Polytope of Type {12,12}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,12}*864c
if this polytope has a name.
Group : SmallGroup(864,1130)
Rank : 3
Schlafli Type : {12,12}
Number of vertices, edges, etc : 36, 216, 36
Order of s0s1s2 : 12
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{12,12,2} of size 1728
Vertex Figure Of :
{2,12,12} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {6,12}*432b, {12,6}*432b
3-fold quotients : {12,12}*288a
4-fold quotients : {6,6}*216b
6-fold quotients : {6,12}*144a, {12,6}*144a
8-fold quotients : {6,6}*108
9-fold quotients : {4,12}*96a, {12,4}*96a
12-fold quotients : {6,6}*72a
18-fold quotients : {2,12}*48, {12,2}*48, {4,6}*48a, {6,4}*48a
27-fold quotients : {4,4}*32
36-fold quotients : {2,6}*24, {6,2}*24
54-fold quotients : {2,4}*16, {4,2}*16
72-fold quotients : {2,3}*12, {3,2}*12
108-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {12,12}*1728c, {12,24}*1728d, {24,12}*1728d, {12,24}*1728f, {24,12}*1728f
Permutation Representation (GAP) :
s0 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)( 11, 12)( 13, 16)( 14, 18)( 15, 17)
( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 29, 30)( 31, 34)( 32, 36)( 33, 35)
( 38, 39)( 40, 43)( 41, 45)( 42, 44)( 47, 48)( 49, 52)( 50, 54)( 51, 53)
( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 65, 66)( 67, 70)( 68, 72)( 69, 71)
( 74, 75)( 76, 79)( 77, 81)( 78, 80)( 83, 84)( 85, 88)( 86, 90)( 87, 89)
( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)(103,106)(104,108)(105,107)
(109,136)(110,138)(111,137)(112,142)(113,144)(114,143)(115,139)(116,141)
(117,140)(118,145)(119,147)(120,146)(121,151)(122,153)(123,152)(124,148)
(125,150)(126,149)(127,154)(128,156)(129,155)(130,160)(131,162)(132,161)
(133,157)(134,159)(135,158)(163,190)(164,192)(165,191)(166,196)(167,198)
(168,197)(169,193)(170,195)(171,194)(172,199)(173,201)(174,200)(175,205)
(176,207)(177,206)(178,202)(179,204)(180,203)(181,208)(182,210)(183,209)
(184,214)(185,216)(186,215)(187,211)(188,213)(189,212);;
s1 := ( 1,112)( 2,113)( 3,114)( 4,109)( 5,110)( 6,111)( 7,115)( 8,116)
( 9,117)( 10,130)( 11,131)( 12,132)( 13,127)( 14,128)( 15,129)( 16,133)
( 17,134)( 18,135)( 19,121)( 20,122)( 21,123)( 22,118)( 23,119)( 24,120)
( 25,124)( 26,125)( 27,126)( 28,139)( 29,140)( 30,141)( 31,136)( 32,137)
( 33,138)( 34,142)( 35,143)( 36,144)( 37,157)( 38,158)( 39,159)( 40,154)
( 41,155)( 42,156)( 43,160)( 44,161)( 45,162)( 46,148)( 47,149)( 48,150)
( 49,145)( 50,146)( 51,147)( 52,151)( 53,152)( 54,153)( 55,166)( 56,167)
( 57,168)( 58,163)( 59,164)( 60,165)( 61,169)( 62,170)( 63,171)( 64,184)
( 65,185)( 66,186)( 67,181)( 68,182)( 69,183)( 70,187)( 71,188)( 72,189)
( 73,175)( 74,176)( 75,177)( 76,172)( 77,173)( 78,174)( 79,178)( 80,179)
( 81,180)( 82,193)( 83,194)( 84,195)( 85,190)( 86,191)( 87,192)( 88,196)
( 89,197)( 90,198)( 91,211)( 92,212)( 93,213)( 94,208)( 95,209)( 96,210)
( 97,214)( 98,215)( 99,216)(100,202)(101,203)(102,204)(103,199)(104,200)
(105,201)(106,205)(107,206)(108,207);;
s2 := ( 1, 10)( 2, 12)( 3, 11)( 4, 14)( 5, 13)( 6, 15)( 7, 18)( 8, 17)
( 9, 16)( 20, 21)( 22, 23)( 25, 27)( 28, 37)( 29, 39)( 30, 38)( 31, 41)
( 32, 40)( 33, 42)( 34, 45)( 35, 44)( 36, 43)( 47, 48)( 49, 50)( 52, 54)
( 55, 64)( 56, 66)( 57, 65)( 58, 68)( 59, 67)( 60, 69)( 61, 72)( 62, 71)
( 63, 70)( 74, 75)( 76, 77)( 79, 81)( 82, 91)( 83, 93)( 84, 92)( 85, 95)
( 86, 94)( 87, 96)( 88, 99)( 89, 98)( 90, 97)(101,102)(103,104)(106,108)
(109,172)(110,174)(111,173)(112,176)(113,175)(114,177)(115,180)(116,179)
(117,178)(118,163)(119,165)(120,164)(121,167)(122,166)(123,168)(124,171)
(125,170)(126,169)(127,181)(128,183)(129,182)(130,185)(131,184)(132,186)
(133,189)(134,188)(135,187)(136,199)(137,201)(138,200)(139,203)(140,202)
(141,204)(142,207)(143,206)(144,205)(145,190)(146,192)(147,191)(148,194)
(149,193)(150,195)(151,198)(152,197)(153,196)(154,208)(155,210)(156,209)
(157,212)(158,211)(159,213)(160,216)(161,215)(162,214);;
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, s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1,
s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(216)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)( 11, 12)( 13, 16)( 14, 18)
( 15, 17)( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 29, 30)( 31, 34)( 32, 36)
( 33, 35)( 38, 39)( 40, 43)( 41, 45)( 42, 44)( 47, 48)( 49, 52)( 50, 54)
( 51, 53)( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 65, 66)( 67, 70)( 68, 72)
( 69, 71)( 74, 75)( 76, 79)( 77, 81)( 78, 80)( 83, 84)( 85, 88)( 86, 90)
( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)(103,106)(104,108)
(105,107)(109,136)(110,138)(111,137)(112,142)(113,144)(114,143)(115,139)
(116,141)(117,140)(118,145)(119,147)(120,146)(121,151)(122,153)(123,152)
(124,148)(125,150)(126,149)(127,154)(128,156)(129,155)(130,160)(131,162)
(132,161)(133,157)(134,159)(135,158)(163,190)(164,192)(165,191)(166,196)
(167,198)(168,197)(169,193)(170,195)(171,194)(172,199)(173,201)(174,200)
(175,205)(176,207)(177,206)(178,202)(179,204)(180,203)(181,208)(182,210)
(183,209)(184,214)(185,216)(186,215)(187,211)(188,213)(189,212);
s1 := Sym(216)!( 1,112)( 2,113)( 3,114)( 4,109)( 5,110)( 6,111)( 7,115)
( 8,116)( 9,117)( 10,130)( 11,131)( 12,132)( 13,127)( 14,128)( 15,129)
( 16,133)( 17,134)( 18,135)( 19,121)( 20,122)( 21,123)( 22,118)( 23,119)
( 24,120)( 25,124)( 26,125)( 27,126)( 28,139)( 29,140)( 30,141)( 31,136)
( 32,137)( 33,138)( 34,142)( 35,143)( 36,144)( 37,157)( 38,158)( 39,159)
( 40,154)( 41,155)( 42,156)( 43,160)( 44,161)( 45,162)( 46,148)( 47,149)
( 48,150)( 49,145)( 50,146)( 51,147)( 52,151)( 53,152)( 54,153)( 55,166)
( 56,167)( 57,168)( 58,163)( 59,164)( 60,165)( 61,169)( 62,170)( 63,171)
( 64,184)( 65,185)( 66,186)( 67,181)( 68,182)( 69,183)( 70,187)( 71,188)
( 72,189)( 73,175)( 74,176)( 75,177)( 76,172)( 77,173)( 78,174)( 79,178)
( 80,179)( 81,180)( 82,193)( 83,194)( 84,195)( 85,190)( 86,191)( 87,192)
( 88,196)( 89,197)( 90,198)( 91,211)( 92,212)( 93,213)( 94,208)( 95,209)
( 96,210)( 97,214)( 98,215)( 99,216)(100,202)(101,203)(102,204)(103,199)
(104,200)(105,201)(106,205)(107,206)(108,207);
s2 := Sym(216)!( 1, 10)( 2, 12)( 3, 11)( 4, 14)( 5, 13)( 6, 15)( 7, 18)
( 8, 17)( 9, 16)( 20, 21)( 22, 23)( 25, 27)( 28, 37)( 29, 39)( 30, 38)
( 31, 41)( 32, 40)( 33, 42)( 34, 45)( 35, 44)( 36, 43)( 47, 48)( 49, 50)
( 52, 54)( 55, 64)( 56, 66)( 57, 65)( 58, 68)( 59, 67)( 60, 69)( 61, 72)
( 62, 71)( 63, 70)( 74, 75)( 76, 77)( 79, 81)( 82, 91)( 83, 93)( 84, 92)
( 85, 95)( 86, 94)( 87, 96)( 88, 99)( 89, 98)( 90, 97)(101,102)(103,104)
(106,108)(109,172)(110,174)(111,173)(112,176)(113,175)(114,177)(115,180)
(116,179)(117,178)(118,163)(119,165)(120,164)(121,167)(122,166)(123,168)
(124,171)(125,170)(126,169)(127,181)(128,183)(129,182)(130,185)(131,184)
(132,186)(133,189)(134,188)(135,187)(136,199)(137,201)(138,200)(139,203)
(140,202)(141,204)(142,207)(143,206)(144,205)(145,190)(146,192)(147,191)
(148,194)(149,193)(150,195)(151,198)(152,197)(153,196)(154,208)(155,210)
(156,209)(157,212)(158,211)(159,213)(160,216)(161,215)(162,214);
poly := sub<Sym(216)|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, s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1,
s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
References : None.
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