# Polytope of Type {6,24}

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

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

```
References : None.
to this polytope