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# Polytope of Type {2,6,12}

Atlas Canonical Name : {2,6,12}*1728b
if this polytope has a name.
Group : SmallGroup(1728,46116)
Rank : 4
Schlafli Type : {2,6,12}
Number of vertices, edges, etc : 2, 36, 216, 72
Order of s0s1s2s3 : 6
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
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 : {2,6,12}*864d
3-fold quotients : {2,6,12}*576a
4-fold quotients : {2,6,6}*432b
6-fold quotients : {2,6,12}*288d
8-fold quotients : {2,6,6}*216
9-fold quotients : {2,6,4}*192
12-fold quotients : {2,6,6}*144a
18-fold quotients : {2,3,4}*96, {2,6,4}*96b, {2,6,4}*96c
36-fold quotients : {2,3,4}*48, {2,2,6}*48, {2,6,2}*48
72-fold quotients : {2,2,3}*24, {2,3,2}*24
108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (1,2);;
s1 := (  4,  5)(  7, 11)(  8, 13)(  9, 12)( 10, 14)( 16, 17)( 19, 23)( 20, 25)
( 21, 24)( 22, 26)( 28, 29)( 31, 35)( 32, 37)( 33, 36)( 34, 38)( 39, 75)
( 40, 77)( 41, 76)( 42, 78)( 43, 83)( 44, 85)( 45, 84)( 46, 86)( 47, 79)
( 48, 81)( 49, 80)( 50, 82)( 51, 87)( 52, 89)( 53, 88)( 54, 90)( 55, 95)
( 56, 97)( 57, 96)( 58, 98)( 59, 91)( 60, 93)( 61, 92)( 62, 94)( 63, 99)
( 64,101)( 65,100)( 66,102)( 67,107)( 68,109)( 69,108)( 70,110)( 71,103)
( 72,105)( 73,104)( 74,106)(112,113)(115,119)(116,121)(117,120)(118,122)
(124,125)(127,131)(128,133)(129,132)(130,134)(136,137)(139,143)(140,145)
(141,144)(142,146)(147,183)(148,185)(149,184)(150,186)(151,191)(152,193)
(153,192)(154,194)(155,187)(156,189)(157,188)(158,190)(159,195)(160,197)
(161,196)(162,198)(163,203)(164,205)(165,204)(166,206)(167,199)(168,201)
(169,200)(170,202)(171,207)(172,209)(173,208)(174,210)(175,215)(176,217)
(177,216)(178,218)(179,211)(180,213)(181,212)(182,214);;
s2 := (  3, 39)(  4, 40)(  5, 42)(  6, 41)(  7, 43)(  8, 44)(  9, 46)( 10, 45)
( 11, 47)( 12, 48)( 13, 50)( 14, 49)( 15, 71)( 16, 72)( 17, 74)( 18, 73)
( 19, 63)( 20, 64)( 21, 66)( 22, 65)( 23, 67)( 24, 68)( 25, 70)( 26, 69)
( 27, 55)( 28, 56)( 29, 58)( 30, 57)( 31, 59)( 32, 60)( 33, 62)( 34, 61)
( 35, 51)( 36, 52)( 37, 54)( 38, 53)( 77, 78)( 81, 82)( 85, 86)( 87,107)
( 88,108)( 89,110)( 90,109)( 91, 99)( 92,100)( 93,102)( 94,101)( 95,103)
( 96,104)( 97,106)( 98,105)(111,147)(112,148)(113,150)(114,149)(115,151)
(116,152)(117,154)(118,153)(119,155)(120,156)(121,158)(122,157)(123,179)
(124,180)(125,182)(126,181)(127,171)(128,172)(129,174)(130,173)(131,175)
(132,176)(133,178)(134,177)(135,163)(136,164)(137,166)(138,165)(139,167)
(140,168)(141,170)(142,169)(143,159)(144,160)(145,162)(146,161)(185,186)
(189,190)(193,194)(195,215)(196,216)(197,218)(198,217)(199,207)(200,208)
(201,210)(202,209)(203,211)(204,212)(205,214)(206,213);;
s3 := (  3,126)(  4,125)(  5,124)(  6,123)(  7,134)(  8,133)(  9,132)( 10,131)
( 11,130)( 12,129)( 13,128)( 14,127)( 15,114)( 16,113)( 17,112)( 18,111)
( 19,122)( 20,121)( 21,120)( 22,119)( 23,118)( 24,117)( 25,116)( 26,115)
( 27,138)( 28,137)( 29,136)( 30,135)( 31,146)( 32,145)( 33,144)( 34,143)
( 35,142)( 36,141)( 37,140)( 38,139)( 39,162)( 40,161)( 41,160)( 42,159)
( 43,170)( 44,169)( 45,168)( 46,167)( 47,166)( 48,165)( 49,164)( 50,163)
( 51,150)( 52,149)( 53,148)( 54,147)( 55,158)( 56,157)( 57,156)( 58,155)
( 59,154)( 60,153)( 61,152)( 62,151)( 63,174)( 64,173)( 65,172)( 66,171)
( 67,182)( 68,181)( 69,180)( 70,179)( 71,178)( 72,177)( 73,176)( 74,175)
( 75,198)( 76,197)( 77,196)( 78,195)( 79,206)( 80,205)( 81,204)( 82,203)
( 83,202)( 84,201)( 85,200)( 86,199)( 87,186)( 88,185)( 89,184)( 90,183)
( 91,194)( 92,193)( 93,192)( 94,191)( 95,190)( 96,189)( 97,188)( 98,187)
( 99,210)(100,209)(101,208)(102,207)(103,218)(104,217)(105,216)(106,215)
(107,214)(108,213)(109,212)(110,211);;
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*s1*s0*s1,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s3*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s1*s2,
s3*s1*s2*s3*s2*s3*s1*s2*s3*s1*s2*s3*s2*s3*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(218)!(1,2);
s1 := Sym(218)!(  4,  5)(  7, 11)(  8, 13)(  9, 12)( 10, 14)( 16, 17)( 19, 23)
( 20, 25)( 21, 24)( 22, 26)( 28, 29)( 31, 35)( 32, 37)( 33, 36)( 34, 38)
( 39, 75)( 40, 77)( 41, 76)( 42, 78)( 43, 83)( 44, 85)( 45, 84)( 46, 86)
( 47, 79)( 48, 81)( 49, 80)( 50, 82)( 51, 87)( 52, 89)( 53, 88)( 54, 90)
( 55, 95)( 56, 97)( 57, 96)( 58, 98)( 59, 91)( 60, 93)( 61, 92)( 62, 94)
( 63, 99)( 64,101)( 65,100)( 66,102)( 67,107)( 68,109)( 69,108)( 70,110)
( 71,103)( 72,105)( 73,104)( 74,106)(112,113)(115,119)(116,121)(117,120)
(118,122)(124,125)(127,131)(128,133)(129,132)(130,134)(136,137)(139,143)
(140,145)(141,144)(142,146)(147,183)(148,185)(149,184)(150,186)(151,191)
(152,193)(153,192)(154,194)(155,187)(156,189)(157,188)(158,190)(159,195)
(160,197)(161,196)(162,198)(163,203)(164,205)(165,204)(166,206)(167,199)
(168,201)(169,200)(170,202)(171,207)(172,209)(173,208)(174,210)(175,215)
(176,217)(177,216)(178,218)(179,211)(180,213)(181,212)(182,214);
s2 := Sym(218)!(  3, 39)(  4, 40)(  5, 42)(  6, 41)(  7, 43)(  8, 44)(  9, 46)
( 10, 45)( 11, 47)( 12, 48)( 13, 50)( 14, 49)( 15, 71)( 16, 72)( 17, 74)
( 18, 73)( 19, 63)( 20, 64)( 21, 66)( 22, 65)( 23, 67)( 24, 68)( 25, 70)
( 26, 69)( 27, 55)( 28, 56)( 29, 58)( 30, 57)( 31, 59)( 32, 60)( 33, 62)
( 34, 61)( 35, 51)( 36, 52)( 37, 54)( 38, 53)( 77, 78)( 81, 82)( 85, 86)
( 87,107)( 88,108)( 89,110)( 90,109)( 91, 99)( 92,100)( 93,102)( 94,101)
( 95,103)( 96,104)( 97,106)( 98,105)(111,147)(112,148)(113,150)(114,149)
(115,151)(116,152)(117,154)(118,153)(119,155)(120,156)(121,158)(122,157)
(123,179)(124,180)(125,182)(126,181)(127,171)(128,172)(129,174)(130,173)
(131,175)(132,176)(133,178)(134,177)(135,163)(136,164)(137,166)(138,165)
(139,167)(140,168)(141,170)(142,169)(143,159)(144,160)(145,162)(146,161)
(185,186)(189,190)(193,194)(195,215)(196,216)(197,218)(198,217)(199,207)
(200,208)(201,210)(202,209)(203,211)(204,212)(205,214)(206,213);
s3 := Sym(218)!(  3,126)(  4,125)(  5,124)(  6,123)(  7,134)(  8,133)(  9,132)
( 10,131)( 11,130)( 12,129)( 13,128)( 14,127)( 15,114)( 16,113)( 17,112)
( 18,111)( 19,122)( 20,121)( 21,120)( 22,119)( 23,118)( 24,117)( 25,116)
( 26,115)( 27,138)( 28,137)( 29,136)( 30,135)( 31,146)( 32,145)( 33,144)
( 34,143)( 35,142)( 36,141)( 37,140)( 38,139)( 39,162)( 40,161)( 41,160)
( 42,159)( 43,170)( 44,169)( 45,168)( 46,167)( 47,166)( 48,165)( 49,164)
( 50,163)( 51,150)( 52,149)( 53,148)( 54,147)( 55,158)( 56,157)( 57,156)
( 58,155)( 59,154)( 60,153)( 61,152)( 62,151)( 63,174)( 64,173)( 65,172)
( 66,171)( 67,182)( 68,181)( 69,180)( 70,179)( 71,178)( 72,177)( 73,176)
( 74,175)( 75,198)( 76,197)( 77,196)( 78,195)( 79,206)( 80,205)( 81,204)
( 82,203)( 83,202)( 84,201)( 85,200)( 86,199)( 87,186)( 88,185)( 89,184)
( 90,183)( 91,194)( 92,193)( 93,192)( 94,191)( 95,190)( 96,189)( 97,188)
( 98,187)( 99,210)(100,209)(101,208)(102,207)(103,218)(104,217)(105,216)
(106,215)(107,214)(108,213)(109,212)(110,211);
poly := sub<Sym(218)|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*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s3*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s1*s2,
s3*s1*s2*s3*s2*s3*s1*s2*s3*s1*s2*s3*s2*s3*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

```

to this polytope