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# Polytope of Type {8,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,12}*768k
if this polytope has a name.
Group : SmallGroup(768,1086301)
Rank : 3
Schlafli Type : {8,12}
Number of vertices, edges, etc : 32, 192, 48
Order of s0s1s2 : 6
Order of s0s1s2s1 : 8
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {8,12}*384c, {8,12}*384d, {8,6}*384d
4-fold quotients : {8,6}*192a
8-fold quotients : {4,6}*96
16-fold quotients : {4,3}*48, {4,6}*48b, {4,6}*48c
32-fold quotients : {4,3}*24, {2,6}*24
64-fold quotients : {2,3}*12
96-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (  1,124)(  2,123)(  3,122)(  4,121)(  5,128)(  6,127)(  7,126)(  8,125)
(  9,116)( 10,115)( 11,114)( 12,113)( 13,120)( 14,119)( 15,118)( 16,117)
( 17,110)( 18,109)( 19,112)( 20,111)( 21,106)( 22,105)( 23,108)( 24,107)
( 25,102)( 26,101)( 27,104)( 28,103)( 29, 98)( 30, 97)( 31,100)( 32, 99)
( 33,156)( 34,155)( 35,154)( 36,153)( 37,160)( 38,159)( 39,158)( 40,157)
( 41,148)( 42,147)( 43,146)( 44,145)( 45,152)( 46,151)( 47,150)( 48,149)
( 49,142)( 50,141)( 51,144)( 52,143)( 53,138)( 54,137)( 55,140)( 56,139)
( 57,134)( 58,133)( 59,136)( 60,135)( 61,130)( 62,129)( 63,132)( 64,131)
( 65,188)( 66,187)( 67,186)( 68,185)( 69,192)( 70,191)( 71,190)( 72,189)
( 73,180)( 74,179)( 75,178)( 76,177)( 77,184)( 78,183)( 79,182)( 80,181)
( 81,174)( 82,173)( 83,176)( 84,175)( 85,170)( 86,169)( 87,172)( 88,171)
( 89,166)( 90,165)( 91,168)( 92,167)( 93,162)( 94,161)( 95,164)( 96,163)
(193,316)(194,315)(195,314)(196,313)(197,320)(198,319)(199,318)(200,317)
(201,308)(202,307)(203,306)(204,305)(205,312)(206,311)(207,310)(208,309)
(209,302)(210,301)(211,304)(212,303)(213,298)(214,297)(215,300)(216,299)
(217,294)(218,293)(219,296)(220,295)(221,290)(222,289)(223,292)(224,291)
(225,348)(226,347)(227,346)(228,345)(229,352)(230,351)(231,350)(232,349)
(233,340)(234,339)(235,338)(236,337)(237,344)(238,343)(239,342)(240,341)
(241,334)(242,333)(243,336)(244,335)(245,330)(246,329)(247,332)(248,331)
(249,326)(250,325)(251,328)(252,327)(253,322)(254,321)(255,324)(256,323)
(257,380)(258,379)(259,378)(260,377)(261,384)(262,383)(263,382)(264,381)
(265,372)(266,371)(267,370)(268,369)(269,376)(270,375)(271,374)(272,373)
(273,366)(274,365)(275,368)(276,367)(277,362)(278,361)(279,364)(280,363)
(281,358)(282,357)(283,360)(284,359)(285,354)(286,353)(287,356)(288,355);;
s1 := (  3,  4)(  5,  6)(  9, 15)( 10, 16)( 11, 14)( 12, 13)( 17, 28)( 18, 27)
( 19, 25)( 20, 26)( 21, 31)( 22, 32)( 23, 30)( 24, 29)( 33, 65)( 34, 66)
( 35, 68)( 36, 67)( 37, 70)( 38, 69)( 39, 71)( 40, 72)( 41, 79)( 42, 80)
( 43, 78)( 44, 77)( 45, 76)( 46, 75)( 47, 73)( 48, 74)( 49, 92)( 50, 91)
( 51, 89)( 52, 90)( 53, 95)( 54, 96)( 55, 94)( 56, 93)( 57, 83)( 58, 84)
( 59, 82)( 60, 81)( 61, 88)( 62, 87)( 63, 85)( 64, 86)( 97,103)( 98,104)
( 99,102)(100,101)(107,108)(109,110)(113,126)(114,125)(115,127)(116,128)
(117,121)(118,122)(119,124)(120,123)(129,167)(130,168)(131,166)(132,165)
(133,164)(134,163)(135,161)(136,162)(137,169)(138,170)(139,172)(140,171)
(141,174)(142,173)(143,175)(144,176)(145,190)(146,189)(147,191)(148,192)
(149,185)(150,186)(151,188)(152,187)(153,181)(154,182)(155,184)(156,183)
(157,178)(158,177)(159,179)(160,180)(195,196)(197,198)(201,207)(202,208)
(203,206)(204,205)(209,220)(210,219)(211,217)(212,218)(213,223)(214,224)
(215,222)(216,221)(225,257)(226,258)(227,260)(228,259)(229,262)(230,261)
(231,263)(232,264)(233,271)(234,272)(235,270)(236,269)(237,268)(238,267)
(239,265)(240,266)(241,284)(242,283)(243,281)(244,282)(245,287)(246,288)
(247,286)(248,285)(249,275)(250,276)(251,274)(252,273)(253,280)(254,279)
(255,277)(256,278)(289,295)(290,296)(291,294)(292,293)(299,300)(301,302)
(305,318)(306,317)(307,319)(308,320)(309,313)(310,314)(311,316)(312,315)
(321,359)(322,360)(323,358)(324,357)(325,356)(326,355)(327,353)(328,354)
(329,361)(330,362)(331,364)(332,363)(333,366)(334,365)(335,367)(336,368)
(337,382)(338,381)(339,383)(340,384)(341,377)(342,378)(343,380)(344,379)
(345,373)(346,374)(347,376)(348,375)(349,370)(350,369)(351,371)(352,372);;
s2 := (  1,353)(  2,355)(  3,354)(  4,356)(  5,360)(  6,358)(  7,359)(  8,357)
(  9,374)( 10,376)( 11,373)( 12,375)( 13,371)( 14,369)( 15,372)( 16,370)
( 17,366)( 18,368)( 19,365)( 20,367)( 21,363)( 22,361)( 23,364)( 24,362)
( 25,383)( 26,381)( 27,384)( 28,382)( 29,378)( 30,380)( 31,377)( 32,379)
( 33,321)( 34,323)( 35,322)( 36,324)( 37,328)( 38,326)( 39,327)( 40,325)
( 41,342)( 42,344)( 43,341)( 44,343)( 45,339)( 46,337)( 47,340)( 48,338)
( 49,334)( 50,336)( 51,333)( 52,335)( 53,331)( 54,329)( 55,332)( 56,330)
( 57,351)( 58,349)( 59,352)( 60,350)( 61,346)( 62,348)( 63,345)( 64,347)
( 65,289)( 66,291)( 67,290)( 68,292)( 69,296)( 70,294)( 71,295)( 72,293)
( 73,310)( 74,312)( 75,309)( 76,311)( 77,307)( 78,305)( 79,308)( 80,306)
( 81,302)( 82,304)( 83,301)( 84,303)( 85,299)( 86,297)( 87,300)( 88,298)
( 89,319)( 90,317)( 91,320)( 92,318)( 93,314)( 94,316)( 95,313)( 96,315)
( 97,257)( 98,259)( 99,258)(100,260)(101,264)(102,262)(103,263)(104,261)
(105,278)(106,280)(107,277)(108,279)(109,275)(110,273)(111,276)(112,274)
(113,270)(114,272)(115,269)(116,271)(117,267)(118,265)(119,268)(120,266)
(121,287)(122,285)(123,288)(124,286)(125,282)(126,284)(127,281)(128,283)
(129,225)(130,227)(131,226)(132,228)(133,232)(134,230)(135,231)(136,229)
(137,246)(138,248)(139,245)(140,247)(141,243)(142,241)(143,244)(144,242)
(145,238)(146,240)(147,237)(148,239)(149,235)(150,233)(151,236)(152,234)
(153,255)(154,253)(155,256)(156,254)(157,250)(158,252)(159,249)(160,251)
(161,193)(162,195)(163,194)(164,196)(165,200)(166,198)(167,199)(168,197)
(169,214)(170,216)(171,213)(172,215)(173,211)(174,209)(175,212)(176,210)
(177,206)(178,208)(179,205)(180,207)(181,203)(182,201)(183,204)(184,202)
(185,223)(186,221)(187,224)(188,222)(189,218)(190,220)(191,217)(192,219);;
poly := Group([s0,s1,s2]);;

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

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

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

```
References : None.
to this polytope