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# Polytope of Type {72,6}

Atlas Canonical Name : {72,6}*864b
if this polytope has a name.
Group : SmallGroup(864,770)
Rank : 3
Schlafli Type : {72,6}
Number of vertices, edges, etc : 72, 216, 6
Order of s0s1s2 : 72
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{72,6,2} of size 1728
Vertex Figure Of :
{2,72,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {36,6}*432b
3-fold quotients : {72,2}*288, {24,6}*288b
4-fold quotients : {18,6}*216b
6-fold quotients : {36,2}*144, {12,6}*144b
8-fold quotients : {9,6}*108
9-fold quotients : {24,2}*96
12-fold quotients : {18,2}*72, {6,6}*72c
18-fold quotients : {12,2}*48
24-fold quotients : {9,2}*36, {3,6}*36
27-fold quotients : {8,2}*32
36-fold quotients : {6,2}*24
54-fold quotients : {4,2}*16
72-fold quotients : {3,2}*12
108-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {144,6}*1728b, {72,12}*1728b
Permutation Representation (GAP) :
```s0 := (  2,  3)(  4,  7)(  5,  9)(  6,  8)( 10, 21)( 11, 20)( 12, 19)( 13, 27)
( 14, 26)( 15, 25)( 16, 24)( 17, 23)( 18, 22)( 29, 30)( 31, 34)( 32, 36)
( 33, 35)( 37, 48)( 38, 47)( 39, 46)( 40, 54)( 41, 53)( 42, 52)( 43, 51)
( 44, 50)( 45, 49)( 55, 82)( 56, 84)( 57, 83)( 58, 88)( 59, 90)( 60, 89)
( 61, 85)( 62, 87)( 63, 86)( 64,102)( 65,101)( 66,100)( 67,108)( 68,107)
( 69,106)( 70,105)( 71,104)( 72,103)( 73, 93)( 74, 92)( 75, 91)( 76, 99)
( 77, 98)( 78, 97)( 79, 96)( 80, 95)( 81, 94)(109,163)(110,165)(111,164)
(112,169)(113,171)(114,170)(115,166)(116,168)(117,167)(118,183)(119,182)
(120,181)(121,189)(122,188)(123,187)(124,186)(125,185)(126,184)(127,174)
(128,173)(129,172)(130,180)(131,179)(132,178)(133,177)(134,176)(135,175)
(136,190)(137,192)(138,191)(139,196)(140,198)(141,197)(142,193)(143,195)
(144,194)(145,210)(146,209)(147,208)(148,216)(149,215)(150,214)(151,213)
(152,212)(153,211)(154,201)(155,200)(156,199)(157,207)(158,206)(159,205)
(160,204)(161,203)(162,202);;
s1 := (  1,121)(  2,123)(  3,122)(  4,118)(  5,120)(  6,119)(  7,124)(  8,126)
(  9,125)( 10,112)( 11,114)( 12,113)( 13,109)( 14,111)( 15,110)( 16,115)
( 17,117)( 18,116)( 19,132)( 20,131)( 21,130)( 22,129)( 23,128)( 24,127)
( 25,135)( 26,134)( 27,133)( 28,148)( 29,150)( 30,149)( 31,145)( 32,147)
( 33,146)( 34,151)( 35,153)( 36,152)( 37,139)( 38,141)( 39,140)( 40,136)
( 41,138)( 42,137)( 43,142)( 44,144)( 45,143)( 46,159)( 47,158)( 48,157)
( 49,156)( 50,155)( 51,154)( 52,162)( 53,161)( 54,160)( 55,202)( 56,204)
( 57,203)( 58,199)( 59,201)( 60,200)( 61,205)( 62,207)( 63,206)( 64,193)
( 65,195)( 66,194)( 67,190)( 68,192)( 69,191)( 70,196)( 71,198)( 72,197)
( 73,213)( 74,212)( 75,211)( 76,210)( 77,209)( 78,208)( 79,216)( 80,215)
( 81,214)( 82,175)( 83,177)( 84,176)( 85,172)( 86,174)( 87,173)( 88,178)
( 89,180)( 90,179)( 91,166)( 92,168)( 93,167)( 94,163)( 95,165)( 96,164)
( 97,169)( 98,171)( 99,170)(100,186)(101,185)(102,184)(103,183)(104,182)
(105,181)(106,189)(107,188)(108,187);;
s2 := (  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)(112,115)(113,116)(114,117)(121,124)
(122,125)(123,126)(130,133)(131,134)(132,135)(139,142)(140,143)(141,144)
(148,151)(149,152)(150,153)(157,160)(158,161)(159,162)(166,169)(167,170)
(168,171)(175,178)(176,179)(177,180)(184,187)(185,188)(186,189)(193,196)
(194,197)(195,198)(202,205)(203,206)(204,207)(211,214)(212,215)(213,216);;
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, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1,
s0*s1*s2*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*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*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*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*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 ];;
poly := F / rels;;

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

```
References : None.
to this polytope