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# Polytope of Type {6,6}

Atlas Canonical Name : {6,6}*864a
if this polytope has a name.
Group : SmallGroup(864,4000)
Rank : 3
Schlafli Type : {6,6}
Number of vertices, edges, etc : 72, 216, 72
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 :
{6,6,2} of size 1728
Vertex Figure Of :
{2,6,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {3,6}*432
3-fold quotients : {6,6}*288b
4-fold quotients : {6,6}*216c
6-fold quotients : {3,6}*144
8-fold quotients : {3,6}*108
9-fold quotients : {6,6}*96
12-fold quotients : {6,6}*72c
18-fold quotients : {3,6}*48, {6,3}*48
24-fold quotients : {3,6}*36
36-fold quotients : {3,3}*24, {6,2}*24
72-fold quotients : {3,2}*12
108-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {12,6}*1728a, {6,12}*1728c, {6,6}*1728b, {12,6}*1728d, {6,12}*1728e
Permutation Representation (GAP) :
```s0 := (  2,  3)(  6,  7)( 10, 11)( 13, 25)( 14, 27)( 15, 26)( 16, 28)( 17, 29)
( 18, 31)( 19, 30)( 20, 32)( 21, 33)( 22, 35)( 23, 34)( 24, 36)( 37, 73)
( 38, 75)( 39, 74)( 40, 76)( 41, 77)( 42, 79)( 43, 78)( 44, 80)( 45, 81)
( 46, 83)( 47, 82)( 48, 84)( 49, 97)( 50, 99)( 51, 98)( 52,100)( 53,101)
( 54,103)( 55,102)( 56,104)( 57,105)( 58,107)( 59,106)( 60,108)( 61, 85)
( 62, 87)( 63, 86)( 64, 88)( 65, 89)( 66, 91)( 67, 90)( 68, 92)( 69, 93)
( 70, 95)( 71, 94)( 72, 96)(110,111)(114,115)(118,119)(121,133)(122,135)
(123,134)(124,136)(125,137)(126,139)(127,138)(128,140)(129,141)(130,143)
(131,142)(132,144)(145,181)(146,183)(147,182)(148,184)(149,185)(150,187)
(151,186)(152,188)(153,189)(154,191)(155,190)(156,192)(157,205)(158,207)
(159,206)(160,208)(161,209)(162,211)(163,210)(164,212)(165,213)(166,215)
(167,214)(168,216)(169,193)(170,195)(171,194)(172,196)(173,197)(174,199)
(175,198)(176,200)(177,201)(178,203)(179,202)(180,204);;
s1 := (  1,197)(  2,200)(  3,199)(  4,198)(  5,201)(  6,204)(  7,203)(  8,202)
(  9,193)( 10,196)( 11,195)( 12,194)( 13,189)( 14,192)( 15,191)( 16,190)
( 17,181)( 18,184)( 19,183)( 20,182)( 21,185)( 22,188)( 23,187)( 24,186)
( 25,205)( 26,208)( 27,207)( 28,206)( 29,209)( 30,212)( 31,211)( 32,210)
( 33,213)( 34,216)( 35,215)( 36,214)( 37,161)( 38,164)( 39,163)( 40,162)
( 41,165)( 42,168)( 43,167)( 44,166)( 45,157)( 46,160)( 47,159)( 48,158)
( 49,153)( 50,156)( 51,155)( 52,154)( 53,145)( 54,148)( 55,147)( 56,146)
( 57,149)( 58,152)( 59,151)( 60,150)( 61,169)( 62,172)( 63,171)( 64,170)
( 65,173)( 66,176)( 67,175)( 68,174)( 69,177)( 70,180)( 71,179)( 72,178)
( 73,125)( 74,128)( 75,127)( 76,126)( 77,129)( 78,132)( 79,131)( 80,130)
( 81,121)( 82,124)( 83,123)( 84,122)( 85,117)( 86,120)( 87,119)( 88,118)
( 89,109)( 90,112)( 91,111)( 92,110)( 93,113)( 94,116)( 95,115)( 96,114)
( 97,133)( 98,136)( 99,135)(100,134)(101,137)(102,140)(103,139)(104,138)
(105,141)(106,144)(107,143)(108,142);;
s2 := (  1,  4)(  5, 12)(  6, 10)(  7, 11)(  8,  9)( 13, 16)( 17, 24)( 18, 22)
( 19, 23)( 20, 21)( 25, 28)( 29, 36)( 30, 34)( 31, 35)( 32, 33)( 37, 76)
( 38, 74)( 39, 75)( 40, 73)( 41, 84)( 42, 82)( 43, 83)( 44, 81)( 45, 80)
( 46, 78)( 47, 79)( 48, 77)( 49, 88)( 50, 86)( 51, 87)( 52, 85)( 53, 96)
( 54, 94)( 55, 95)( 56, 93)( 57, 92)( 58, 90)( 59, 91)( 60, 89)( 61,100)
( 62, 98)( 63, 99)( 64, 97)( 65,108)( 66,106)( 67,107)( 68,105)( 69,104)
( 70,102)( 71,103)( 72,101)(109,112)(113,120)(114,118)(115,119)(116,117)
(121,124)(125,132)(126,130)(127,131)(128,129)(133,136)(137,144)(138,142)
(139,143)(140,141)(145,184)(146,182)(147,183)(148,181)(149,192)(150,190)
(151,191)(152,189)(153,188)(154,186)(155,187)(156,185)(157,196)(158,194)
(159,195)(160,193)(161,204)(162,202)(163,203)(164,201)(165,200)(166,198)
(167,199)(168,197)(169,208)(170,206)(171,207)(172,205)(173,216)(174,214)
(175,215)(176,213)(177,212)(178,210)(179,211)(180,209);;
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*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope