# Polytope of Type {12,4}

Atlas Canonical Name : {12,4}*864c
if this polytope has a name.
Group : SmallGroup(864,4080)
Rank : 3
Schlafli Type : {12,4}
Number of vertices, edges, etc : 108, 216, 36
Order of s0s1s2 : 12
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Skewing Operation
Facet Of :
{12,4,2} of size 1728
Vertex Figure Of :
{2,12,4} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {12,4}*432b
3-fold quotients : {4,4}*288
6-fold quotients : {4,4}*144
9-fold quotients : {12,4}*96a
12-fold quotients : {4,4}*72
18-fold quotients : {12,2}*48, {6,4}*48a
27-fold quotients : {4,4}*32
36-fold quotients : {6,2}*24
54-fold quotients : {2,4}*16, {4,2}*16
72-fold quotients : {3,2}*12
108-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {24,4}*1728e, {12,8}*1728e, {24,4}*1728h, {12,8}*1728f, {12,4}*1728d
Permutation Representation (GAP) :
```s0 := (  2,  3)(  4, 22)(  5, 24)(  6, 23)(  7, 16)(  8, 18)(  9, 17)( 10, 19)
( 11, 21)( 12, 20)( 14, 15)( 26, 27)( 29, 30)( 31, 49)( 32, 51)( 33, 50)
( 34, 43)( 35, 45)( 36, 44)( 37, 46)( 38, 48)( 39, 47)( 41, 42)( 53, 54)
( 56, 57)( 58, 76)( 59, 78)( 60, 77)( 61, 70)( 62, 72)( 63, 71)( 64, 73)
( 65, 75)( 66, 74)( 68, 69)( 80, 81)( 83, 84)( 85,103)( 86,105)( 87,104)
( 88, 97)( 89, 99)( 90, 98)( 91,100)( 92,102)( 93,101)( 95, 96)(107,108)
(109,190)(110,192)(111,191)(112,211)(113,213)(114,212)(115,205)(116,207)
(117,206)(118,208)(119,210)(120,209)(121,202)(122,204)(123,203)(124,196)
(125,198)(126,197)(127,199)(128,201)(129,200)(130,193)(131,195)(132,194)
(133,214)(134,216)(135,215)(136,163)(137,165)(138,164)(139,184)(140,186)
(141,185)(142,178)(143,180)(144,179)(145,181)(146,183)(147,182)(148,175)
(149,177)(150,176)(151,169)(152,171)(153,170)(154,172)(155,174)(156,173)
(157,166)(158,168)(159,167)(160,187)(161,189)(162,188);;
s1 := (  1,110)(  2,109)(  3,111)(  4,113)(  5,112)(  6,114)(  7,116)(  8,115)
(  9,117)( 10,134)( 11,133)( 12,135)( 13,128)( 14,127)( 15,129)( 16,131)
( 17,130)( 18,132)( 19,122)( 20,121)( 21,123)( 22,125)( 23,124)( 24,126)
( 25,119)( 26,118)( 27,120)( 28,137)( 29,136)( 30,138)( 31,140)( 32,139)
( 33,141)( 34,143)( 35,142)( 36,144)( 37,161)( 38,160)( 39,162)( 40,155)
( 41,154)( 42,156)( 43,158)( 44,157)( 45,159)( 46,149)( 47,148)( 48,150)
( 49,152)( 50,151)( 51,153)( 52,146)( 53,145)( 54,147)( 55,164)( 56,163)
( 57,165)( 58,167)( 59,166)( 60,168)( 61,170)( 62,169)( 63,171)( 64,188)
( 65,187)( 66,189)( 67,182)( 68,181)( 69,183)( 70,185)( 71,184)( 72,186)
( 73,176)( 74,175)( 75,177)( 76,179)( 77,178)( 78,180)( 79,173)( 80,172)
( 81,174)( 82,191)( 83,190)( 84,192)( 85,194)( 86,193)( 87,195)( 88,197)
( 89,196)( 90,198)( 91,215)( 92,214)( 93,216)( 94,209)( 95,208)( 96,210)
( 97,212)( 98,211)( 99,213)(100,203)(101,202)(102,204)(103,206)(104,205)
(105,207)(106,200)(107,199)(108,201);;
s2 := (  1, 67)(  2, 68)(  3, 69)(  4, 73)(  5, 74)(  6, 75)(  7, 61)(  8, 62)
(  9, 63)( 10, 76)( 11, 77)( 12, 78)( 13, 55)( 14, 56)( 15, 57)( 16, 70)
( 17, 71)( 18, 72)( 19, 58)( 20, 59)( 21, 60)( 22, 64)( 23, 65)( 24, 66)
( 25, 79)( 26, 80)( 27, 81)( 28, 94)( 29, 95)( 30, 96)( 31,100)( 32,101)
( 33,102)( 34, 88)( 35, 89)( 36, 90)( 37,103)( 38,104)( 39,105)( 40, 82)
( 41, 83)( 42, 84)( 43, 97)( 44, 98)( 45, 99)( 46, 85)( 47, 86)( 48, 87)
( 49, 91)( 50, 92)( 51, 93)( 52,106)( 53,107)( 54,108)(109,121)(110,122)
(111,123)(112,127)(113,128)(114,129)(118,130)(119,131)(120,132)(136,148)
(137,149)(138,150)(139,154)(140,155)(141,156)(145,157)(146,158)(147,159)
(163,175)(164,176)(165,177)(166,181)(167,182)(168,183)(172,184)(173,185)
(174,186)(190,202)(191,203)(192,204)(193,208)(194,209)(195,210)(199,211)
(200,212)(201,213);;
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, s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*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*s0*s1*s0*s1,
s0*s1*s2*s1*s0*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(216)!(  2,  3)(  4, 22)(  5, 24)(  6, 23)(  7, 16)(  8, 18)(  9, 17)
( 10, 19)( 11, 21)( 12, 20)( 14, 15)( 26, 27)( 29, 30)( 31, 49)( 32, 51)
( 33, 50)( 34, 43)( 35, 45)( 36, 44)( 37, 46)( 38, 48)( 39, 47)( 41, 42)
( 53, 54)( 56, 57)( 58, 76)( 59, 78)( 60, 77)( 61, 70)( 62, 72)( 63, 71)
( 64, 73)( 65, 75)( 66, 74)( 68, 69)( 80, 81)( 83, 84)( 85,103)( 86,105)
( 87,104)( 88, 97)( 89, 99)( 90, 98)( 91,100)( 92,102)( 93,101)( 95, 96)
(107,108)(109,190)(110,192)(111,191)(112,211)(113,213)(114,212)(115,205)
(116,207)(117,206)(118,208)(119,210)(120,209)(121,202)(122,204)(123,203)
(124,196)(125,198)(126,197)(127,199)(128,201)(129,200)(130,193)(131,195)
(132,194)(133,214)(134,216)(135,215)(136,163)(137,165)(138,164)(139,184)
(140,186)(141,185)(142,178)(143,180)(144,179)(145,181)(146,183)(147,182)
(148,175)(149,177)(150,176)(151,169)(152,171)(153,170)(154,172)(155,174)
(156,173)(157,166)(158,168)(159,167)(160,187)(161,189)(162,188);
s1 := Sym(216)!(  1,110)(  2,109)(  3,111)(  4,113)(  5,112)(  6,114)(  7,116)
(  8,115)(  9,117)( 10,134)( 11,133)( 12,135)( 13,128)( 14,127)( 15,129)
( 16,131)( 17,130)( 18,132)( 19,122)( 20,121)( 21,123)( 22,125)( 23,124)
( 24,126)( 25,119)( 26,118)( 27,120)( 28,137)( 29,136)( 30,138)( 31,140)
( 32,139)( 33,141)( 34,143)( 35,142)( 36,144)( 37,161)( 38,160)( 39,162)
( 40,155)( 41,154)( 42,156)( 43,158)( 44,157)( 45,159)( 46,149)( 47,148)
( 48,150)( 49,152)( 50,151)( 51,153)( 52,146)( 53,145)( 54,147)( 55,164)
( 56,163)( 57,165)( 58,167)( 59,166)( 60,168)( 61,170)( 62,169)( 63,171)
( 64,188)( 65,187)( 66,189)( 67,182)( 68,181)( 69,183)( 70,185)( 71,184)
( 72,186)( 73,176)( 74,175)( 75,177)( 76,179)( 77,178)( 78,180)( 79,173)
( 80,172)( 81,174)( 82,191)( 83,190)( 84,192)( 85,194)( 86,193)( 87,195)
( 88,197)( 89,196)( 90,198)( 91,215)( 92,214)( 93,216)( 94,209)( 95,208)
( 96,210)( 97,212)( 98,211)( 99,213)(100,203)(101,202)(102,204)(103,206)
(104,205)(105,207)(106,200)(107,199)(108,201);
s2 := Sym(216)!(  1, 67)(  2, 68)(  3, 69)(  4, 73)(  5, 74)(  6, 75)(  7, 61)
(  8, 62)(  9, 63)( 10, 76)( 11, 77)( 12, 78)( 13, 55)( 14, 56)( 15, 57)
( 16, 70)( 17, 71)( 18, 72)( 19, 58)( 20, 59)( 21, 60)( 22, 64)( 23, 65)
( 24, 66)( 25, 79)( 26, 80)( 27, 81)( 28, 94)( 29, 95)( 30, 96)( 31,100)
( 32,101)( 33,102)( 34, 88)( 35, 89)( 36, 90)( 37,103)( 38,104)( 39,105)
( 40, 82)( 41, 83)( 42, 84)( 43, 97)( 44, 98)( 45, 99)( 46, 85)( 47, 86)
( 48, 87)( 49, 91)( 50, 92)( 51, 93)( 52,106)( 53,107)( 54,108)(109,121)
(110,122)(111,123)(112,127)(113,128)(114,129)(118,130)(119,131)(120,132)
(136,148)(137,149)(138,150)(139,154)(140,155)(141,156)(145,157)(146,158)
(147,159)(163,175)(164,176)(165,177)(166,181)(167,182)(168,183)(172,184)
(173,185)(174,186)(190,202)(191,203)(192,204)(193,208)(194,209)(195,210)
(199,211)(200,212)(201,213);
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, s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*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*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 >;

```
References : None.
to this polytope