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# Polytope of Type {4,6,8}

Atlas Canonical Name : {4,6,8}*768a
if this polytope has a name.
Group : SmallGroup(768,1089270)
Rank : 4
Schlafli Type : {4,6,8}
Number of vertices, edges, etc : 8, 24, 48, 8
Order of s0s1s2s3 : 24
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 : {4,6,8}*384b, {4,6,4}*384b
4-fold quotients : {2,6,8}*192, {4,6,4}*192c, {4,6,2}*192
8-fold quotients : {2,6,4}*96a, {4,3,2}*96, {4,6,2}*96b, {4,6,2}*96c
12-fold quotients : {2,2,8}*64
16-fold quotients : {4,3,2}*48, {2,6,2}*48
24-fold quotients : {2,2,4}*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 := (  1,196)(  2,195)(  3,194)(  4,193)(  5,200)(  6,199)(  7,198)(  8,197)
(  9,204)( 10,203)( 11,202)( 12,201)( 13,208)( 14,207)( 15,206)( 16,205)
( 17,212)( 18,211)( 19,210)( 20,209)( 21,216)( 22,215)( 23,214)( 24,213)
( 25,220)( 26,219)( 27,218)( 28,217)( 29,224)( 30,223)( 31,222)( 32,221)
( 33,228)( 34,227)( 35,226)( 36,225)( 37,232)( 38,231)( 39,230)( 40,229)
( 41,236)( 42,235)( 43,234)( 44,233)( 45,240)( 46,239)( 47,238)( 48,237)
( 49,244)( 50,243)( 51,242)( 52,241)( 53,248)( 54,247)( 55,246)( 56,245)
( 57,252)( 58,251)( 59,250)( 60,249)( 61,256)( 62,255)( 63,254)( 64,253)
( 65,260)( 66,259)( 67,258)( 68,257)( 69,264)( 70,263)( 71,262)( 72,261)
( 73,268)( 74,267)( 75,266)( 76,265)( 77,272)( 78,271)( 79,270)( 80,269)
( 81,276)( 82,275)( 83,274)( 84,273)( 85,280)( 86,279)( 87,278)( 88,277)
( 89,284)( 90,283)( 91,282)( 92,281)( 93,288)( 94,287)( 95,286)( 96,285)
( 97,292)( 98,291)( 99,290)(100,289)(101,296)(102,295)(103,294)(104,293)
(105,300)(106,299)(107,298)(108,297)(109,304)(110,303)(111,302)(112,301)
(113,308)(114,307)(115,306)(116,305)(117,312)(118,311)(119,310)(120,309)
(121,316)(122,315)(123,314)(124,313)(125,320)(126,319)(127,318)(128,317)
(129,324)(130,323)(131,322)(132,321)(133,328)(134,327)(135,326)(136,325)
(137,332)(138,331)(139,330)(140,329)(141,336)(142,335)(143,334)(144,333)
(145,340)(146,339)(147,338)(148,337)(149,344)(150,343)(151,342)(152,341)
(153,348)(154,347)(155,346)(156,345)(157,352)(158,351)(159,350)(160,349)
(161,356)(162,355)(163,354)(164,353)(165,360)(166,359)(167,358)(168,357)
(169,364)(170,363)(171,362)(172,361)(173,368)(174,367)(175,366)(176,365)
(177,372)(178,371)(179,370)(180,369)(181,376)(182,375)(183,374)(184,373)
(185,380)(186,379)(187,378)(188,377)(189,384)(190,383)(191,382)(192,381);;
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)
(195,196)(197,201)(198,202)(199,204)(200,203)(207,208)(209,213)(210,214)
(211,216)(212,215)(219,220)(221,225)(222,226)(223,228)(224,227)(231,232)
(233,237)(234,238)(235,240)(236,239)(243,244)(245,249)(246,250)(247,252)
(248,251)(255,256)(257,261)(258,262)(259,264)(260,263)(267,268)(269,273)
(270,274)(271,276)(272,275)(279,280)(281,285)(282,286)(283,288)(284,287)
(291,292)(293,297)(294,298)(295,300)(296,299)(303,304)(305,309)(306,310)
(307,312)(308,311)(315,316)(317,321)(318,322)(319,324)(320,323)(327,328)
(329,333)(330,334)(331,336)(332,335)(339,340)(341,345)(342,346)(343,348)
(344,347)(351,352)(353,357)(354,358)(355,360)(356,359)(363,364)(365,369)
(366,370)(367,372)(368,371)(375,376)(377,381)(378,382)(379,384)(380,383);;
s2 := (  1,105)(  2,107)(  3,106)(  4,108)(  5,101)(  6,103)(  7,102)(  8,104)
(  9, 97)( 10, 99)( 11, 98)( 12,100)( 13,117)( 14,119)( 15,118)( 16,120)
( 17,113)( 18,115)( 19,114)( 20,116)( 21,109)( 22,111)( 23,110)( 24,112)
( 25,141)( 26,143)( 27,142)( 28,144)( 29,137)( 30,139)( 31,138)( 32,140)
( 33,133)( 34,135)( 35,134)( 36,136)( 37,129)( 38,131)( 39,130)( 40,132)
( 41,125)( 42,127)( 43,126)( 44,128)( 45,121)( 46,123)( 47,122)( 48,124)
( 49,177)( 50,179)( 51,178)( 52,180)( 53,173)( 54,175)( 55,174)( 56,176)
( 57,169)( 58,171)( 59,170)( 60,172)( 61,189)( 62,191)( 63,190)( 64,192)
( 65,185)( 66,187)( 67,186)( 68,188)( 69,181)( 70,183)( 71,182)( 72,184)
( 73,153)( 74,155)( 75,154)( 76,156)( 77,149)( 78,151)( 79,150)( 80,152)
( 81,145)( 82,147)( 83,146)( 84,148)( 85,165)( 86,167)( 87,166)( 88,168)
( 89,161)( 90,163)( 91,162)( 92,164)( 93,157)( 94,159)( 95,158)( 96,160)
(193,297)(194,299)(195,298)(196,300)(197,293)(198,295)(199,294)(200,296)
(201,289)(202,291)(203,290)(204,292)(205,309)(206,311)(207,310)(208,312)
(209,305)(210,307)(211,306)(212,308)(213,301)(214,303)(215,302)(216,304)
(217,333)(218,335)(219,334)(220,336)(221,329)(222,331)(223,330)(224,332)
(225,325)(226,327)(227,326)(228,328)(229,321)(230,323)(231,322)(232,324)
(233,317)(234,319)(235,318)(236,320)(237,313)(238,315)(239,314)(240,316)
(241,369)(242,371)(243,370)(244,372)(245,365)(246,367)(247,366)(248,368)
(249,361)(250,363)(251,362)(252,364)(253,381)(254,383)(255,382)(256,384)
(257,377)(258,379)(259,378)(260,380)(261,373)(262,375)(263,374)(264,376)
(265,345)(266,347)(267,346)(268,348)(269,341)(270,343)(271,342)(272,344)
(273,337)(274,339)(275,338)(276,340)(277,357)(278,359)(279,358)(280,360)
(281,353)(282,355)(283,354)(284,356)(285,349)(286,351)(287,350)(288,352);;
s3 := (  1, 49)(  2, 50)(  3, 51)(  4, 52)(  5, 53)(  6, 54)(  7, 55)(  8, 56)
(  9, 57)( 10, 58)( 11, 59)( 12, 60)( 13, 61)( 14, 62)( 15, 63)( 16, 64)
( 17, 65)( 18, 66)( 19, 67)( 20, 68)( 21, 69)( 22, 70)( 23, 71)( 24, 72)
( 25, 85)( 26, 86)( 27, 87)( 28, 88)( 29, 89)( 30, 90)( 31, 91)( 32, 92)
( 33, 93)( 34, 94)( 35, 95)( 36, 96)( 37, 73)( 38, 74)( 39, 75)( 40, 76)
( 41, 77)( 42, 78)( 43, 79)( 44, 80)( 45, 81)( 46, 82)( 47, 83)( 48, 84)
( 97,145)( 98,146)( 99,147)(100,148)(101,149)(102,150)(103,151)(104,152)
(105,153)(106,154)(107,155)(108,156)(109,157)(110,158)(111,159)(112,160)
(113,161)(114,162)(115,163)(116,164)(117,165)(118,166)(119,167)(120,168)
(121,181)(122,182)(123,183)(124,184)(125,185)(126,186)(127,187)(128,188)
(129,189)(130,190)(131,191)(132,192)(133,169)(134,170)(135,171)(136,172)
(137,173)(138,174)(139,175)(140,176)(141,177)(142,178)(143,179)(144,180)
(193,241)(194,242)(195,243)(196,244)(197,245)(198,246)(199,247)(200,248)
(201,249)(202,250)(203,251)(204,252)(205,253)(206,254)(207,255)(208,256)
(209,257)(210,258)(211,259)(212,260)(213,261)(214,262)(215,263)(216,264)
(217,277)(218,278)(219,279)(220,280)(221,281)(222,282)(223,283)(224,284)
(225,285)(226,286)(227,287)(228,288)(229,265)(230,266)(231,267)(232,268)
(233,269)(234,270)(235,271)(236,272)(237,273)(238,274)(239,275)(240,276)
(289,337)(290,338)(291,339)(292,340)(293,341)(294,342)(295,343)(296,344)
(297,345)(298,346)(299,347)(300,348)(301,349)(302,350)(303,351)(304,352)
(305,353)(306,354)(307,355)(308,356)(309,357)(310,358)(311,359)(312,360)
(313,373)(314,374)(315,375)(316,376)(317,377)(318,378)(319,379)(320,380)
(321,381)(322,382)(323,383)(324,384)(325,361)(326,362)(327,363)(328,364)
(329,365)(330,366)(331,367)(332,368)(333,369)(334,370)(335,371)(336,372);;
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,
s1*s2*s3*s2*s1*s2*s3*s2, s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

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

```
References : None.
to this polytope