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# Polytope of Type {6,6,4}

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

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

```
References : None.
to this polytope