Polytope of Type {12,18,2}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,18,2}*1728a
if this polytope has a name.
Group : SmallGroup(1728,46115)
Rank : 4
Schlafli Type : {12,18,2}
Number of vertices, edges, etc : 24, 216, 36, 2
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 : {12,18,2}*864c
3-fold quotients : {4,18,2}*576, {12,6,2}*576a
4-fold quotients : {6,18,2}*432a
6-fold quotients : {4,9,2}*288, {4,18,2}*288b, {4,18,2}*288c, {12,6,2}*288d
9-fold quotients : {4,6,2}*192
12-fold quotients : {4,9,2}*144, {2,18,2}*144, {6,6,2}*144a
18-fold quotients : {4,3,2}*96, {4,6,2}*96b, {4,6,2}*96c
24-fold quotients : {2,9,2}*72
36-fold quotients : {4,3,2}*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 := ( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9, 11)( 10, 12)( 13, 27)( 14, 28)
( 15, 25)( 16, 26)( 17, 31)( 18, 32)( 19, 29)( 20, 30)( 21, 35)( 22, 36)
( 23, 33)( 24, 34)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)( 46, 48)
( 49, 63)( 50, 64)( 51, 61)( 52, 62)( 53, 67)( 54, 68)( 55, 65)( 56, 66)
( 57, 71)( 58, 72)( 59, 69)( 60, 70)( 73, 75)( 74, 76)( 77, 79)( 78, 80)
( 81, 83)( 82, 84)( 85, 99)( 86,100)( 87, 97)( 88, 98)( 89,103)( 90,104)
( 91,101)( 92,102)( 93,107)( 94,108)( 95,105)( 96,106)(109,111)(110,112)
(113,115)(114,116)(117,119)(118,120)(121,135)(122,136)(123,133)(124,134)
(125,139)(126,140)(127,137)(128,138)(129,143)(130,144)(131,141)(132,142)
(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,171)(158,172)
(159,169)(160,170)(161,175)(162,176)(163,173)(164,174)(165,179)(166,180)
(167,177)(168,178)(181,183)(182,184)(185,187)(186,188)(189,191)(190,192)
(193,207)(194,208)(195,205)(196,206)(197,211)(198,212)(199,209)(200,210)
(201,215)(202,216)(203,213)(204,214);;
s1 := ( 1, 13)( 2, 15)( 3, 14)( 4, 16)( 5, 21)( 6, 23)( 7, 22)( 8, 24)
( 9, 17)( 10, 19)( 11, 18)( 12, 20)( 26, 27)( 29, 33)( 30, 35)( 31, 34)
( 32, 36)( 37, 89)( 38, 91)( 39, 90)( 40, 92)( 41, 85)( 42, 87)( 43, 86)
( 44, 88)( 45, 93)( 46, 95)( 47, 94)( 48, 96)( 49, 77)( 50, 79)( 51, 78)
( 52, 80)( 53, 73)( 54, 75)( 55, 74)( 56, 76)( 57, 81)( 58, 83)( 59, 82)
( 60, 84)( 61,101)( 62,103)( 63,102)( 64,104)( 65, 97)( 66, 99)( 67, 98)
( 68,100)( 69,105)( 70,107)( 71,106)( 72,108)(109,121)(110,123)(111,122)
(112,124)(113,129)(114,131)(115,130)(116,132)(117,125)(118,127)(119,126)
(120,128)(134,135)(137,141)(138,143)(139,142)(140,144)(145,197)(146,199)
(147,198)(148,200)(149,193)(150,195)(151,194)(152,196)(153,201)(154,203)
(155,202)(156,204)(157,185)(158,187)(159,186)(160,188)(161,181)(162,183)
(163,182)(164,184)(165,189)(166,191)(167,190)(168,192)(169,209)(170,211)
(171,210)(172,212)(173,205)(174,207)(175,206)(176,208)(177,213)(178,215)
(179,214)(180,216);;
s2 := ( 1,181)( 2,184)( 3,183)( 4,182)( 5,189)( 6,192)( 7,191)( 8,190)
( 9,185)( 10,188)( 11,187)( 12,186)( 13,193)( 14,196)( 15,195)( 16,194)
( 17,201)( 18,204)( 19,203)( 20,202)( 21,197)( 22,200)( 23,199)( 24,198)
( 25,205)( 26,208)( 27,207)( 28,206)( 29,213)( 30,216)( 31,215)( 32,214)
( 33,209)( 34,212)( 35,211)( 36,210)( 37,145)( 38,148)( 39,147)( 40,146)
( 41,153)( 42,156)( 43,155)( 44,154)( 45,149)( 46,152)( 47,151)( 48,150)
( 49,157)( 50,160)( 51,159)( 52,158)( 53,165)( 54,168)( 55,167)( 56,166)
( 57,161)( 58,164)( 59,163)( 60,162)( 61,169)( 62,172)( 63,171)( 64,170)
( 65,177)( 66,180)( 67,179)( 68,178)( 69,173)( 70,176)( 71,175)( 72,174)
( 73,109)( 74,112)( 75,111)( 76,110)( 77,117)( 78,120)( 79,119)( 80,118)
( 81,113)( 82,116)( 83,115)( 84,114)( 85,121)( 86,124)( 87,123)( 88,122)
( 89,129)( 90,132)( 91,131)( 92,130)( 93,125)( 94,128)( 95,127)( 96,126)
( 97,133)( 98,136)( 99,135)(100,134)(101,141)(102,144)(103,143)(104,142)
(105,137)(106,140)(107,139)(108,138);;
s3 := (217,218);;
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*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(218)!( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9, 11)( 10, 12)( 13, 27)
( 14, 28)( 15, 25)( 16, 26)( 17, 31)( 18, 32)( 19, 29)( 20, 30)( 21, 35)
( 22, 36)( 23, 33)( 24, 34)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)
( 46, 48)( 49, 63)( 50, 64)( 51, 61)( 52, 62)( 53, 67)( 54, 68)( 55, 65)
( 56, 66)( 57, 71)( 58, 72)( 59, 69)( 60, 70)( 73, 75)( 74, 76)( 77, 79)
( 78, 80)( 81, 83)( 82, 84)( 85, 99)( 86,100)( 87, 97)( 88, 98)( 89,103)
( 90,104)( 91,101)( 92,102)( 93,107)( 94,108)( 95,105)( 96,106)(109,111)
(110,112)(113,115)(114,116)(117,119)(118,120)(121,135)(122,136)(123,133)
(124,134)(125,139)(126,140)(127,137)(128,138)(129,143)(130,144)(131,141)
(132,142)(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,171)
(158,172)(159,169)(160,170)(161,175)(162,176)(163,173)(164,174)(165,179)
(166,180)(167,177)(168,178)(181,183)(182,184)(185,187)(186,188)(189,191)
(190,192)(193,207)(194,208)(195,205)(196,206)(197,211)(198,212)(199,209)
(200,210)(201,215)(202,216)(203,213)(204,214);
s1 := Sym(218)!( 1, 13)( 2, 15)( 3, 14)( 4, 16)( 5, 21)( 6, 23)( 7, 22)
( 8, 24)( 9, 17)( 10, 19)( 11, 18)( 12, 20)( 26, 27)( 29, 33)( 30, 35)
( 31, 34)( 32, 36)( 37, 89)( 38, 91)( 39, 90)( 40, 92)( 41, 85)( 42, 87)
( 43, 86)( 44, 88)( 45, 93)( 46, 95)( 47, 94)( 48, 96)( 49, 77)( 50, 79)
( 51, 78)( 52, 80)( 53, 73)( 54, 75)( 55, 74)( 56, 76)( 57, 81)( 58, 83)
( 59, 82)( 60, 84)( 61,101)( 62,103)( 63,102)( 64,104)( 65, 97)( 66, 99)
( 67, 98)( 68,100)( 69,105)( 70,107)( 71,106)( 72,108)(109,121)(110,123)
(111,122)(112,124)(113,129)(114,131)(115,130)(116,132)(117,125)(118,127)
(119,126)(120,128)(134,135)(137,141)(138,143)(139,142)(140,144)(145,197)
(146,199)(147,198)(148,200)(149,193)(150,195)(151,194)(152,196)(153,201)
(154,203)(155,202)(156,204)(157,185)(158,187)(159,186)(160,188)(161,181)
(162,183)(163,182)(164,184)(165,189)(166,191)(167,190)(168,192)(169,209)
(170,211)(171,210)(172,212)(173,205)(174,207)(175,206)(176,208)(177,213)
(178,215)(179,214)(180,216);
s2 := Sym(218)!( 1,181)( 2,184)( 3,183)( 4,182)( 5,189)( 6,192)( 7,191)
( 8,190)( 9,185)( 10,188)( 11,187)( 12,186)( 13,193)( 14,196)( 15,195)
( 16,194)( 17,201)( 18,204)( 19,203)( 20,202)( 21,197)( 22,200)( 23,199)
( 24,198)( 25,205)( 26,208)( 27,207)( 28,206)( 29,213)( 30,216)( 31,215)
( 32,214)( 33,209)( 34,212)( 35,211)( 36,210)( 37,145)( 38,148)( 39,147)
( 40,146)( 41,153)( 42,156)( 43,155)( 44,154)( 45,149)( 46,152)( 47,151)
( 48,150)( 49,157)( 50,160)( 51,159)( 52,158)( 53,165)( 54,168)( 55,167)
( 56,166)( 57,161)( 58,164)( 59,163)( 60,162)( 61,169)( 62,172)( 63,171)
( 64,170)( 65,177)( 66,180)( 67,179)( 68,178)( 69,173)( 70,176)( 71,175)
( 72,174)( 73,109)( 74,112)( 75,111)( 76,110)( 77,117)( 78,120)( 79,119)
( 80,118)( 81,113)( 82,116)( 83,115)( 84,114)( 85,121)( 86,124)( 87,123)
( 88,122)( 89,129)( 90,132)( 91,131)( 92,130)( 93,125)( 94,128)( 95,127)
( 96,126)( 97,133)( 98,136)( 99,135)(100,134)(101,141)(102,144)(103,143)
(104,142)(105,137)(106,140)(107,139)(108,138);
s3 := Sym(218)!(217,218);
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*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s2*s3*s2*s3, s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
to this polytope