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# Polytope of Type {4,12}

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