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# Polytope of Type {6,24}

Atlas Canonical Name : {6,24}*864b
if this polytope has a name.
Group : SmallGroup(864,814)
Rank : 3
Schlafli Type : {6,24}
Number of vertices, edges, etc : 18, 216, 72
Order of s0s1s2 : 24
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{6,24,2} of size 1728
Vertex Figure Of :
{2,6,24} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {6,12}*432b
3-fold quotients : {6,24}*288a
4-fold quotients : {6,6}*216b
6-fold quotients : {6,12}*144a
8-fold quotients : {6,6}*108
9-fold quotients : {2,24}*96, {6,8}*96
12-fold quotients : {6,6}*72a
18-fold quotients : {2,12}*48, {6,4}*48a
27-fold quotients : {2,8}*32
36-fold quotients : {2,6}*24, {6,2}*24
54-fold quotients : {2,4}*16
72-fold quotients : {2,3}*12, {3,2}*12
108-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,48}*1728b, {12,24}*1728d
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)
( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 65, 66)( 67, 70)( 68, 72)( 69, 71)
( 74, 75)( 76, 79)( 77, 81)( 78, 80)( 83, 84)( 85, 88)( 86, 90)( 87, 89)
( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)(103,106)(104,108)(105,107)
(110,111)(112,115)(113,117)(114,116)(119,120)(121,124)(122,126)(123,125)
(128,129)(130,133)(131,135)(132,134)(137,138)(139,142)(140,144)(141,143)
(146,147)(148,151)(149,153)(150,152)(155,156)(157,160)(158,162)(159,161)
(164,165)(166,169)(167,171)(168,170)(173,174)(175,178)(176,180)(177,179)
(182,183)(184,187)(185,189)(186,188)(191,192)(193,196)(194,198)(195,197)
(200,201)(202,205)(203,207)(204,206)(209,210)(211,214)(212,216)(213,215);;
s1 := (  1,  4)(  2,  5)(  3,  6)( 10, 22)( 11, 23)( 12, 24)( 13, 19)( 14, 20)
( 15, 21)( 16, 25)( 17, 26)( 18, 27)( 28, 31)( 29, 32)( 30, 33)( 37, 49)
( 38, 50)( 39, 51)( 40, 46)( 41, 47)( 42, 48)( 43, 52)( 44, 53)( 45, 54)
( 55, 85)( 56, 86)( 57, 87)( 58, 82)( 59, 83)( 60, 84)( 61, 88)( 62, 89)
( 63, 90)( 64,103)( 65,104)( 66,105)( 67,100)( 68,101)( 69,102)( 70,106)
( 71,107)( 72,108)( 73, 94)( 74, 95)( 75, 96)( 76, 91)( 77, 92)( 78, 93)
( 79, 97)( 80, 98)( 81, 99)(109,166)(110,167)(111,168)(112,163)(113,164)
(114,165)(115,169)(116,170)(117,171)(118,184)(119,185)(120,186)(121,181)
(122,182)(123,183)(124,187)(125,188)(126,189)(127,175)(128,176)(129,177)
(130,172)(131,173)(132,174)(133,178)(134,179)(135,180)(136,193)(137,194)
(138,195)(139,190)(140,191)(141,192)(142,196)(143,197)(144,198)(145,211)
(146,212)(147,213)(148,208)(149,209)(150,210)(151,214)(152,215)(153,216)
(154,202)(155,203)(156,204)(157,199)(158,200)(159,201)(160,205)(161,206)
(162,207);;
s2 := (  1,118)(  2,120)(  3,119)(  4,122)(  5,121)(  6,123)(  7,126)(  8,125)
(  9,124)( 10,109)( 11,111)( 12,110)( 13,113)( 14,112)( 15,114)( 16,117)
( 17,116)( 18,115)( 19,127)( 20,129)( 21,128)( 22,131)( 23,130)( 24,132)
( 25,135)( 26,134)( 27,133)( 28,145)( 29,147)( 30,146)( 31,149)( 32,148)
( 33,150)( 34,153)( 35,152)( 36,151)( 37,136)( 38,138)( 39,137)( 40,140)
( 41,139)( 42,141)( 43,144)( 44,143)( 45,142)( 46,154)( 47,156)( 48,155)
( 49,158)( 50,157)( 51,159)( 52,162)( 53,161)( 54,160)( 55,199)( 56,201)
( 57,200)( 58,203)( 59,202)( 60,204)( 61,207)( 62,206)( 63,205)( 64,190)
( 65,192)( 66,191)( 67,194)( 68,193)( 69,195)( 70,198)( 71,197)( 72,196)
( 73,208)( 74,210)( 75,209)( 76,212)( 77,211)( 78,213)( 79,216)( 80,215)
( 81,214)( 82,172)( 83,174)( 84,173)( 85,176)( 86,175)( 87,177)( 88,180)
( 89,179)( 90,178)( 91,163)( 92,165)( 93,164)( 94,167)( 95,166)( 96,168)
( 97,171)( 98,170)( 99,169)(100,181)(101,183)(102,182)(103,185)(104,184)
(105,186)(106,189)(107,188)(108,187);;
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,
s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*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 ];;
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)( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 65, 66)( 67, 70)( 68, 72)
( 69, 71)( 74, 75)( 76, 79)( 77, 81)( 78, 80)( 83, 84)( 85, 88)( 86, 90)
( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)(103,106)(104,108)
(105,107)(110,111)(112,115)(113,117)(114,116)(119,120)(121,124)(122,126)
(123,125)(128,129)(130,133)(131,135)(132,134)(137,138)(139,142)(140,144)
(141,143)(146,147)(148,151)(149,153)(150,152)(155,156)(157,160)(158,162)
(159,161)(164,165)(166,169)(167,171)(168,170)(173,174)(175,178)(176,180)
(177,179)(182,183)(184,187)(185,189)(186,188)(191,192)(193,196)(194,198)
(195,197)(200,201)(202,205)(203,207)(204,206)(209,210)(211,214)(212,216)
(213,215);
s1 := Sym(216)!(  1,  4)(  2,  5)(  3,  6)( 10, 22)( 11, 23)( 12, 24)( 13, 19)
( 14, 20)( 15, 21)( 16, 25)( 17, 26)( 18, 27)( 28, 31)( 29, 32)( 30, 33)
( 37, 49)( 38, 50)( 39, 51)( 40, 46)( 41, 47)( 42, 48)( 43, 52)( 44, 53)
( 45, 54)( 55, 85)( 56, 86)( 57, 87)( 58, 82)( 59, 83)( 60, 84)( 61, 88)
( 62, 89)( 63, 90)( 64,103)( 65,104)( 66,105)( 67,100)( 68,101)( 69,102)
( 70,106)( 71,107)( 72,108)( 73, 94)( 74, 95)( 75, 96)( 76, 91)( 77, 92)
( 78, 93)( 79, 97)( 80, 98)( 81, 99)(109,166)(110,167)(111,168)(112,163)
(113,164)(114,165)(115,169)(116,170)(117,171)(118,184)(119,185)(120,186)
(121,181)(122,182)(123,183)(124,187)(125,188)(126,189)(127,175)(128,176)
(129,177)(130,172)(131,173)(132,174)(133,178)(134,179)(135,180)(136,193)
(137,194)(138,195)(139,190)(140,191)(141,192)(142,196)(143,197)(144,198)
(145,211)(146,212)(147,213)(148,208)(149,209)(150,210)(151,214)(152,215)
(153,216)(154,202)(155,203)(156,204)(157,199)(158,200)(159,201)(160,205)
(161,206)(162,207);
s2 := Sym(216)!(  1,118)(  2,120)(  3,119)(  4,122)(  5,121)(  6,123)(  7,126)
(  8,125)(  9,124)( 10,109)( 11,111)( 12,110)( 13,113)( 14,112)( 15,114)
( 16,117)( 17,116)( 18,115)( 19,127)( 20,129)( 21,128)( 22,131)( 23,130)
( 24,132)( 25,135)( 26,134)( 27,133)( 28,145)( 29,147)( 30,146)( 31,149)
( 32,148)( 33,150)( 34,153)( 35,152)( 36,151)( 37,136)( 38,138)( 39,137)
( 40,140)( 41,139)( 42,141)( 43,144)( 44,143)( 45,142)( 46,154)( 47,156)
( 48,155)( 49,158)( 50,157)( 51,159)( 52,162)( 53,161)( 54,160)( 55,199)
( 56,201)( 57,200)( 58,203)( 59,202)( 60,204)( 61,207)( 62,206)( 63,205)
( 64,190)( 65,192)( 66,191)( 67,194)( 68,193)( 69,195)( 70,198)( 71,197)
( 72,196)( 73,208)( 74,210)( 75,209)( 76,212)( 77,211)( 78,213)( 79,216)
( 80,215)( 81,214)( 82,172)( 83,174)( 84,173)( 85,176)( 86,175)( 87,177)
( 88,180)( 89,179)( 90,178)( 91,163)( 92,165)( 93,164)( 94,167)( 95,166)
( 96,168)( 97,171)( 98,170)( 99,169)(100,181)(101,183)(102,182)(103,185)
(104,184)(105,186)(106,189)(107,188)(108,187);
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,
s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*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 >;

```
References : None.
to this polytope