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# Polytope of Type {4,12,4}

Atlas Canonical Name : {4,12,4}*768h
if this polytope has a name.
Group : SmallGroup(768,1088921)
Rank : 4
Schlafli Type : {4,12,4}
Number of vertices, edges, etc : 8, 48, 48, 4
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}*384b, {4,12,2}*384c
4-fold quotients : {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
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)
( 49,256)( 50,255)( 51,254)( 52,253)( 53,260)( 54,259)( 55,258)( 56,257)
( 57,264)( 58,263)( 59,262)( 60,261)( 61,244)( 62,243)( 63,242)( 64,241)
( 65,248)( 66,247)( 67,246)( 68,245)( 69,252)( 70,251)( 71,250)( 72,249)
( 73,280)( 74,279)( 75,278)( 76,277)( 77,284)( 78,283)( 79,282)( 80,281)
( 81,288)( 82,287)( 83,286)( 84,285)( 85,268)( 86,267)( 87,266)( 88,265)
( 89,272)( 90,271)( 91,270)( 92,269)( 93,276)( 94,275)( 95,274)( 96,273)
( 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)
(145,352)(146,351)(147,350)(148,349)(149,356)(150,355)(151,354)(152,353)
(153,360)(154,359)(155,358)(156,357)(157,340)(158,339)(159,338)(160,337)
(161,344)(162,343)(163,342)(164,341)(165,348)(166,347)(167,346)(168,345)
(169,376)(170,375)(171,374)(172,373)(173,380)(174,379)(175,378)(176,377)
(177,384)(178,383)(179,382)(180,381)(181,364)(182,363)(183,362)(184,361)
(185,368)(186,367)(187,366)(188,365)(189,372)(190,371)(191,370)(192,369);;
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 := (  1,249)(  2,251)(  3,250)(  4,252)(  5,245)(  6,247)(  7,246)(  8,248)
(  9,241)( 10,243)( 11,242)( 12,244)( 13,261)( 14,263)( 15,262)( 16,264)
( 17,257)( 18,259)( 19,258)( 20,260)( 21,253)( 22,255)( 23,254)( 24,256)
( 25,273)( 26,275)( 27,274)( 28,276)( 29,269)( 30,271)( 31,270)( 32,272)
( 33,265)( 34,267)( 35,266)( 36,268)( 37,285)( 38,287)( 39,286)( 40,288)
( 41,281)( 42,283)( 43,282)( 44,284)( 45,277)( 46,279)( 47,278)( 48,280)
( 49,213)( 50,215)( 51,214)( 52,216)( 53,209)( 54,211)( 55,210)( 56,212)
( 57,205)( 58,207)( 59,206)( 60,208)( 61,201)( 62,203)( 63,202)( 64,204)
( 65,197)( 66,199)( 67,198)( 68,200)( 69,193)( 70,195)( 71,194)( 72,196)
( 73,237)( 74,239)( 75,238)( 76,240)( 77,233)( 78,235)( 79,234)( 80,236)
( 81,229)( 82,231)( 83,230)( 84,232)( 85,225)( 86,227)( 87,226)( 88,228)
( 89,221)( 90,223)( 91,222)( 92,224)( 93,217)( 94,219)( 95,218)( 96,220)
( 97,345)( 98,347)( 99,346)(100,348)(101,341)(102,343)(103,342)(104,344)
(105,337)(106,339)(107,338)(108,340)(109,357)(110,359)(111,358)(112,360)
(113,353)(114,355)(115,354)(116,356)(117,349)(118,351)(119,350)(120,352)
(121,369)(122,371)(123,370)(124,372)(125,365)(126,367)(127,366)(128,368)
(129,361)(130,363)(131,362)(132,364)(133,381)(134,383)(135,382)(136,384)
(137,377)(138,379)(139,378)(140,380)(141,373)(142,375)(143,374)(144,376)
(145,309)(146,311)(147,310)(148,312)(149,305)(150,307)(151,306)(152,308)
(153,301)(154,303)(155,302)(156,304)(157,297)(158,299)(159,298)(160,300)
(161,293)(162,295)(163,294)(164,296)(165,289)(166,291)(167,290)(168,292)
(169,333)(170,335)(171,334)(172,336)(173,329)(174,331)(175,330)(176,332)
(177,325)(178,327)(179,326)(180,328)(181,321)(182,323)(183,322)(184,324)
(185,317)(186,319)(187,318)(188,320)(189,313)(190,315)(191,314)(192,316);;
s3 := (  1, 97)(  2, 98)(  3, 99)(  4,100)(  5,101)(  6,102)(  7,103)(  8,104)
(  9,105)( 10,106)( 11,107)( 12,108)( 13,109)( 14,110)( 15,111)( 16,112)
( 17,113)( 18,114)( 19,115)( 20,116)( 21,117)( 22,118)( 23,119)( 24,120)
( 25,121)( 26,122)( 27,123)( 28,124)( 29,125)( 30,126)( 31,127)( 32,128)
( 33,129)( 34,130)( 35,131)( 36,132)( 37,133)( 38,134)( 39,135)( 40,136)
( 41,137)( 42,138)( 43,139)( 44,140)( 45,141)( 46,142)( 47,143)( 48,144)
( 49,169)( 50,170)( 51,171)( 52,172)( 53,173)( 54,174)( 55,175)( 56,176)
( 57,177)( 58,178)( 59,179)( 60,180)( 61,181)( 62,182)( 63,183)( 64,184)
( 65,185)( 66,186)( 67,187)( 68,188)( 69,189)( 70,190)( 71,191)( 72,192)
( 73,145)( 74,146)( 75,147)( 76,148)( 77,149)( 78,150)( 79,151)( 80,152)
( 81,153)( 82,154)( 83,155)( 84,156)( 85,157)( 86,158)( 87,159)( 88,160)
( 89,161)( 90,162)( 91,163)( 92,164)( 93,165)( 94,166)( 95,167)( 96,168)
(193,289)(194,290)(195,291)(196,292)(197,293)(198,294)(199,295)(200,296)
(201,297)(202,298)(203,299)(204,300)(205,301)(206,302)(207,303)(208,304)
(209,305)(210,306)(211,307)(212,308)(213,309)(214,310)(215,311)(216,312)
(217,313)(218,314)(219,315)(220,316)(221,317)(222,318)(223,319)(224,320)
(225,321)(226,322)(227,323)(228,324)(229,325)(230,326)(231,327)(232,328)
(233,329)(234,330)(235,331)(236,332)(237,333)(238,334)(239,335)(240,336)
(241,361)(242,362)(243,363)(244,364)(245,365)(246,366)(247,367)(248,368)
(249,369)(250,370)(251,371)(252,372)(253,373)(254,374)(255,375)(256,376)
(257,377)(258,378)(259,379)(260,380)(261,381)(262,382)(263,383)(264,384)
(265,337)(266,338)(267,339)(268,340)(269,341)(270,342)(271,343)(272,344)
(273,345)(274,346)(275,347)(276,348)(277,349)(278,350)(279,351)(280,352)
(281,353)(282,354)(283,355)(284,356)(285,357)(286,358)(287,359)(288,360);;
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, s2*s3*s2*s3*s2*s3*s2*s3,
s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 ];;
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)( 49,256)( 50,255)( 51,254)( 52,253)( 53,260)( 54,259)( 55,258)
( 56,257)( 57,264)( 58,263)( 59,262)( 60,261)( 61,244)( 62,243)( 63,242)
( 64,241)( 65,248)( 66,247)( 67,246)( 68,245)( 69,252)( 70,251)( 71,250)
( 72,249)( 73,280)( 74,279)( 75,278)( 76,277)( 77,284)( 78,283)( 79,282)
( 80,281)( 81,288)( 82,287)( 83,286)( 84,285)( 85,268)( 86,267)( 87,266)
( 88,265)( 89,272)( 90,271)( 91,270)( 92,269)( 93,276)( 94,275)( 95,274)
( 96,273)( 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)(145,352)(146,351)(147,350)(148,349)(149,356)(150,355)(151,354)
(152,353)(153,360)(154,359)(155,358)(156,357)(157,340)(158,339)(159,338)
(160,337)(161,344)(162,343)(163,342)(164,341)(165,348)(166,347)(167,346)
(168,345)(169,376)(170,375)(171,374)(172,373)(173,380)(174,379)(175,378)
(176,377)(177,384)(178,383)(179,382)(180,381)(181,364)(182,363)(183,362)
(184,361)(185,368)(186,367)(187,366)(188,365)(189,372)(190,371)(191,370)
(192,369);
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)!(  1,249)(  2,251)(  3,250)(  4,252)(  5,245)(  6,247)(  7,246)
(  8,248)(  9,241)( 10,243)( 11,242)( 12,244)( 13,261)( 14,263)( 15,262)
( 16,264)( 17,257)( 18,259)( 19,258)( 20,260)( 21,253)( 22,255)( 23,254)
( 24,256)( 25,273)( 26,275)( 27,274)( 28,276)( 29,269)( 30,271)( 31,270)
( 32,272)( 33,265)( 34,267)( 35,266)( 36,268)( 37,285)( 38,287)( 39,286)
( 40,288)( 41,281)( 42,283)( 43,282)( 44,284)( 45,277)( 46,279)( 47,278)
( 48,280)( 49,213)( 50,215)( 51,214)( 52,216)( 53,209)( 54,211)( 55,210)
( 56,212)( 57,205)( 58,207)( 59,206)( 60,208)( 61,201)( 62,203)( 63,202)
( 64,204)( 65,197)( 66,199)( 67,198)( 68,200)( 69,193)( 70,195)( 71,194)
( 72,196)( 73,237)( 74,239)( 75,238)( 76,240)( 77,233)( 78,235)( 79,234)
( 80,236)( 81,229)( 82,231)( 83,230)( 84,232)( 85,225)( 86,227)( 87,226)
( 88,228)( 89,221)( 90,223)( 91,222)( 92,224)( 93,217)( 94,219)( 95,218)
( 96,220)( 97,345)( 98,347)( 99,346)(100,348)(101,341)(102,343)(103,342)
(104,344)(105,337)(106,339)(107,338)(108,340)(109,357)(110,359)(111,358)
(112,360)(113,353)(114,355)(115,354)(116,356)(117,349)(118,351)(119,350)
(120,352)(121,369)(122,371)(123,370)(124,372)(125,365)(126,367)(127,366)
(128,368)(129,361)(130,363)(131,362)(132,364)(133,381)(134,383)(135,382)
(136,384)(137,377)(138,379)(139,378)(140,380)(141,373)(142,375)(143,374)
(144,376)(145,309)(146,311)(147,310)(148,312)(149,305)(150,307)(151,306)
(152,308)(153,301)(154,303)(155,302)(156,304)(157,297)(158,299)(159,298)
(160,300)(161,293)(162,295)(163,294)(164,296)(165,289)(166,291)(167,290)
(168,292)(169,333)(170,335)(171,334)(172,336)(173,329)(174,331)(175,330)
(176,332)(177,325)(178,327)(179,326)(180,328)(181,321)(182,323)(183,322)
(184,324)(185,317)(186,319)(187,318)(188,320)(189,313)(190,315)(191,314)
(192,316);
s3 := Sym(384)!(  1, 97)(  2, 98)(  3, 99)(  4,100)(  5,101)(  6,102)(  7,103)
(  8,104)(  9,105)( 10,106)( 11,107)( 12,108)( 13,109)( 14,110)( 15,111)
( 16,112)( 17,113)( 18,114)( 19,115)( 20,116)( 21,117)( 22,118)( 23,119)
( 24,120)( 25,121)( 26,122)( 27,123)( 28,124)( 29,125)( 30,126)( 31,127)
( 32,128)( 33,129)( 34,130)( 35,131)( 36,132)( 37,133)( 38,134)( 39,135)
( 40,136)( 41,137)( 42,138)( 43,139)( 44,140)( 45,141)( 46,142)( 47,143)
( 48,144)( 49,169)( 50,170)( 51,171)( 52,172)( 53,173)( 54,174)( 55,175)
( 56,176)( 57,177)( 58,178)( 59,179)( 60,180)( 61,181)( 62,182)( 63,183)
( 64,184)( 65,185)( 66,186)( 67,187)( 68,188)( 69,189)( 70,190)( 71,191)
( 72,192)( 73,145)( 74,146)( 75,147)( 76,148)( 77,149)( 78,150)( 79,151)
( 80,152)( 81,153)( 82,154)( 83,155)( 84,156)( 85,157)( 86,158)( 87,159)
( 88,160)( 89,161)( 90,162)( 91,163)( 92,164)( 93,165)( 94,166)( 95,167)
( 96,168)(193,289)(194,290)(195,291)(196,292)(197,293)(198,294)(199,295)
(200,296)(201,297)(202,298)(203,299)(204,300)(205,301)(206,302)(207,303)
(208,304)(209,305)(210,306)(211,307)(212,308)(213,309)(214,310)(215,311)
(216,312)(217,313)(218,314)(219,315)(220,316)(221,317)(222,318)(223,319)
(224,320)(225,321)(226,322)(227,323)(228,324)(229,325)(230,326)(231,327)
(232,328)(233,329)(234,330)(235,331)(236,332)(237,333)(238,334)(239,335)
(240,336)(241,361)(242,362)(243,363)(244,364)(245,365)(246,366)(247,367)
(248,368)(249,369)(250,370)(251,371)(252,372)(253,373)(254,374)(255,375)
(256,376)(257,377)(258,378)(259,379)(260,380)(261,381)(262,382)(263,383)
(264,384)(265,337)(266,338)(267,339)(268,340)(269,341)(270,342)(271,343)
(272,344)(273,345)(274,346)(275,347)(276,348)(277,349)(278,350)(279,351)
(280,352)(281,353)(282,354)(283,355)(284,356)(285,357)(286,358)(287,359)
(288,360);
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,
s2*s3*s2*s3*s2*s3*s2*s3, s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 >;

```
References : None.
to this polytope