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# Polytope of Type {6,6,4}

Atlas Canonical Name : {6,6,4}*1728c
if this polytope has a name.
Group : SmallGroup(1728,47874)
Rank : 4
Schlafli Type : {6,6,4}
Number of vertices, edges, etc : 18, 108, 72, 8
Order of s0s1s2s3 : 6
Order of s0s1s2s3s2s1 : 2
Special Properties :
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {6,6,4}*864i
3-fold quotients : {6,6,4}*576a, {6,6,4}*576b
4-fold quotients : {6,6,2}*432d
6-fold quotients : {6,3,4}*288, {6,6,4}*288d, {6,6,4}*288e, {6,6,4}*288f
9-fold quotients : {2,6,4}*192
12-fold quotients : {6,3,4}*144, {6,6,2}*144a, {6,6,2}*144b, {6,6,2}*144c
18-fold quotients : {2,3,4}*96, {2,6,4}*96b, {2,6,4}*96c
24-fold quotients : {3,6,2}*72, {6,3,2}*72
36-fold quotients : {2,3,4}*48, {2,6,2}*48, {6,2,2}*48
72-fold quotients : {2,3,2}*24, {3,2,2}*24
108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
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, 17)(  2, 18)(  3, 20)(  4, 19)(  5, 13)(  6, 14)(  7, 16)(  8, 15)
(  9, 21)( 10, 22)( 11, 24)( 12, 23)( 25, 29)( 26, 30)( 27, 32)( 28, 31)
( 35, 36)( 37, 89)( 38, 90)( 39, 92)( 40, 91)( 41, 85)( 42, 86)( 43, 88)
( 44, 87)( 45, 93)( 46, 94)( 47, 96)( 48, 95)( 49, 77)( 50, 78)( 51, 80)
( 52, 79)( 53, 73)( 54, 74)( 55, 76)( 56, 75)( 57, 81)( 58, 82)( 59, 84)
( 60, 83)( 61,101)( 62,102)( 63,104)( 64,103)( 65, 97)( 66, 98)( 67,100)
( 68, 99)( 69,105)( 70,106)( 71,108)( 72,107)(109,125)(110,126)(111,128)
(112,127)(113,121)(114,122)(115,124)(116,123)(117,129)(118,130)(119,132)
(120,131)(133,137)(134,138)(135,140)(136,139)(143,144)(145,197)(146,198)
(147,200)(148,199)(149,193)(150,194)(151,196)(152,195)(153,201)(154,202)
(155,204)(156,203)(157,185)(158,186)(159,188)(160,187)(161,181)(162,182)
(163,184)(164,183)(165,189)(166,190)(167,192)(168,191)(169,209)(170,210)
(171,212)(172,211)(173,205)(174,206)(175,208)(176,207)(177,213)(178,214)
(179,216)(180,215);;
s2 := (  1, 37)(  2, 40)(  3, 39)(  4, 38)(  5, 41)(  6, 44)(  7, 43)(  8, 42)
(  9, 45)( 10, 48)( 11, 47)( 12, 46)( 13, 61)( 14, 64)( 15, 63)( 16, 62)
( 17, 65)( 18, 68)( 19, 67)( 20, 66)( 21, 69)( 22, 72)( 23, 71)( 24, 70)
( 25, 49)( 26, 52)( 27, 51)( 28, 50)( 29, 53)( 30, 56)( 31, 55)( 32, 54)
( 33, 57)( 34, 60)( 35, 59)( 36, 58)( 74, 76)( 78, 80)( 82, 84)( 85, 97)
( 86,100)( 87, 99)( 88, 98)( 89,101)( 90,104)( 91,103)( 92,102)( 93,105)
( 94,108)( 95,107)( 96,106)(109,145)(110,148)(111,147)(112,146)(113,149)
(114,152)(115,151)(116,150)(117,153)(118,156)(119,155)(120,154)(121,169)
(122,172)(123,171)(124,170)(125,173)(126,176)(127,175)(128,174)(129,177)
(130,180)(131,179)(132,178)(133,157)(134,160)(135,159)(136,158)(137,161)
(138,164)(139,163)(140,162)(141,165)(142,168)(143,167)(144,166)(182,184)
(186,188)(190,192)(193,205)(194,208)(195,207)(196,206)(197,209)(198,212)
(199,211)(200,210)(201,213)(202,216)(203,215)(204,214);;
s3 := (  1,110)(  2,109)(  3,112)(  4,111)(  5,114)(  6,113)(  7,116)(  8,115)
(  9,118)( 10,117)( 11,120)( 12,119)( 13,122)( 14,121)( 15,124)( 16,123)
( 17,126)( 18,125)( 19,128)( 20,127)( 21,130)( 22,129)( 23,132)( 24,131)
( 25,134)( 26,133)( 27,136)( 28,135)( 29,138)( 30,137)( 31,140)( 32,139)
( 33,142)( 34,141)( 35,144)( 36,143)( 37,146)( 38,145)( 39,148)( 40,147)
( 41,150)( 42,149)( 43,152)( 44,151)( 45,154)( 46,153)( 47,156)( 48,155)
( 49,158)( 50,157)( 51,160)( 52,159)( 53,162)( 54,161)( 55,164)( 56,163)
( 57,166)( 58,165)( 59,168)( 60,167)( 61,170)( 62,169)( 63,172)( 64,171)
( 65,174)( 66,173)( 67,176)( 68,175)( 69,178)( 70,177)( 71,180)( 72,179)
( 73,182)( 74,181)( 75,184)( 76,183)( 77,186)( 78,185)( 79,188)( 80,187)
( 81,190)( 82,189)( 83,192)( 84,191)( 85,194)( 86,193)( 87,196)( 88,195)
( 89,198)( 90,197)( 91,200)( 92,199)( 93,202)( 94,201)( 95,204)( 96,203)
( 97,206)( 98,205)( 99,208)(100,207)(101,210)(102,209)(103,212)(104,211)
(105,214)(106,213)(107,216)(108,215);;
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,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s3*s2*s1*s0*s1*s2*s3*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*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, 17)(  2, 18)(  3, 20)(  4, 19)(  5, 13)(  6, 14)(  7, 16)
(  8, 15)(  9, 21)( 10, 22)( 11, 24)( 12, 23)( 25, 29)( 26, 30)( 27, 32)
( 28, 31)( 35, 36)( 37, 89)( 38, 90)( 39, 92)( 40, 91)( 41, 85)( 42, 86)
( 43, 88)( 44, 87)( 45, 93)( 46, 94)( 47, 96)( 48, 95)( 49, 77)( 50, 78)
( 51, 80)( 52, 79)( 53, 73)( 54, 74)( 55, 76)( 56, 75)( 57, 81)( 58, 82)
( 59, 84)( 60, 83)( 61,101)( 62,102)( 63,104)( 64,103)( 65, 97)( 66, 98)
( 67,100)( 68, 99)( 69,105)( 70,106)( 71,108)( 72,107)(109,125)(110,126)
(111,128)(112,127)(113,121)(114,122)(115,124)(116,123)(117,129)(118,130)
(119,132)(120,131)(133,137)(134,138)(135,140)(136,139)(143,144)(145,197)
(146,198)(147,200)(148,199)(149,193)(150,194)(151,196)(152,195)(153,201)
(154,202)(155,204)(156,203)(157,185)(158,186)(159,188)(160,187)(161,181)
(162,182)(163,184)(164,183)(165,189)(166,190)(167,192)(168,191)(169,209)
(170,210)(171,212)(172,211)(173,205)(174,206)(175,208)(176,207)(177,213)
(178,214)(179,216)(180,215);
s2 := Sym(216)!(  1, 37)(  2, 40)(  3, 39)(  4, 38)(  5, 41)(  6, 44)(  7, 43)
(  8, 42)(  9, 45)( 10, 48)( 11, 47)( 12, 46)( 13, 61)( 14, 64)( 15, 63)
( 16, 62)( 17, 65)( 18, 68)( 19, 67)( 20, 66)( 21, 69)( 22, 72)( 23, 71)
( 24, 70)( 25, 49)( 26, 52)( 27, 51)( 28, 50)( 29, 53)( 30, 56)( 31, 55)
( 32, 54)( 33, 57)( 34, 60)( 35, 59)( 36, 58)( 74, 76)( 78, 80)( 82, 84)
( 85, 97)( 86,100)( 87, 99)( 88, 98)( 89,101)( 90,104)( 91,103)( 92,102)
( 93,105)( 94,108)( 95,107)( 96,106)(109,145)(110,148)(111,147)(112,146)
(113,149)(114,152)(115,151)(116,150)(117,153)(118,156)(119,155)(120,154)
(121,169)(122,172)(123,171)(124,170)(125,173)(126,176)(127,175)(128,174)
(129,177)(130,180)(131,179)(132,178)(133,157)(134,160)(135,159)(136,158)
(137,161)(138,164)(139,163)(140,162)(141,165)(142,168)(143,167)(144,166)
(182,184)(186,188)(190,192)(193,205)(194,208)(195,207)(196,206)(197,209)
(198,212)(199,211)(200,210)(201,213)(202,216)(203,215)(204,214);
s3 := Sym(216)!(  1,110)(  2,109)(  3,112)(  4,111)(  5,114)(  6,113)(  7,116)
(  8,115)(  9,118)( 10,117)( 11,120)( 12,119)( 13,122)( 14,121)( 15,124)
( 16,123)( 17,126)( 18,125)( 19,128)( 20,127)( 21,130)( 22,129)( 23,132)
( 24,131)( 25,134)( 26,133)( 27,136)( 28,135)( 29,138)( 30,137)( 31,140)
( 32,139)( 33,142)( 34,141)( 35,144)( 36,143)( 37,146)( 38,145)( 39,148)
( 40,147)( 41,150)( 42,149)( 43,152)( 44,151)( 45,154)( 46,153)( 47,156)
( 48,155)( 49,158)( 50,157)( 51,160)( 52,159)( 53,162)( 54,161)( 55,164)
( 56,163)( 57,166)( 58,165)( 59,168)( 60,167)( 61,170)( 62,169)( 63,172)
( 64,171)( 65,174)( 66,173)( 67,176)( 68,175)( 69,178)( 70,177)( 71,180)
( 72,179)( 73,182)( 74,181)( 75,184)( 76,183)( 77,186)( 78,185)( 79,188)
( 80,187)( 81,190)( 82,189)( 83,192)( 84,191)( 85,194)( 86,193)( 87,196)
( 88,195)( 89,198)( 90,197)( 91,200)( 92,199)( 93,202)( 94,201)( 95,204)
( 96,203)( 97,206)( 98,205)( 99,208)(100,207)(101,210)(102,209)(103,212)
(104,211)(105,214)(106,213)(107,216)(108,215);
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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s3*s2*s1*s0*s1*s2*s3*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1 >;

```
References : None.
to this polytope