Polytope of Type {4,3,6}

Atlas Canonical Name : {4,3,6}*864
if this polytope has a name.
Group : SmallGroup(864,4000)
Rank : 4
Schlafli Type : {4,3,6}
Number of vertices, edges, etc : 8, 36, 54, 18
Order of s0s1s2s3 : 6
Order of s0s1s2s3s2s1 : 2
Special Properties :
Locally Toroidal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,3,6,2} of size 1728
Vertex Figure Of :
{2,4,3,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,3,6}*432
3-fold quotients : {4,3,6}*288
4-fold quotients : {2,3,6}*216
6-fold quotients : {4,3,6}*144
9-fold quotients : {4,3,2}*96
12-fold quotients : {2,3,6}*72
18-fold quotients : {4,3,2}*48
36-fold quotients : {2,3,2}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {8,3,6}*1728, {4,6,6}*1728a
Permutation Representation (GAP) :
```s0 := (  1,111)(  2,112)(  3,109)(  4,110)(  5,115)(  6,116)(  7,113)(  8,114)
(  9,119)( 10,120)( 11,117)( 12,118)( 13,123)( 14,124)( 15,121)( 16,122)
( 17,127)( 18,128)( 19,125)( 20,126)( 21,131)( 22,132)( 23,129)( 24,130)
( 25,135)( 26,136)( 27,133)( 28,134)( 29,139)( 30,140)( 31,137)( 32,138)
( 33,143)( 34,144)( 35,141)( 36,142)( 37,147)( 38,148)( 39,145)( 40,146)
( 41,151)( 42,152)( 43,149)( 44,150)( 45,155)( 46,156)( 47,153)( 48,154)
( 49,159)( 50,160)( 51,157)( 52,158)( 53,163)( 54,164)( 55,161)( 56,162)
( 57,167)( 58,168)( 59,165)( 60,166)( 61,171)( 62,172)( 63,169)( 64,170)
( 65,175)( 66,176)( 67,173)( 68,174)( 69,179)( 70,180)( 71,177)( 72,178)
( 73,183)( 74,184)( 75,181)( 76,182)( 77,187)( 78,188)( 79,185)( 80,186)
( 81,191)( 82,192)( 83,189)( 84,190)( 85,195)( 86,196)( 87,193)( 88,194)
( 89,199)( 90,200)( 91,197)( 92,198)( 93,203)( 94,204)( 95,201)( 96,202)
( 97,207)( 98,208)( 99,205)(100,206)(101,211)(102,212)(103,209)(104,210)
(105,215)(106,216)(107,213)(108,214);;
s1 := (  2,  3)(  6,  7)( 10, 11)( 13, 25)( 14, 27)( 15, 26)( 16, 28)( 17, 29)
( 18, 31)( 19, 30)( 20, 32)( 21, 33)( 22, 35)( 23, 34)( 24, 36)( 37, 73)
( 38, 75)( 39, 74)( 40, 76)( 41, 77)( 42, 79)( 43, 78)( 44, 80)( 45, 81)
( 46, 83)( 47, 82)( 48, 84)( 49, 97)( 50, 99)( 51, 98)( 52,100)( 53,101)
( 54,103)( 55,102)( 56,104)( 57,105)( 58,107)( 59,106)( 60,108)( 61, 85)
( 62, 87)( 63, 86)( 64, 88)( 65, 89)( 66, 91)( 67, 90)( 68, 92)( 69, 93)
( 70, 95)( 71, 94)( 72, 96)(110,111)(114,115)(118,119)(121,133)(122,135)
(123,134)(124,136)(125,137)(126,139)(127,138)(128,140)(129,141)(130,143)
(131,142)(132,144)(145,181)(146,183)(147,182)(148,184)(149,185)(150,187)
(151,186)(152,188)(153,189)(154,191)(155,190)(156,192)(157,205)(158,207)
(159,206)(160,208)(161,209)(162,211)(163,210)(164,212)(165,213)(166,215)
(167,214)(168,216)(169,193)(170,195)(171,194)(172,196)(173,197)(174,199)
(175,198)(176,200)(177,201)(178,203)(179,202)(180,204);;
s2 := (  1, 89)(  2, 92)(  3, 91)(  4, 90)(  5, 93)(  6, 96)(  7, 95)(  8, 94)
(  9, 85)( 10, 88)( 11, 87)( 12, 86)( 13, 81)( 14, 84)( 15, 83)( 16, 82)
( 17, 73)( 18, 76)( 19, 75)( 20, 74)( 21, 77)( 22, 80)( 23, 79)( 24, 78)
( 25, 97)( 26,100)( 27, 99)( 28, 98)( 29,101)( 30,104)( 31,103)( 32,102)
( 33,105)( 34,108)( 35,107)( 36,106)( 37, 53)( 38, 56)( 39, 55)( 40, 54)
( 41, 57)( 42, 60)( 43, 59)( 44, 58)( 45, 49)( 46, 52)( 47, 51)( 48, 50)
( 62, 64)( 66, 68)( 70, 72)(109,197)(110,200)(111,199)(112,198)(113,201)
(114,204)(115,203)(116,202)(117,193)(118,196)(119,195)(120,194)(121,189)
(122,192)(123,191)(124,190)(125,181)(126,184)(127,183)(128,182)(129,185)
(130,188)(131,187)(132,186)(133,205)(134,208)(135,207)(136,206)(137,209)
(138,212)(139,211)(140,210)(141,213)(142,216)(143,215)(144,214)(145,161)
(146,164)(147,163)(148,162)(149,165)(150,168)(151,167)(152,166)(153,157)
(154,160)(155,159)(156,158)(170,172)(174,176)(178,180);;
s3 := (  5,  9)(  6, 10)(  7, 11)(  8, 12)( 13, 25)( 14, 26)( 15, 27)( 16, 28)
( 17, 33)( 18, 34)( 19, 35)( 20, 36)( 21, 29)( 22, 30)( 23, 31)( 24, 32)
( 41, 45)( 42, 46)( 43, 47)( 44, 48)( 49, 61)( 50, 62)( 51, 63)( 52, 64)
( 53, 69)( 54, 70)( 55, 71)( 56, 72)( 57, 65)( 58, 66)( 59, 67)( 60, 68)
( 77, 81)( 78, 82)( 79, 83)( 80, 84)( 85, 97)( 86, 98)( 87, 99)( 88,100)
( 89,105)( 90,106)( 91,107)( 92,108)( 93,101)( 94,102)( 95,103)( 96,104)
(113,117)(114,118)(115,119)(116,120)(121,133)(122,134)(123,135)(124,136)
(125,141)(126,142)(127,143)(128,144)(129,137)(130,138)(131,139)(132,140)
(149,153)(150,154)(151,155)(152,156)(157,169)(158,170)(159,171)(160,172)
(161,177)(162,178)(163,179)(164,180)(165,173)(166,174)(167,175)(168,176)
(185,189)(186,190)(187,191)(188,192)(193,205)(194,206)(195,207)(196,208)
(197,213)(198,214)(199,215)(200,216)(201,209)(202,210)(203,211)(204,212);;
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, s1*s2*s1*s2*s1*s2,
s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s3*s2*s1*s0*s1*s2*s3*s2*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(216)!(  1,111)(  2,112)(  3,109)(  4,110)(  5,115)(  6,116)(  7,113)
(  8,114)(  9,119)( 10,120)( 11,117)( 12,118)( 13,123)( 14,124)( 15,121)
( 16,122)( 17,127)( 18,128)( 19,125)( 20,126)( 21,131)( 22,132)( 23,129)
( 24,130)( 25,135)( 26,136)( 27,133)( 28,134)( 29,139)( 30,140)( 31,137)
( 32,138)( 33,143)( 34,144)( 35,141)( 36,142)( 37,147)( 38,148)( 39,145)
( 40,146)( 41,151)( 42,152)( 43,149)( 44,150)( 45,155)( 46,156)( 47,153)
( 48,154)( 49,159)( 50,160)( 51,157)( 52,158)( 53,163)( 54,164)( 55,161)
( 56,162)( 57,167)( 58,168)( 59,165)( 60,166)( 61,171)( 62,172)( 63,169)
( 64,170)( 65,175)( 66,176)( 67,173)( 68,174)( 69,179)( 70,180)( 71,177)
( 72,178)( 73,183)( 74,184)( 75,181)( 76,182)( 77,187)( 78,188)( 79,185)
( 80,186)( 81,191)( 82,192)( 83,189)( 84,190)( 85,195)( 86,196)( 87,193)
( 88,194)( 89,199)( 90,200)( 91,197)( 92,198)( 93,203)( 94,204)( 95,201)
( 96,202)( 97,207)( 98,208)( 99,205)(100,206)(101,211)(102,212)(103,209)
(104,210)(105,215)(106,216)(107,213)(108,214);
s1 := Sym(216)!(  2,  3)(  6,  7)( 10, 11)( 13, 25)( 14, 27)( 15, 26)( 16, 28)
( 17, 29)( 18, 31)( 19, 30)( 20, 32)( 21, 33)( 22, 35)( 23, 34)( 24, 36)
( 37, 73)( 38, 75)( 39, 74)( 40, 76)( 41, 77)( 42, 79)( 43, 78)( 44, 80)
( 45, 81)( 46, 83)( 47, 82)( 48, 84)( 49, 97)( 50, 99)( 51, 98)( 52,100)
( 53,101)( 54,103)( 55,102)( 56,104)( 57,105)( 58,107)( 59,106)( 60,108)
( 61, 85)( 62, 87)( 63, 86)( 64, 88)( 65, 89)( 66, 91)( 67, 90)( 68, 92)
( 69, 93)( 70, 95)( 71, 94)( 72, 96)(110,111)(114,115)(118,119)(121,133)
(122,135)(123,134)(124,136)(125,137)(126,139)(127,138)(128,140)(129,141)
(130,143)(131,142)(132,144)(145,181)(146,183)(147,182)(148,184)(149,185)
(150,187)(151,186)(152,188)(153,189)(154,191)(155,190)(156,192)(157,205)
(158,207)(159,206)(160,208)(161,209)(162,211)(163,210)(164,212)(165,213)
(166,215)(167,214)(168,216)(169,193)(170,195)(171,194)(172,196)(173,197)
(174,199)(175,198)(176,200)(177,201)(178,203)(179,202)(180,204);
s2 := Sym(216)!(  1, 89)(  2, 92)(  3, 91)(  4, 90)(  5, 93)(  6, 96)(  7, 95)
(  8, 94)(  9, 85)( 10, 88)( 11, 87)( 12, 86)( 13, 81)( 14, 84)( 15, 83)
( 16, 82)( 17, 73)( 18, 76)( 19, 75)( 20, 74)( 21, 77)( 22, 80)( 23, 79)
( 24, 78)( 25, 97)( 26,100)( 27, 99)( 28, 98)( 29,101)( 30,104)( 31,103)
( 32,102)( 33,105)( 34,108)( 35,107)( 36,106)( 37, 53)( 38, 56)( 39, 55)
( 40, 54)( 41, 57)( 42, 60)( 43, 59)( 44, 58)( 45, 49)( 46, 52)( 47, 51)
( 48, 50)( 62, 64)( 66, 68)( 70, 72)(109,197)(110,200)(111,199)(112,198)
(113,201)(114,204)(115,203)(116,202)(117,193)(118,196)(119,195)(120,194)
(121,189)(122,192)(123,191)(124,190)(125,181)(126,184)(127,183)(128,182)
(129,185)(130,188)(131,187)(132,186)(133,205)(134,208)(135,207)(136,206)
(137,209)(138,212)(139,211)(140,210)(141,213)(142,216)(143,215)(144,214)
(145,161)(146,164)(147,163)(148,162)(149,165)(150,168)(151,167)(152,166)
(153,157)(154,160)(155,159)(156,158)(170,172)(174,176)(178,180);
s3 := Sym(216)!(  5,  9)(  6, 10)(  7, 11)(  8, 12)( 13, 25)( 14, 26)( 15, 27)
( 16, 28)( 17, 33)( 18, 34)( 19, 35)( 20, 36)( 21, 29)( 22, 30)( 23, 31)
( 24, 32)( 41, 45)( 42, 46)( 43, 47)( 44, 48)( 49, 61)( 50, 62)( 51, 63)
( 52, 64)( 53, 69)( 54, 70)( 55, 71)( 56, 72)( 57, 65)( 58, 66)( 59, 67)
( 60, 68)( 77, 81)( 78, 82)( 79, 83)( 80, 84)( 85, 97)( 86, 98)( 87, 99)
( 88,100)( 89,105)( 90,106)( 91,107)( 92,108)( 93,101)( 94,102)( 95,103)
( 96,104)(113,117)(114,118)(115,119)(116,120)(121,133)(122,134)(123,135)
(124,136)(125,141)(126,142)(127,143)(128,144)(129,137)(130,138)(131,139)
(132,140)(149,153)(150,154)(151,155)(152,156)(157,169)(158,170)(159,171)
(160,172)(161,177)(162,178)(163,179)(164,180)(165,173)(166,174)(167,175)
(168,176)(185,189)(186,190)(187,191)(188,192)(193,205)(194,206)(195,207)
(196,208)(197,213)(198,214)(199,215)(200,216)(201,209)(202,210)(203,211)
(204,212);
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,
s1*s2*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s3*s2*s1*s0*s1*s2*s3*s2*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2 >;

```
References : None.
to this polytope