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# Polytope of Type {36,6}

Atlas Canonical Name : {36,6}*864
if this polytope has a name.
Group : SmallGroup(864,3998)
Rank : 3
Schlafli Type : {36,6}
Number of vertices, edges, etc : 72, 216, 12
Order of s0s1s2 : 18
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{36,6,2} of size 1728
Vertex Figure Of :
{2,36,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {36,6}*432c
3-fold quotients : {12,6}*288a
4-fold quotients : {18,6}*216a
6-fold quotients : {12,6}*144d
9-fold quotients : {4,6}*96
12-fold quotients : {18,2}*72, {6,6}*72a
18-fold quotients : {4,3}*48, {4,6}*48b, {4,6}*48c
24-fold quotients : {9,2}*36
36-fold quotients : {4,3}*24, {2,6}*24, {6,2}*24
72-fold quotients : {2,3}*12, {3,2}*12
108-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {36,12}*1728c, {36,6}*1728b, {72,6}*1728b, {72,6}*1728c, {36,12}*1728d
Permutation Representation (GAP) :
```s0 := (  1,  3)(  2,  4)(  5, 11)(  6, 12)(  7,  9)(  8, 10)( 13, 31)( 14, 32)
( 15, 29)( 16, 30)( 17, 27)( 18, 28)( 19, 25)( 20, 26)( 21, 35)( 22, 36)
( 23, 33)( 24, 34)( 37, 39)( 38, 40)( 41, 47)( 42, 48)( 43, 45)( 44, 46)
( 49, 67)( 50, 68)( 51, 65)( 52, 66)( 53, 63)( 54, 64)( 55, 61)( 56, 62)
( 57, 71)( 58, 72)( 59, 69)( 60, 70)( 73, 75)( 74, 76)( 77, 83)( 78, 84)
( 79, 81)( 80, 82)( 85,103)( 86,104)( 87,101)( 88,102)( 89, 99)( 90,100)
( 91, 97)( 92, 98)( 93,107)( 94,108)( 95,105)( 96,106)(109,111)(110,112)
(113,119)(114,120)(115,117)(116,118)(121,139)(122,140)(123,137)(124,138)
(125,135)(126,136)(127,133)(128,134)(129,143)(130,144)(131,141)(132,142)
(145,147)(146,148)(149,155)(150,156)(151,153)(152,154)(157,175)(158,176)
(159,173)(160,174)(161,171)(162,172)(163,169)(164,170)(165,179)(166,180)
(167,177)(168,178)(181,183)(182,184)(185,191)(186,192)(187,189)(188,190)
(193,211)(194,212)(195,209)(196,210)(197,207)(198,208)(199,205)(200,206)
(201,215)(202,216)(203,213)(204,214);;
s1 := (  1, 13)(  2, 15)(  3, 14)(  4, 16)(  5, 21)(  6, 23)(  7, 22)(  8, 24)
(  9, 17)( 10, 19)( 11, 18)( 12, 20)( 25, 29)( 26, 31)( 27, 30)( 28, 32)
( 34, 35)( 37, 85)( 38, 87)( 39, 86)( 40, 88)( 41, 93)( 42, 95)( 43, 94)
( 44, 96)( 45, 89)( 46, 91)( 47, 90)( 48, 92)( 49, 73)( 50, 75)( 51, 74)
( 52, 76)( 53, 81)( 54, 83)( 55, 82)( 56, 84)( 57, 77)( 58, 79)( 59, 78)
( 60, 80)( 61,101)( 62,103)( 63,102)( 64,104)( 65, 97)( 66, 99)( 67, 98)
( 68,100)( 69,105)( 70,107)( 71,106)( 72,108)(109,121)(110,123)(111,122)
(112,124)(113,129)(114,131)(115,130)(116,132)(117,125)(118,127)(119,126)
(120,128)(133,137)(134,139)(135,138)(136,140)(142,143)(145,193)(146,195)
(147,194)(148,196)(149,201)(150,203)(151,202)(152,204)(153,197)(154,199)
(155,198)(156,200)(157,181)(158,183)(159,182)(160,184)(161,189)(162,191)
(163,190)(164,192)(165,185)(166,187)(167,186)(168,188)(169,209)(170,211)
(171,210)(172,212)(173,205)(174,207)(175,206)(176,208)(177,213)(178,215)
(179,214)(180,216);;
s2 := (  1,181)(  2,184)(  3,183)(  4,182)(  5,185)(  6,188)(  7,187)(  8,186)
(  9,189)( 10,192)( 11,191)( 12,190)( 13,193)( 14,196)( 15,195)( 16,194)
( 17,197)( 18,200)( 19,199)( 20,198)( 21,201)( 22,204)( 23,203)( 24,202)
( 25,205)( 26,208)( 27,207)( 28,206)( 29,209)( 30,212)( 31,211)( 32,210)
( 33,213)( 34,216)( 35,215)( 36,214)( 37,145)( 38,148)( 39,147)( 40,146)
( 41,149)( 42,152)( 43,151)( 44,150)( 45,153)( 46,156)( 47,155)( 48,154)
( 49,157)( 50,160)( 51,159)( 52,158)( 53,161)( 54,164)( 55,163)( 56,162)
( 57,165)( 58,168)( 59,167)( 60,166)( 61,169)( 62,172)( 63,171)( 64,170)
( 65,173)( 66,176)( 67,175)( 68,174)( 69,177)( 70,180)( 71,179)( 72,178)
( 73,109)( 74,112)( 75,111)( 76,110)( 77,113)( 78,116)( 79,115)( 80,114)
( 81,117)( 82,120)( 83,119)( 84,118)( 85,121)( 86,124)( 87,123)( 88,122)
( 89,125)( 90,128)( 91,127)( 92,126)( 93,129)( 94,132)( 95,131)( 96,130)
( 97,133)( 98,136)( 99,135)(100,134)(101,137)(102,140)(103,139)(104,138)
(105,141)(106,144)(107,143)(108,142);;
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*s1*s2*s1*s2,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*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*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(216)!(  1,  3)(  2,  4)(  5, 11)(  6, 12)(  7,  9)(  8, 10)( 13, 31)
( 14, 32)( 15, 29)( 16, 30)( 17, 27)( 18, 28)( 19, 25)( 20, 26)( 21, 35)
( 22, 36)( 23, 33)( 24, 34)( 37, 39)( 38, 40)( 41, 47)( 42, 48)( 43, 45)
( 44, 46)( 49, 67)( 50, 68)( 51, 65)( 52, 66)( 53, 63)( 54, 64)( 55, 61)
( 56, 62)( 57, 71)( 58, 72)( 59, 69)( 60, 70)( 73, 75)( 74, 76)( 77, 83)
( 78, 84)( 79, 81)( 80, 82)( 85,103)( 86,104)( 87,101)( 88,102)( 89, 99)
( 90,100)( 91, 97)( 92, 98)( 93,107)( 94,108)( 95,105)( 96,106)(109,111)
(110,112)(113,119)(114,120)(115,117)(116,118)(121,139)(122,140)(123,137)
(124,138)(125,135)(126,136)(127,133)(128,134)(129,143)(130,144)(131,141)
(132,142)(145,147)(146,148)(149,155)(150,156)(151,153)(152,154)(157,175)
(158,176)(159,173)(160,174)(161,171)(162,172)(163,169)(164,170)(165,179)
(166,180)(167,177)(168,178)(181,183)(182,184)(185,191)(186,192)(187,189)
(188,190)(193,211)(194,212)(195,209)(196,210)(197,207)(198,208)(199,205)
(200,206)(201,215)(202,216)(203,213)(204,214);
s1 := Sym(216)!(  1, 13)(  2, 15)(  3, 14)(  4, 16)(  5, 21)(  6, 23)(  7, 22)
(  8, 24)(  9, 17)( 10, 19)( 11, 18)( 12, 20)( 25, 29)( 26, 31)( 27, 30)
( 28, 32)( 34, 35)( 37, 85)( 38, 87)( 39, 86)( 40, 88)( 41, 93)( 42, 95)
( 43, 94)( 44, 96)( 45, 89)( 46, 91)( 47, 90)( 48, 92)( 49, 73)( 50, 75)
( 51, 74)( 52, 76)( 53, 81)( 54, 83)( 55, 82)( 56, 84)( 57, 77)( 58, 79)
( 59, 78)( 60, 80)( 61,101)( 62,103)( 63,102)( 64,104)( 65, 97)( 66, 99)
( 67, 98)( 68,100)( 69,105)( 70,107)( 71,106)( 72,108)(109,121)(110,123)
(111,122)(112,124)(113,129)(114,131)(115,130)(116,132)(117,125)(118,127)
(119,126)(120,128)(133,137)(134,139)(135,138)(136,140)(142,143)(145,193)
(146,195)(147,194)(148,196)(149,201)(150,203)(151,202)(152,204)(153,197)
(154,199)(155,198)(156,200)(157,181)(158,183)(159,182)(160,184)(161,189)
(162,191)(163,190)(164,192)(165,185)(166,187)(167,186)(168,188)(169,209)
(170,211)(171,210)(172,212)(173,205)(174,207)(175,206)(176,208)(177,213)
(178,215)(179,214)(180,216);
s2 := Sym(216)!(  1,181)(  2,184)(  3,183)(  4,182)(  5,185)(  6,188)(  7,187)
(  8,186)(  9,189)( 10,192)( 11,191)( 12,190)( 13,193)( 14,196)( 15,195)
( 16,194)( 17,197)( 18,200)( 19,199)( 20,198)( 21,201)( 22,204)( 23,203)
( 24,202)( 25,205)( 26,208)( 27,207)( 28,206)( 29,209)( 30,212)( 31,211)
( 32,210)( 33,213)( 34,216)( 35,215)( 36,214)( 37,145)( 38,148)( 39,147)
( 40,146)( 41,149)( 42,152)( 43,151)( 44,150)( 45,153)( 46,156)( 47,155)
( 48,154)( 49,157)( 50,160)( 51,159)( 52,158)( 53,161)( 54,164)( 55,163)
( 56,162)( 57,165)( 58,168)( 59,167)( 60,166)( 61,169)( 62,172)( 63,171)
( 64,170)( 65,173)( 66,176)( 67,175)( 68,174)( 69,177)( 70,180)( 71,179)
( 72,178)( 73,109)( 74,112)( 75,111)( 76,110)( 77,113)( 78,116)( 79,115)
( 80,114)( 81,117)( 82,120)( 83,119)( 84,118)( 85,121)( 86,124)( 87,123)
( 88,122)( 89,125)( 90,128)( 91,127)( 92,126)( 93,129)( 94,132)( 95,131)
( 96,130)( 97,133)( 98,136)( 99,135)(100,134)(101,137)(102,140)(103,139)
(104,138)(105,141)(106,144)(107,143)(108,142);
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*s1*s2*s1*s2,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*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*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1 >;

```
References : None.
to this polytope