# Polytope of Type {24,6,3}

Atlas Canonical Name : {24,6,3}*864a
if this polytope has a name.
Group : SmallGroup(864,814)
Rank : 4
Schlafli Type : {24,6,3}
Number of vertices, edges, etc : 24, 72, 9, 3
Order of s0s1s2s3 : 24
Order of s0s1s2s3s2s1 : 6
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{24,6,3,2} of size 1728
Vertex Figure Of :
{2,24,6,3} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {12,6,3}*432a
3-fold quotients : {24,2,3}*288
4-fold quotients : {6,6,3}*216a
6-fold quotients : {12,2,3}*144
8-fold quotients : {3,6,3}*108
9-fold quotients : {8,2,3}*96
12-fold quotients : {6,2,3}*72
18-fold quotients : {4,2,3}*48
24-fold quotients : {3,2,3}*36
36-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {48,6,3}*1728a, {24,6,6}*1728a
Permutation Representation (GAP) :
```s0 := (  2,  3)(  5,  6)(  8,  9)( 10, 19)( 11, 21)( 12, 20)( 13, 22)( 14, 24)
( 15, 23)( 16, 25)( 17, 27)( 18, 26)( 29, 30)( 32, 33)( 35, 36)( 37, 46)
( 38, 48)( 39, 47)( 40, 49)( 41, 51)( 42, 50)( 43, 52)( 44, 54)( 45, 53)
( 55, 82)( 56, 84)( 57, 83)( 58, 85)( 59, 87)( 60, 86)( 61, 88)( 62, 90)
( 63, 89)( 64,100)( 65,102)( 66,101)( 67,103)( 68,105)( 69,104)( 70,106)
( 71,108)( 72,107)( 73, 91)( 74, 93)( 75, 92)( 76, 94)( 77, 96)( 78, 95)
( 79, 97)( 80, 99)( 81, 98)(109,163)(110,165)(111,164)(112,166)(113,168)
(114,167)(115,169)(116,171)(117,170)(118,181)(119,183)(120,182)(121,184)
(122,186)(123,185)(124,187)(125,189)(126,188)(127,172)(128,174)(129,173)
(130,175)(131,177)(132,176)(133,178)(134,180)(135,179)(136,190)(137,192)
(138,191)(139,193)(140,195)(141,194)(142,196)(143,198)(144,197)(145,208)
(146,210)(147,209)(148,211)(149,213)(150,212)(151,214)(152,216)(153,215)
(154,199)(155,201)(156,200)(157,202)(158,204)(159,203)(160,205)(161,207)
(162,206);;
s1 := (  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);;
s2 := (  1,  4)(  2,  6)(  3,  5)(  8,  9)( 10, 13)( 11, 15)( 12, 14)( 17, 18)
( 19, 22)( 20, 24)( 21, 23)( 26, 27)( 28, 31)( 29, 33)( 30, 32)( 35, 36)
( 37, 40)( 38, 42)( 39, 41)( 44, 45)( 46, 49)( 47, 51)( 48, 50)( 53, 54)
( 55, 58)( 56, 60)( 57, 59)( 62, 63)( 64, 67)( 65, 69)( 66, 68)( 71, 72)
( 73, 76)( 74, 78)( 75, 77)( 80, 81)( 82, 85)( 83, 87)( 84, 86)( 89, 90)
( 91, 94)( 92, 96)( 93, 95)( 98, 99)(100,103)(101,105)(102,104)(107,108)
(109,112)(110,114)(111,113)(116,117)(118,121)(119,123)(120,122)(125,126)
(127,130)(128,132)(129,131)(134,135)(136,139)(137,141)(138,140)(143,144)
(145,148)(146,150)(147,149)(152,153)(154,157)(155,159)(156,158)(161,162)
(163,166)(164,168)(165,167)(170,171)(172,175)(173,177)(174,176)(179,180)
(181,184)(182,186)(183,185)(188,189)(190,193)(191,195)(192,194)(197,198)
(199,202)(200,204)(201,203)(206,207)(208,211)(209,213)(210,212)(215,216);;
s3 := (  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);;
poly := Group([s0,s1,s2,s3]);;

```
Finitely Presented Group Representation (GAP) :
```F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*s3*s2*s3,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2,
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)(  5,  6)(  8,  9)( 10, 19)( 11, 21)( 12, 20)( 13, 22)
( 14, 24)( 15, 23)( 16, 25)( 17, 27)( 18, 26)( 29, 30)( 32, 33)( 35, 36)
( 37, 46)( 38, 48)( 39, 47)( 40, 49)( 41, 51)( 42, 50)( 43, 52)( 44, 54)
( 45, 53)( 55, 82)( 56, 84)( 57, 83)( 58, 85)( 59, 87)( 60, 86)( 61, 88)
( 62, 90)( 63, 89)( 64,100)( 65,102)( 66,101)( 67,103)( 68,105)( 69,104)
( 70,106)( 71,108)( 72,107)( 73, 91)( 74, 93)( 75, 92)( 76, 94)( 77, 96)
( 78, 95)( 79, 97)( 80, 99)( 81, 98)(109,163)(110,165)(111,164)(112,166)
(113,168)(114,167)(115,169)(116,171)(117,170)(118,181)(119,183)(120,182)
(121,184)(122,186)(123,185)(124,187)(125,189)(126,188)(127,172)(128,174)
(129,173)(130,175)(131,177)(132,176)(133,178)(134,180)(135,179)(136,190)
(137,192)(138,191)(139,193)(140,195)(141,194)(142,196)(143,198)(144,197)
(145,208)(146,210)(147,209)(148,211)(149,213)(150,212)(151,214)(152,216)
(153,215)(154,199)(155,201)(156,200)(157,202)(158,204)(159,203)(160,205)
(161,207)(162,206);
s1 := 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);
s2 := Sym(216)!(  1,  4)(  2,  6)(  3,  5)(  8,  9)( 10, 13)( 11, 15)( 12, 14)
( 17, 18)( 19, 22)( 20, 24)( 21, 23)( 26, 27)( 28, 31)( 29, 33)( 30, 32)
( 35, 36)( 37, 40)( 38, 42)( 39, 41)( 44, 45)( 46, 49)( 47, 51)( 48, 50)
( 53, 54)( 55, 58)( 56, 60)( 57, 59)( 62, 63)( 64, 67)( 65, 69)( 66, 68)
( 71, 72)( 73, 76)( 74, 78)( 75, 77)( 80, 81)( 82, 85)( 83, 87)( 84, 86)
( 89, 90)( 91, 94)( 92, 96)( 93, 95)( 98, 99)(100,103)(101,105)(102,104)
(107,108)(109,112)(110,114)(111,113)(116,117)(118,121)(119,123)(120,122)
(125,126)(127,130)(128,132)(129,131)(134,135)(136,139)(137,141)(138,140)
(143,144)(145,148)(146,150)(147,149)(152,153)(154,157)(155,159)(156,158)
(161,162)(163,166)(164,168)(165,167)(170,171)(172,175)(173,177)(174,176)
(179,180)(181,184)(182,186)(183,185)(188,189)(190,193)(191,195)(192,194)
(197,198)(199,202)(200,204)(201,203)(206,207)(208,211)(209,213)(210,212)
(215,216);
s3 := 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);
poly := sub<Sym(216)|s0,s1,s2,s3>;

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2,
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s2*s3*s2*s3*s2*s3, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1,
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, 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