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# Polytope of Type {12,36}

Atlas Canonical Name : {12,36}*864b
if this polytope has a name.
Group : SmallGroup(864,1018)
Rank : 3
Schlafli Type : {12,36}
Number of vertices, edges, etc : 12, 216, 36
Order of s0s1s2 : 36
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{12,36,2} of size 1728
Vertex Figure Of :
{2,12,36} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {6,36}*432b, {12,18}*432b
3-fold quotients : {4,36}*288a, {12,12}*288b
4-fold quotients : {6,18}*216b
6-fold quotients : {2,36}*144, {4,18}*144a, {6,12}*144b, {12,6}*144c
8-fold quotients : {6,9}*108
9-fold quotients : {4,12}*96a
12-fold quotients : {2,18}*72, {6,6}*72b
18-fold quotients : {2,12}*48, {4,6}*48a
24-fold quotients : {2,9}*36, {6,3}*36
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 : {24,36}*1728a, {12,36}*1728b, {24,36}*1728b, {12,72}*1728b, {12,72}*1728d
Permutation Representation (GAP) :
```s0 := (  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)( 23, 26)
( 24, 27)( 31, 34)( 32, 35)( 33, 36)( 40, 43)( 41, 44)( 42, 45)( 49, 52)
( 50, 53)( 51, 54)( 58, 61)( 59, 62)( 60, 63)( 67, 70)( 68, 71)( 69, 72)
( 76, 79)( 77, 80)( 78, 81)( 85, 88)( 86, 89)( 87, 90)( 94, 97)( 95, 98)
( 96, 99)(103,106)(104,107)(105,108)(109,136)(110,137)(111,138)(112,142)
(113,143)(114,144)(115,139)(116,140)(117,141)(118,145)(119,146)(120,147)
(121,151)(122,152)(123,153)(124,148)(125,149)(126,150)(127,154)(128,155)
(129,156)(130,160)(131,161)(132,162)(133,157)(134,158)(135,159)(163,190)
(164,191)(165,192)(166,196)(167,197)(168,198)(169,193)(170,194)(171,195)
(172,199)(173,200)(174,201)(175,205)(176,206)(177,207)(178,202)(179,203)
(180,204)(181,208)(182,209)(183,210)(184,214)(185,215)(186,216)(187,211)
(188,212)(189,213);;
s1 := (  1,112)(  2,114)(  3,113)(  4,109)(  5,111)(  6,110)(  7,115)(  8,117)
(  9,116)( 10,132)( 11,131)( 12,130)( 13,129)( 14,128)( 15,127)( 16,135)
( 17,134)( 18,133)( 19,123)( 20,122)( 21,121)( 22,120)( 23,119)( 24,118)
( 25,126)( 26,125)( 27,124)( 28,139)( 29,141)( 30,140)( 31,136)( 32,138)
( 33,137)( 34,142)( 35,144)( 36,143)( 37,159)( 38,158)( 39,157)( 40,156)
( 41,155)( 42,154)( 43,162)( 44,161)( 45,160)( 46,150)( 47,149)( 48,148)
( 49,147)( 50,146)( 51,145)( 52,153)( 53,152)( 54,151)( 55,166)( 56,168)
( 57,167)( 58,163)( 59,165)( 60,164)( 61,169)( 62,171)( 63,170)( 64,186)
( 65,185)( 66,184)( 67,183)( 68,182)( 69,181)( 70,189)( 71,188)( 72,187)
( 73,177)( 74,176)( 75,175)( 76,174)( 77,173)( 78,172)( 79,180)( 80,179)
( 81,178)( 82,193)( 83,195)( 84,194)( 85,190)( 86,192)( 87,191)( 88,196)
( 89,198)( 90,197)( 91,213)( 92,212)( 93,211)( 94,210)( 95,209)( 96,208)
( 97,216)( 98,215)( 99,214)(100,204)(101,203)(102,202)(103,201)(104,200)
(105,199)(106,207)(107,206)(108,205);;
s2 := (  1, 10)(  2, 12)(  3, 11)(  4, 16)(  5, 18)(  6, 17)(  7, 13)(  8, 15)
(  9, 14)( 19, 21)( 22, 27)( 23, 26)( 24, 25)( 28, 37)( 29, 39)( 30, 38)
( 31, 43)( 32, 45)( 33, 44)( 34, 40)( 35, 42)( 36, 41)( 46, 48)( 49, 54)
( 50, 53)( 51, 52)( 55, 64)( 56, 66)( 57, 65)( 58, 70)( 59, 72)( 60, 71)
( 61, 67)( 62, 69)( 63, 68)( 73, 75)( 76, 81)( 77, 80)( 78, 79)( 82, 91)
( 83, 93)( 84, 92)( 85, 97)( 86, 99)( 87, 98)( 88, 94)( 89, 96)( 90, 95)
(100,102)(103,108)(104,107)(105,106)(109,199)(110,201)(111,200)(112,205)
(113,207)(114,206)(115,202)(116,204)(117,203)(118,190)(119,192)(120,191)
(121,196)(122,198)(123,197)(124,193)(125,195)(126,194)(127,210)(128,209)
(129,208)(130,216)(131,215)(132,214)(133,213)(134,212)(135,211)(136,172)
(137,174)(138,173)(139,178)(140,180)(141,179)(142,175)(143,177)(144,176)
(145,163)(146,165)(147,164)(148,169)(149,171)(150,170)(151,166)(152,168)
(153,167)(154,183)(155,182)(156,181)(157,189)(158,188)(159,187)(160,186)
(161,185)(162,184);;
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*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*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)
( 23, 26)( 24, 27)( 31, 34)( 32, 35)( 33, 36)( 40, 43)( 41, 44)( 42, 45)
( 49, 52)( 50, 53)( 51, 54)( 58, 61)( 59, 62)( 60, 63)( 67, 70)( 68, 71)
( 69, 72)( 76, 79)( 77, 80)( 78, 81)( 85, 88)( 86, 89)( 87, 90)( 94, 97)
( 95, 98)( 96, 99)(103,106)(104,107)(105,108)(109,136)(110,137)(111,138)
(112,142)(113,143)(114,144)(115,139)(116,140)(117,141)(118,145)(119,146)
(120,147)(121,151)(122,152)(123,153)(124,148)(125,149)(126,150)(127,154)
(128,155)(129,156)(130,160)(131,161)(132,162)(133,157)(134,158)(135,159)
(163,190)(164,191)(165,192)(166,196)(167,197)(168,198)(169,193)(170,194)
(171,195)(172,199)(173,200)(174,201)(175,205)(176,206)(177,207)(178,202)
(179,203)(180,204)(181,208)(182,209)(183,210)(184,214)(185,215)(186,216)
(187,211)(188,212)(189,213);
s1 := Sym(216)!(  1,112)(  2,114)(  3,113)(  4,109)(  5,111)(  6,110)(  7,115)
(  8,117)(  9,116)( 10,132)( 11,131)( 12,130)( 13,129)( 14,128)( 15,127)
( 16,135)( 17,134)( 18,133)( 19,123)( 20,122)( 21,121)( 22,120)( 23,119)
( 24,118)( 25,126)( 26,125)( 27,124)( 28,139)( 29,141)( 30,140)( 31,136)
( 32,138)( 33,137)( 34,142)( 35,144)( 36,143)( 37,159)( 38,158)( 39,157)
( 40,156)( 41,155)( 42,154)( 43,162)( 44,161)( 45,160)( 46,150)( 47,149)
( 48,148)( 49,147)( 50,146)( 51,145)( 52,153)( 53,152)( 54,151)( 55,166)
( 56,168)( 57,167)( 58,163)( 59,165)( 60,164)( 61,169)( 62,171)( 63,170)
( 64,186)( 65,185)( 66,184)( 67,183)( 68,182)( 69,181)( 70,189)( 71,188)
( 72,187)( 73,177)( 74,176)( 75,175)( 76,174)( 77,173)( 78,172)( 79,180)
( 80,179)( 81,178)( 82,193)( 83,195)( 84,194)( 85,190)( 86,192)( 87,191)
( 88,196)( 89,198)( 90,197)( 91,213)( 92,212)( 93,211)( 94,210)( 95,209)
( 96,208)( 97,216)( 98,215)( 99,214)(100,204)(101,203)(102,202)(103,201)
(104,200)(105,199)(106,207)(107,206)(108,205);
s2 := Sym(216)!(  1, 10)(  2, 12)(  3, 11)(  4, 16)(  5, 18)(  6, 17)(  7, 13)
(  8, 15)(  9, 14)( 19, 21)( 22, 27)( 23, 26)( 24, 25)( 28, 37)( 29, 39)
( 30, 38)( 31, 43)( 32, 45)( 33, 44)( 34, 40)( 35, 42)( 36, 41)( 46, 48)
( 49, 54)( 50, 53)( 51, 52)( 55, 64)( 56, 66)( 57, 65)( 58, 70)( 59, 72)
( 60, 71)( 61, 67)( 62, 69)( 63, 68)( 73, 75)( 76, 81)( 77, 80)( 78, 79)
( 82, 91)( 83, 93)( 84, 92)( 85, 97)( 86, 99)( 87, 98)( 88, 94)( 89, 96)
( 90, 95)(100,102)(103,108)(104,107)(105,106)(109,199)(110,201)(111,200)
(112,205)(113,207)(114,206)(115,202)(116,204)(117,203)(118,190)(119,192)
(120,191)(121,196)(122,198)(123,197)(124,193)(125,195)(126,194)(127,210)
(128,209)(129,208)(130,216)(131,215)(132,214)(133,213)(134,212)(135,211)
(136,172)(137,174)(138,173)(139,178)(140,180)(141,179)(142,175)(143,177)
(144,176)(145,163)(146,165)(147,164)(148,169)(149,171)(150,170)(151,166)
(152,168)(153,167)(154,183)(155,182)(156,181)(157,189)(158,188)(159,187)
(160,186)(161,185)(162,184);
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, s2*s0*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*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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