Polytope of Type {12,24}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,24}*1728w
if this polytope has a name.
Group : SmallGroup(1728,33616)
Rank : 3
Schlafli Type : {12,24}
Number of vertices, edges, etc : 36, 432, 72
Order of s0s1s2 : 24
Order of s0s1s2s1 : 12
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
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 : {12,12}*864k
3-fold quotients : {4,24}*576b
4-fold quotients : {12,6}*432i
6-fold quotients : {4,12}*288
8-fold quotients : {12,6}*216c
9-fold quotients : {12,8}*192b
12-fold quotients : {4,6}*144
18-fold quotients : {12,4}*96a
24-fold quotients : {4,6}*72
27-fold quotients : {4,8}*64b
36-fold quotients : {12,2}*48, {6,4}*48a
54-fold quotients : {4,4}*32
72-fold quotients : {6,2}*24
108-fold quotients : {2,4}*16, {4,2}*16
144-fold quotients : {3,2}*12
216-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 2, 8)( 3, 6)( 4, 7)( 10, 19)( 11, 26)( 12, 24)( 13, 25)( 14, 23)
( 15, 21)( 16, 22)( 17, 20)( 18, 27)( 29, 35)( 30, 33)( 31, 34)( 37, 46)
( 38, 53)( 39, 51)( 40, 52)( 41, 50)( 42, 48)( 43, 49)( 44, 47)( 45, 54)
( 55, 82)( 56, 89)( 57, 87)( 58, 88)( 59, 86)( 60, 84)( 61, 85)( 62, 83)
( 63, 90)( 64,100)( 65,107)( 66,105)( 67,106)( 68,104)( 69,102)( 70,103)
( 71,101)( 72,108)( 73, 91)( 74, 98)( 75, 96)( 76, 97)( 77, 95)( 78, 93)
( 79, 94)( 80, 92)( 81, 99)(110,116)(111,114)(112,115)(118,127)(119,134)
(120,132)(121,133)(122,131)(123,129)(124,130)(125,128)(126,135)(137,143)
(138,141)(139,142)(145,154)(146,161)(147,159)(148,160)(149,158)(150,156)
(151,157)(152,155)(153,162)(163,190)(164,197)(165,195)(166,196)(167,194)
(168,192)(169,193)(170,191)(171,198)(172,208)(173,215)(174,213)(175,214)
(176,212)(177,210)(178,211)(179,209)(180,216)(181,199)(182,206)(183,204)
(184,205)(185,203)(186,201)(187,202)(188,200)(189,207);;
s1 := ( 1, 10)( 2, 11)( 3, 12)( 4, 18)( 5, 16)( 6, 17)( 7, 14)( 8, 15)
( 9, 13)( 22, 27)( 23, 25)( 24, 26)( 28, 37)( 29, 38)( 30, 39)( 31, 45)
( 32, 43)( 33, 44)( 34, 41)( 35, 42)( 36, 40)( 49, 54)( 50, 52)( 51, 53)
( 55, 91)( 56, 92)( 57, 93)( 58, 99)( 59, 97)( 60, 98)( 61, 95)( 62, 96)
( 63, 94)( 64, 82)( 65, 83)( 66, 84)( 67, 90)( 68, 88)( 69, 89)( 70, 86)
( 71, 87)( 72, 85)( 73,100)( 74,101)( 75,102)( 76,108)( 77,106)( 78,107)
( 79,104)( 80,105)( 81,103)(109,172)(110,173)(111,174)(112,180)(113,178)
(114,179)(115,176)(116,177)(117,175)(118,163)(119,164)(120,165)(121,171)
(122,169)(123,170)(124,167)(125,168)(126,166)(127,181)(128,182)(129,183)
(130,189)(131,187)(132,188)(133,185)(134,186)(135,184)(136,199)(137,200)
(138,201)(139,207)(140,205)(141,206)(142,203)(143,204)(144,202)(145,190)
(146,191)(147,192)(148,198)(149,196)(150,197)(151,194)(152,195)(153,193)
(154,208)(155,209)(156,210)(157,216)(158,214)(159,215)(160,212)(161,213)
(162,211);;
s2 := ( 1,140)( 2,139)( 3,141)( 4,137)( 5,136)( 6,138)( 7,143)( 8,142)
( 9,144)( 10,149)( 11,148)( 12,150)( 13,146)( 14,145)( 15,147)( 16,152)
( 17,151)( 18,153)( 19,158)( 20,157)( 21,159)( 22,155)( 23,154)( 24,156)
( 25,161)( 26,160)( 27,162)( 28,113)( 29,112)( 30,114)( 31,110)( 32,109)
( 33,111)( 34,116)( 35,115)( 36,117)( 37,122)( 38,121)( 39,123)( 40,119)
( 41,118)( 42,120)( 43,125)( 44,124)( 45,126)( 46,131)( 47,130)( 48,132)
( 49,128)( 50,127)( 51,129)( 52,134)( 53,133)( 54,135)( 55,167)( 56,166)
( 57,168)( 58,164)( 59,163)( 60,165)( 61,170)( 62,169)( 63,171)( 64,176)
( 65,175)( 66,177)( 67,173)( 68,172)( 69,174)( 70,179)( 71,178)( 72,180)
( 73,185)( 74,184)( 75,186)( 76,182)( 77,181)( 78,183)( 79,188)( 80,187)
( 81,189)( 82,194)( 83,193)( 84,195)( 85,191)( 86,190)( 87,192)( 88,197)
( 89,196)( 90,198)( 91,203)( 92,202)( 93,204)( 94,200)( 95,199)( 96,201)
( 97,206)( 98,205)( 99,207)(100,212)(101,211)(102,213)(103,209)(104,208)
(105,210)(106,215)(107,214)(108,216);;
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*s0*s1*s2*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1,
s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*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,
s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(216)!( 2, 8)( 3, 6)( 4, 7)( 10, 19)( 11, 26)( 12, 24)( 13, 25)
( 14, 23)( 15, 21)( 16, 22)( 17, 20)( 18, 27)( 29, 35)( 30, 33)( 31, 34)
( 37, 46)( 38, 53)( 39, 51)( 40, 52)( 41, 50)( 42, 48)( 43, 49)( 44, 47)
( 45, 54)( 55, 82)( 56, 89)( 57, 87)( 58, 88)( 59, 86)( 60, 84)( 61, 85)
( 62, 83)( 63, 90)( 64,100)( 65,107)( 66,105)( 67,106)( 68,104)( 69,102)
( 70,103)( 71,101)( 72,108)( 73, 91)( 74, 98)( 75, 96)( 76, 97)( 77, 95)
( 78, 93)( 79, 94)( 80, 92)( 81, 99)(110,116)(111,114)(112,115)(118,127)
(119,134)(120,132)(121,133)(122,131)(123,129)(124,130)(125,128)(126,135)
(137,143)(138,141)(139,142)(145,154)(146,161)(147,159)(148,160)(149,158)
(150,156)(151,157)(152,155)(153,162)(163,190)(164,197)(165,195)(166,196)
(167,194)(168,192)(169,193)(170,191)(171,198)(172,208)(173,215)(174,213)
(175,214)(176,212)(177,210)(178,211)(179,209)(180,216)(181,199)(182,206)
(183,204)(184,205)(185,203)(186,201)(187,202)(188,200)(189,207);
s1 := Sym(216)!( 1, 10)( 2, 11)( 3, 12)( 4, 18)( 5, 16)( 6, 17)( 7, 14)
( 8, 15)( 9, 13)( 22, 27)( 23, 25)( 24, 26)( 28, 37)( 29, 38)( 30, 39)
( 31, 45)( 32, 43)( 33, 44)( 34, 41)( 35, 42)( 36, 40)( 49, 54)( 50, 52)
( 51, 53)( 55, 91)( 56, 92)( 57, 93)( 58, 99)( 59, 97)( 60, 98)( 61, 95)
( 62, 96)( 63, 94)( 64, 82)( 65, 83)( 66, 84)( 67, 90)( 68, 88)( 69, 89)
( 70, 86)( 71, 87)( 72, 85)( 73,100)( 74,101)( 75,102)( 76,108)( 77,106)
( 78,107)( 79,104)( 80,105)( 81,103)(109,172)(110,173)(111,174)(112,180)
(113,178)(114,179)(115,176)(116,177)(117,175)(118,163)(119,164)(120,165)
(121,171)(122,169)(123,170)(124,167)(125,168)(126,166)(127,181)(128,182)
(129,183)(130,189)(131,187)(132,188)(133,185)(134,186)(135,184)(136,199)
(137,200)(138,201)(139,207)(140,205)(141,206)(142,203)(143,204)(144,202)
(145,190)(146,191)(147,192)(148,198)(149,196)(150,197)(151,194)(152,195)
(153,193)(154,208)(155,209)(156,210)(157,216)(158,214)(159,215)(160,212)
(161,213)(162,211);
s2 := Sym(216)!( 1,140)( 2,139)( 3,141)( 4,137)( 5,136)( 6,138)( 7,143)
( 8,142)( 9,144)( 10,149)( 11,148)( 12,150)( 13,146)( 14,145)( 15,147)
( 16,152)( 17,151)( 18,153)( 19,158)( 20,157)( 21,159)( 22,155)( 23,154)
( 24,156)( 25,161)( 26,160)( 27,162)( 28,113)( 29,112)( 30,114)( 31,110)
( 32,109)( 33,111)( 34,116)( 35,115)( 36,117)( 37,122)( 38,121)( 39,123)
( 40,119)( 41,118)( 42,120)( 43,125)( 44,124)( 45,126)( 46,131)( 47,130)
( 48,132)( 49,128)( 50,127)( 51,129)( 52,134)( 53,133)( 54,135)( 55,167)
( 56,166)( 57,168)( 58,164)( 59,163)( 60,165)( 61,170)( 62,169)( 63,171)
( 64,176)( 65,175)( 66,177)( 67,173)( 68,172)( 69,174)( 70,179)( 71,178)
( 72,180)( 73,185)( 74,184)( 75,186)( 76,182)( 77,181)( 78,183)( 79,188)
( 80,187)( 81,189)( 82,194)( 83,193)( 84,195)( 85,191)( 86,190)( 87,192)
( 88,197)( 89,196)( 90,198)( 91,203)( 92,202)( 93,204)( 94,200)( 95,199)
( 96,201)( 97,206)( 98,205)( 99,207)(100,212)(101,211)(102,213)(103,209)
(104,208)(105,210)(106,215)(107,214)(108,216);
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*s0*s1*s2*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1,
s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*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,
s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0 >;
References : None.
to this polytope