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# Polytope of Type {18,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,12}*864b
if this polytope has a name.
Group : SmallGroup(864,3999)
Rank : 3
Schlafli Type : {18,12}
Number of vertices, edges, etc : 36, 216, 24
Order of s0s1s2 : 18
Order of s0s1s2s1 : 12
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{18,12,2} of size 1728
Vertex Figure Of :
{2,18,12} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {9,12}*432
3-fold quotients : {18,4}*288, {6,12}*288b
4-fold quotients : {18,6}*216b
6-fold quotients : {9,4}*144, {18,4}*144b, {18,4}*144c, {3,12}*144
8-fold quotients : {9,6}*108
9-fold quotients : {6,4}*96
12-fold quotients : {9,4}*72, {18,2}*72, {6,6}*72c
18-fold quotients : {3,4}*48, {6,4}*48b, {6,4}*48c
24-fold quotients : {9,2}*36, {3,6}*36
36-fold quotients : {3,4}*24, {6,2}*24
72-fold quotients : {3,2}*12
108-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {36,12}*1728f, {36,12}*1728g, {18,24}*1728b, {18,24}*1728d, {18,12}*1728d
Permutation Representation (GAP) :
```s0 := (  2,  3)(  5,  9)(  6, 11)(  7, 10)(  8, 12)( 13, 25)( 14, 27)( 15, 26)
( 16, 28)( 17, 33)( 18, 35)( 19, 34)( 20, 36)( 21, 29)( 22, 31)( 23, 30)
( 24, 32)( 37, 77)( 38, 79)( 39, 78)( 40, 80)( 41, 73)( 42, 75)( 43, 74)
( 44, 76)( 45, 81)( 46, 83)( 47, 82)( 48, 84)( 49,101)( 50,103)( 51,102)
( 52,104)( 53, 97)( 54, 99)( 55, 98)( 56,100)( 57,105)( 58,107)( 59,106)
( 60,108)( 61, 89)( 62, 91)( 63, 90)( 64, 92)( 65, 85)( 66, 87)( 67, 86)
( 68, 88)( 69, 93)( 70, 95)( 71, 94)( 72, 96)(110,111)(113,117)(114,119)
(115,118)(116,120)(121,133)(122,135)(123,134)(124,136)(125,141)(126,143)
(127,142)(128,144)(129,137)(130,139)(131,138)(132,140)(145,185)(146,187)
(147,186)(148,188)(149,181)(150,183)(151,182)(152,184)(153,189)(154,191)
(155,190)(156,192)(157,209)(158,211)(159,210)(160,212)(161,205)(162,207)
(163,206)(164,208)(165,213)(166,215)(167,214)(168,216)(169,197)(170,199)
(171,198)(172,200)(173,193)(174,195)(175,194)(176,196)(177,201)(178,203)
(179,202)(180,204);;
s1 := (  1,157)(  2,158)(  3,160)(  4,159)(  5,165)(  6,166)(  7,168)(  8,167)
(  9,161)( 10,162)( 11,164)( 12,163)( 13,145)( 14,146)( 15,148)( 16,147)
( 17,153)( 18,154)( 19,156)( 20,155)( 21,149)( 22,150)( 23,152)( 24,151)
( 25,169)( 26,170)( 27,172)( 28,171)( 29,177)( 30,178)( 31,180)( 32,179)
( 33,173)( 34,174)( 35,176)( 36,175)( 37,121)( 38,122)( 39,124)( 40,123)
( 41,129)( 42,130)( 43,132)( 44,131)( 45,125)( 46,126)( 47,128)( 48,127)
( 49,109)( 50,110)( 51,112)( 52,111)( 53,117)( 54,118)( 55,120)( 56,119)
( 57,113)( 58,114)( 59,116)( 60,115)( 61,133)( 62,134)( 63,136)( 64,135)
( 65,141)( 66,142)( 67,144)( 68,143)( 69,137)( 70,138)( 71,140)( 72,139)
( 73,197)( 74,198)( 75,200)( 76,199)( 77,193)( 78,194)( 79,196)( 80,195)
( 81,201)( 82,202)( 83,204)( 84,203)( 85,185)( 86,186)( 87,188)( 88,187)
( 89,181)( 90,182)( 91,184)( 92,183)( 93,189)( 94,190)( 95,192)( 96,191)
( 97,209)( 98,210)( 99,212)(100,211)(101,205)(102,206)(103,208)(104,207)
(105,213)(106,214)(107,216)(108,215);;
s2 := (  1,  4)(  2,  3)(  5,  8)(  6,  7)(  9, 12)( 10, 11)( 13, 28)( 14, 27)
( 15, 26)( 16, 25)( 17, 32)( 18, 31)( 19, 30)( 20, 29)( 21, 36)( 22, 35)
( 23, 34)( 24, 33)( 37, 40)( 38, 39)( 41, 44)( 42, 43)( 45, 48)( 46, 47)
( 49, 64)( 50, 63)( 51, 62)( 52, 61)( 53, 68)( 54, 67)( 55, 66)( 56, 65)
( 57, 72)( 58, 71)( 59, 70)( 60, 69)( 73, 76)( 74, 75)( 77, 80)( 78, 79)
( 81, 84)( 82, 83)( 85,100)( 86, 99)( 87, 98)( 88, 97)( 89,104)( 90,103)
( 91,102)( 92,101)( 93,108)( 94,107)( 95,106)( 96,105)(109,112)(110,111)
(113,116)(114,115)(117,120)(118,119)(121,136)(122,135)(123,134)(124,133)
(125,140)(126,139)(127,138)(128,137)(129,144)(130,143)(131,142)(132,141)
(145,148)(146,147)(149,152)(150,151)(153,156)(154,155)(157,172)(158,171)
(159,170)(160,169)(161,176)(162,175)(163,174)(164,173)(165,180)(166,179)
(167,178)(168,177)(181,184)(182,183)(185,188)(186,187)(189,192)(190,191)
(193,208)(194,207)(195,206)(196,205)(197,212)(198,211)(199,210)(200,209)
(201,216)(202,215)(203,214)(204,213);;
poly := Group([s0,s1,s2]);;

```
Finitely Presented Group Representation (GAP) :
```F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1,
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

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

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2,
s0*s2*s0*s2, s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1,
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

```
References : None.
to this polytope