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# Polytope of Type {2,12,18}

Atlas Canonical Name : {2,12,18}*1728b
if this polytope has a name.
Group : SmallGroup(1728,46115)
Rank : 4
Schlafli Type : {2,12,18}
Number of vertices, edges, etc : 2, 24, 216, 36
Order of s0s1s2s3 : 18
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 : {2,12,9}*864
3-fold quotients : {2,4,18}*576, {2,12,6}*576b
4-fold quotients : {2,6,18}*432b
6-fold quotients : {2,4,9}*288, {2,4,18}*288b, {2,4,18}*288c, {2,12,3}*288
8-fold quotients : {2,6,9}*216
9-fold quotients : {2,4,6}*192
12-fold quotients : {2,4,9}*144, {2,2,18}*144, {2,6,6}*144b
18-fold quotients : {2,4,3}*96, {2,4,6}*96b, {2,4,6}*96c
24-fold quotients : {2,2,9}*72, {2,6,3}*72
36-fold quotients : {2,4,3}*48, {2,2,6}*48
72-fold quotients : {2,2,3}*24
108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (1,2);;
s1 := (  3,  5)(  4,  6)(  7,  9)(  8, 10)( 11, 13)( 12, 14)( 15, 29)( 16, 30)
( 17, 27)( 18, 28)( 19, 33)( 20, 34)( 21, 31)( 22, 32)( 23, 37)( 24, 38)
( 25, 35)( 26, 36)( 39, 41)( 40, 42)( 43, 45)( 44, 46)( 47, 49)( 48, 50)
( 51, 65)( 52, 66)( 53, 63)( 54, 64)( 55, 69)( 56, 70)( 57, 67)( 58, 68)
( 59, 73)( 60, 74)( 61, 71)( 62, 72)( 75, 77)( 76, 78)( 79, 81)( 80, 82)
( 83, 85)( 84, 86)( 87,101)( 88,102)( 89, 99)( 90,100)( 91,105)( 92,106)
( 93,103)( 94,104)( 95,109)( 96,110)( 97,107)( 98,108)(111,113)(112,114)
(115,117)(116,118)(119,121)(120,122)(123,137)(124,138)(125,135)(126,136)
(127,141)(128,142)(129,139)(130,140)(131,145)(132,146)(133,143)(134,144)
(147,149)(148,150)(151,153)(152,154)(155,157)(156,158)(159,173)(160,174)
(161,171)(162,172)(163,177)(164,178)(165,175)(166,176)(167,181)(168,182)
(169,179)(170,180)(183,185)(184,186)(187,189)(188,190)(191,193)(192,194)
(195,209)(196,210)(197,207)(198,208)(199,213)(200,214)(201,211)(202,212)
(203,217)(204,218)(205,215)(206,216);;
s2 := (  3, 15)(  4, 17)(  5, 16)(  6, 18)(  7, 23)(  8, 25)(  9, 24)( 10, 26)
( 11, 19)( 12, 21)( 13, 20)( 14, 22)( 28, 29)( 31, 35)( 32, 37)( 33, 36)
( 34, 38)( 39, 91)( 40, 93)( 41, 92)( 42, 94)( 43, 87)( 44, 89)( 45, 88)
( 46, 90)( 47, 95)( 48, 97)( 49, 96)( 50, 98)( 51, 79)( 52, 81)( 53, 80)
( 54, 82)( 55, 75)( 56, 77)( 57, 76)( 58, 78)( 59, 83)( 60, 85)( 61, 84)
( 62, 86)( 63,103)( 64,105)( 65,104)( 66,106)( 67, 99)( 68,101)( 69,100)
( 70,102)( 71,107)( 72,109)( 73,108)( 74,110)(111,123)(112,125)(113,124)
(114,126)(115,131)(116,133)(117,132)(118,134)(119,127)(120,129)(121,128)
(122,130)(136,137)(139,143)(140,145)(141,144)(142,146)(147,199)(148,201)
(149,200)(150,202)(151,195)(152,197)(153,196)(154,198)(155,203)(156,205)
(157,204)(158,206)(159,187)(160,189)(161,188)(162,190)(163,183)(164,185)
(165,184)(166,186)(167,191)(168,193)(169,192)(170,194)(171,211)(172,213)
(173,212)(174,214)(175,207)(176,209)(177,208)(178,210)(179,215)(180,217)
(181,216)(182,218);;
s3 := (  3,183)(  4,186)(  5,185)(  6,184)(  7,191)(  8,194)(  9,193)( 10,192)
( 11,187)( 12,190)( 13,189)( 14,188)( 15,207)( 16,210)( 17,209)( 18,208)
( 19,215)( 20,218)( 21,217)( 22,216)( 23,211)( 24,214)( 25,213)( 26,212)
( 27,195)( 28,198)( 29,197)( 30,196)( 31,203)( 32,206)( 33,205)( 34,204)
( 35,199)( 36,202)( 37,201)( 38,200)( 39,147)( 40,150)( 41,149)( 42,148)
( 43,155)( 44,158)( 45,157)( 46,156)( 47,151)( 48,154)( 49,153)( 50,152)
( 51,171)( 52,174)( 53,173)( 54,172)( 55,179)( 56,182)( 57,181)( 58,180)
( 59,175)( 60,178)( 61,177)( 62,176)( 63,159)( 64,162)( 65,161)( 66,160)
( 67,167)( 68,170)( 69,169)( 70,168)( 71,163)( 72,166)( 73,165)( 74,164)
( 75,111)( 76,114)( 77,113)( 78,112)( 79,119)( 80,122)( 81,121)( 82,120)
( 83,115)( 84,118)( 85,117)( 86,116)( 87,135)( 88,138)( 89,137)( 90,136)
( 91,143)( 92,146)( 93,145)( 94,144)( 95,139)( 96,142)( 97,141)( 98,140)
( 99,123)(100,126)(101,125)(102,124)(103,131)(104,134)(105,133)(106,132)
(107,127)(108,130)(109,129)(110,128);;
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*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2,
s3*s1*s2*s1*s2*s1*s2*s1*s2*s3*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2*s3*s1*s2*s1*s3*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*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)!(  3,  5)(  4,  6)(  7,  9)(  8, 10)( 11, 13)( 12, 14)( 15, 29)
( 16, 30)( 17, 27)( 18, 28)( 19, 33)( 20, 34)( 21, 31)( 22, 32)( 23, 37)
( 24, 38)( 25, 35)( 26, 36)( 39, 41)( 40, 42)( 43, 45)( 44, 46)( 47, 49)
( 48, 50)( 51, 65)( 52, 66)( 53, 63)( 54, 64)( 55, 69)( 56, 70)( 57, 67)
( 58, 68)( 59, 73)( 60, 74)( 61, 71)( 62, 72)( 75, 77)( 76, 78)( 79, 81)
( 80, 82)( 83, 85)( 84, 86)( 87,101)( 88,102)( 89, 99)( 90,100)( 91,105)
( 92,106)( 93,103)( 94,104)( 95,109)( 96,110)( 97,107)( 98,108)(111,113)
(112,114)(115,117)(116,118)(119,121)(120,122)(123,137)(124,138)(125,135)
(126,136)(127,141)(128,142)(129,139)(130,140)(131,145)(132,146)(133,143)
(134,144)(147,149)(148,150)(151,153)(152,154)(155,157)(156,158)(159,173)
(160,174)(161,171)(162,172)(163,177)(164,178)(165,175)(166,176)(167,181)
(168,182)(169,179)(170,180)(183,185)(184,186)(187,189)(188,190)(191,193)
(192,194)(195,209)(196,210)(197,207)(198,208)(199,213)(200,214)(201,211)
(202,212)(203,217)(204,218)(205,215)(206,216);
s2 := Sym(218)!(  3, 15)(  4, 17)(  5, 16)(  6, 18)(  7, 23)(  8, 25)(  9, 24)
( 10, 26)( 11, 19)( 12, 21)( 13, 20)( 14, 22)( 28, 29)( 31, 35)( 32, 37)
( 33, 36)( 34, 38)( 39, 91)( 40, 93)( 41, 92)( 42, 94)( 43, 87)( 44, 89)
( 45, 88)( 46, 90)( 47, 95)( 48, 97)( 49, 96)( 50, 98)( 51, 79)( 52, 81)
( 53, 80)( 54, 82)( 55, 75)( 56, 77)( 57, 76)( 58, 78)( 59, 83)( 60, 85)
( 61, 84)( 62, 86)( 63,103)( 64,105)( 65,104)( 66,106)( 67, 99)( 68,101)
( 69,100)( 70,102)( 71,107)( 72,109)( 73,108)( 74,110)(111,123)(112,125)
(113,124)(114,126)(115,131)(116,133)(117,132)(118,134)(119,127)(120,129)
(121,128)(122,130)(136,137)(139,143)(140,145)(141,144)(142,146)(147,199)
(148,201)(149,200)(150,202)(151,195)(152,197)(153,196)(154,198)(155,203)
(156,205)(157,204)(158,206)(159,187)(160,189)(161,188)(162,190)(163,183)
(164,185)(165,184)(166,186)(167,191)(168,193)(169,192)(170,194)(171,211)
(172,213)(173,212)(174,214)(175,207)(176,209)(177,208)(178,210)(179,215)
(180,217)(181,216)(182,218);
s3 := Sym(218)!(  3,183)(  4,186)(  5,185)(  6,184)(  7,191)(  8,194)(  9,193)
( 10,192)( 11,187)( 12,190)( 13,189)( 14,188)( 15,207)( 16,210)( 17,209)
( 18,208)( 19,215)( 20,218)( 21,217)( 22,216)( 23,211)( 24,214)( 25,213)
( 26,212)( 27,195)( 28,198)( 29,197)( 30,196)( 31,203)( 32,206)( 33,205)
( 34,204)( 35,199)( 36,202)( 37,201)( 38,200)( 39,147)( 40,150)( 41,149)
( 42,148)( 43,155)( 44,158)( 45,157)( 46,156)( 47,151)( 48,154)( 49,153)
( 50,152)( 51,171)( 52,174)( 53,173)( 54,172)( 55,179)( 56,182)( 57,181)
( 58,180)( 59,175)( 60,178)( 61,177)( 62,176)( 63,159)( 64,162)( 65,161)
( 66,160)( 67,167)( 68,170)( 69,169)( 70,168)( 71,163)( 72,166)( 73,165)
( 74,164)( 75,111)( 76,114)( 77,113)( 78,112)( 79,119)( 80,122)( 81,121)
( 82,120)( 83,115)( 84,118)( 85,117)( 86,116)( 87,135)( 88,138)( 89,137)
( 90,136)( 91,143)( 92,146)( 93,145)( 94,144)( 95,139)( 96,142)( 97,141)
( 98,140)( 99,123)(100,126)(101,125)(102,124)(103,131)(104,134)(105,133)
(106,132)(107,127)(108,130)(109,129)(110,128);
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*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2,
s3*s1*s2*s1*s2*s1*s2*s1*s2*s3*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2*s3*s1*s2*s1*s3*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*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

```

to this polytope