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# Polytope of Type {4,6,6}

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

```
References : None.
to this polytope