288)(272,287)(273,281)(274,282)(275,284)
(276,283)(289,301)(290,302)(291,304)(292,303)(293,309)(294,310)(295,312)
(296,311)(297,305)(298,306)(299,308)(300,307)(313,325)(314,326)(315,328)
(316,327)(317,333)(318,334)(319,336)(320,335)(321,329)(322,330)(323,332)
(324,331)(337,349)(338,350)(339,352)(340,351)(341,357)(342,358)(343,360)
(344,359)(345,353)(346,354)(347,356)(348,355)(361,373)(362,374)(363,376)
(364,375)(365,381)(366,382)(367,384)(368,383)(369,377)(370,378)(371,380)
(372,379);
s2 := Sym(384)!(  1, 57)(  2, 59)(  3, 58)(  4, 60)(  5, 53)(  6, 55)(  7, 54)
(  8, 56)(  9, 49)( 10, 51)( 11, 50)( 12, 52)( 13, 69)( 14, 71)( 15, 70)
( 16, 72)( 17, 65)( 18, 67)( 19, 66)( 20, 68)( 21, 61)( 22, 63)( 23, 62)
( 24, 64)( 25, 81)( 26, 83)( 27, 82)( 28, 84)( 29, 77)( 30, 79)( 31, 78)
( 32, 80)( 33, 73)( 34, 75)( 35, 74)( 36, 76)( 37, 93)( 38, 95)( 39, 94)
( 40, 96)( 41, 89)( 42, 91)( 43, 90)( 44, 92)( 45, 85)( 46, 87)( 47, 86)
( 48, 88)( 97,153)( 98,155)( 99,154)(100,156)(101,149)(102,151)(103,150)
(104,152)(105,145)(106,147)(107,146)(108,148)(109,165)(110,167)(111,166)
(112,168)(113,161)(114,163)(115,162)(116,164)(117,157)(118,159)(119,158)
(120,160)(121,177)(122,179)(123,178)(124,180)(125,173)(126,175)(127,174)
(128,176)(129,169)(130,171)(131,170)(132,172)(133,189)(134,191)(135,190)
(136,192)(137,185)(138,187)(139,186)(140,188)(141,181)(142,183)(143,182)
(144,184)(193,261)(194,263)(195,262)(196,264)(197,257)(198,259)(199,258)
(200,260)(201,253)(202,255)(203,254)(204,256)(205,249)(206,251)(207,250)
(208,252)(209,245)(210,247)(211,246)(212,248)(213,241)(214,243)(215,242)
(216,244)(217,285)(218,287)(219,286)(220,288)(221,281)(222,283)(223,282)
(224,284)(225,277)(226,279)(227,278)(228,280)(229,273)(230,275)(231,274)
(232,276)(233,269)(234,271)(235,270)(236,272)(237,265)(238,267)(239,266)
(240,268)(289,357)(290,359)(291,358)(292,360)(293,353)(294,355)(295,354)
(296,356)(297,349)(298,351)(299,350)(300,352)(301,345)(302,347)(303,346)
(304,348)(305,341)(306,343)(307,342)(308,344)(309,337)(310,339)(311,338)
(312,340)(313,381)(314,383)(315,382)(316,384)(317,377)(318,379)(319,378)
(320,380)(321,373)(322,375)(323,374)(324,376)(325,369)(326,371)(327,370)
(328,372)(329,365)(330,367)(331,366)(332,368)(333,361)(334,363)(335,362)
(336,364);
s3 := Sym(384)!(  1, 98)(  2, 97)(  3,100)(  4, 99)(  5,102)(  6,101)(  7,104)
(  8,103)(  9,106)( 10,105)( 11,108)( 12,107)( 13,110)( 14,109)( 15,112)
( 16,111)( 17,114)( 18,113)( 19,116)( 20,115)( 21,118)( 22,117)( 23,120)
( 24,119)( 25,122)( 26,121)( 27,124)( 28,123)( 29,126)( 30,125)( 31,128)
( 32,127)( 33,130)( 34,129)( 35,132)( 36,131)( 37,134)( 38,133)( 39,136)
( 40,135)( 41,138)( 42,137)( 43,140)( 44,139)( 45,142)( 46,141)( 47,144)
( 48,143)( 49,170)( 50,169)( 51,172)( 52,171)( 53,174)( 54,173)( 55,176)
( 56,175)( 57,178)( 58,177)( 59,180)( 60,179)( 61,182)( 62,181)( 63,184)
( 64,183)( 65,186)( 66,185)( 67,188)( 68,187)( 69,190)( 70,189)( 71,192)
( 72,191)( 73,146)( 74,145)( 75,148)( 76,147)( 77,150)( 78,149)( 79,152)
( 80,151)( 81,154)( 82,153)( 83,156)( 84,155)( 85,158)( 86,157)( 87,160)
( 88,159)( 89,162)( 90,161)( 91,164)( 92,163)( 93,166)( 94,165)( 95,168)
( 96,167)(193,290)(194,289)(195,292)(196,291)(197,294)(198,293)(199,296)
(200,295)(201,298)(202,297)(203,300)(204,299)(205,302)(206,301)(207,304)
(208,303)(209,306)(210,305)(211,308)(212,307)(213,310)(214,309)(215,312)
(216,311)(217,314)(218,313)(219,316)(220,315)(221,318)(222,317)(223,320)
(224,319)(225,322)(226,321)(227,324)(228,323)(229,326)(230,325)(231,328)
(232,327)(233,330)(234,329)(235,332)(236,331)(237,334)(238,333)(239,336)
(240,335)(241,362)(242,361)(243,364)(244,363)(245,366)(246,365)(247,368)
(248,367)(249,370)(250,369)(251,372)(252,371)(253,374)(254,373)(255,376)
(256,375)(257,378)(258,377)(259,380)(260,379)(261,382)(262,381)(263,384)
(264,383)(265,338)(266,337)(267,340)(268,339)(269,342)(270,341)(271,344)
(272,343)(273,346)(274,345)(275,348)(276,347)(277,350)(278,349)(279,352)
(280,351)(281,354)(282,353)(283,356)(284,355)(285,358)(286,357)(287,360)
(288,359);
poly := sub<Sym(384)|s0,s1,s2,s3>;

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2,
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1,
s2*s3*s2*s3*s2*s3*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s3*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s1*s2 >;

```
References : None.
to this polytope