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# Polytope of Type {24,6}

Atlas Canonical Name : {24,6}*864c
if this polytope has a name.
Group : SmallGroup(864,1157)
Rank : 3
Schlafli Type : {24,6}
Number of vertices, edges, etc : 72, 216, 18
Order of s0s1s2 : 24
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Self-Petrie
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{24,6,2} of size 1728
Vertex Figure Of :
{2,24,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {12,6}*432c
3-fold quotients : {24,6}*288c
4-fold quotients : {6,6}*216a
6-fold quotients : {12,6}*144c
8-fold quotients : {6,3}*108
9-fold quotients : {8,6}*96
12-fold quotients : {6,6}*72b
18-fold quotients : {4,6}*48a
24-fold quotients : {6,3}*36
27-fold quotients : {8,2}*32
36-fold quotients : {2,6}*24
54-fold quotients : {4,2}*16
72-fold quotients : {2,3}*12
108-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {24,12}*1728b, {48,6}*1728c
Permutation Representation (GAP) :
```s0 := (  2,  3)(  4,  7)(  5,  9)(  6,  8)( 11, 12)( 13, 16)( 14, 18)( 15, 17)
( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 29, 30)( 31, 34)( 32, 36)( 33, 35)
( 38, 39)( 40, 43)( 41, 45)( 42, 44)( 47, 48)( 49, 52)( 50, 54)( 51, 53)
( 55, 82)( 56, 84)( 57, 83)( 58, 88)( 59, 90)( 60, 89)( 61, 85)( 62, 87)
( 63, 86)( 64, 91)( 65, 93)( 66, 92)( 67, 97)( 68, 99)( 69, 98)( 70, 94)
( 71, 96)( 72, 95)( 73,100)( 74,102)( 75,101)( 76,106)( 77,108)( 78,107)
( 79,103)( 80,105)( 81,104)(109,163)(110,165)(111,164)(112,169)(113,171)
(114,170)(115,166)(116,168)(117,167)(118,172)(119,174)(120,173)(121,178)
(122,180)(123,179)(124,175)(125,177)(126,176)(127,181)(128,183)(129,182)
(130,187)(131,189)(132,188)(133,184)(134,186)(135,185)(136,190)(137,192)
(138,191)(139,196)(140,198)(141,197)(142,193)(143,195)(144,194)(145,199)
(146,201)(147,200)(148,205)(149,207)(150,206)(151,202)(152,204)(153,203)
(154,208)(155,210)(156,209)(157,214)(158,216)(159,215)(160,211)(161,213)
(162,212);;
s1 := (  1,112)(  2,113)(  3,114)(  4,109)(  5,110)(  6,111)(  7,115)(  8,116)
(  9,117)( 10,130)( 11,131)( 12,132)( 13,127)( 14,128)( 15,129)( 16,133)
( 17,134)( 18,135)( 19,121)( 20,122)( 21,123)( 22,118)( 23,119)( 24,120)
( 25,124)( 26,125)( 27,126)( 28,139)( 29,140)( 30,141)( 31,136)( 32,137)
( 33,138)( 34,142)( 35,143)( 36,144)( 37,157)( 38,158)( 39,159)( 40,154)
( 41,155)( 42,156)( 43,160)( 44,161)( 45,162)( 46,148)( 47,149)( 48,150)
( 49,145)( 50,146)( 51,147)( 52,151)( 53,152)( 54,153)( 55,193)( 56,194)
( 57,195)( 58,190)( 59,191)( 60,192)( 61,196)( 62,197)( 63,198)( 64,211)
( 65,212)( 66,213)( 67,208)( 68,209)( 69,210)( 70,214)( 71,215)( 72,216)
( 73,202)( 74,203)( 75,204)( 76,199)( 77,200)( 78,201)( 79,205)( 80,206)
( 81,207)( 82,166)( 83,167)( 84,168)( 85,163)( 86,164)( 87,165)( 88,169)
( 89,170)( 90,171)( 91,184)( 92,185)( 93,186)( 94,181)( 95,182)( 96,183)
( 97,187)( 98,188)( 99,189)(100,175)(101,176)(102,177)(103,172)(104,173)
(105,174)(106,178)(107,179)(108,180);;
s2 := (  1, 10)(  2, 11)(  3, 12)(  4, 18)(  5, 16)(  6, 17)(  7, 14)(  8, 15)
(  9, 13)( 22, 27)( 23, 25)( 24, 26)( 28, 37)( 29, 38)( 30, 39)( 31, 45)
( 32, 43)( 33, 44)( 34, 41)( 35, 42)( 36, 40)( 49, 54)( 50, 52)( 51, 53)
( 55, 64)( 56, 65)( 57, 66)( 58, 72)( 59, 70)( 60, 71)( 61, 68)( 62, 69)
( 63, 67)( 76, 81)( 77, 79)( 78, 80)( 82, 91)( 83, 92)( 84, 93)( 85, 99)
( 86, 97)( 87, 98)( 88, 95)( 89, 96)( 90, 94)(103,108)(104,106)(105,107)
(109,118)(110,119)(111,120)(112,126)(113,124)(114,125)(115,122)(116,123)
(117,121)(130,135)(131,133)(132,134)(136,145)(137,146)(138,147)(139,153)
(140,151)(141,152)(142,149)(143,150)(144,148)(157,162)(158,160)(159,161)
(163,172)(164,173)(165,174)(166,180)(167,178)(168,179)(169,176)(170,177)
(171,175)(184,189)(185,187)(186,188)(190,199)(191,200)(192,201)(193,207)
(194,205)(195,206)(196,203)(197,204)(198,202)(211,216)(212,214)(213,215);;
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*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1,
s1*s2*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*s0*s1*s0 ];;
poly := F / rels;;

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

```
References : None.
to this polytope