Polytope of Type {3,12,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,12,6}*864a
if this polytope has a name.
Group : SmallGroup(864,4000)
Rank : 4
Schlafli Type : {3,12,6}
Number of vertices, edges, etc : 6, 36, 72, 6
Order of s0s1s2s3 : 6
Order of s0s1s2s3s2s1 : 6
Special Properties :
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{3,12,6,2} of size 1728
Vertex Figure Of :
{2,3,12,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {3,4,6}*288
4-fold quotients : {3,6,6}*216a
8-fold quotients : {3,6,3}*108
9-fold quotients : {3,4,2}*96
12-fold quotients : {3,2,6}*72
18-fold quotients : {3,4,2}*48
24-fold quotients : {3,2,3}*36
36-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {3,12,12}*1728a, {3,24,6}*1728a, {6,12,6}*1728b
Irregular Quotients (of which this is a minimal cover):
P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2> of order 2.
6 facets:
6 of 2-fold non-regular quotient of {3,12}*144
4 vertex figures:
2 of {6,6}*72b
2 of {12,6}*144c
Permutation Representation (GAP) :
s0 := ( 2, 3)( 5, 9)( 6, 11)( 7, 10)( 8, 12)( 14, 15)( 17, 21)( 18, 23)( 19, 22)( 20, 24)( 26, 27)( 29, 33)( 30, 35)( 31, 34)( 32, 36)( 37, 73)( 38, 75)( 39, 74)( 40, 76)( 41, 81)( 42, 83)( 43, 82)( 44, 84)( 45, 77)( 46, 79)( 47, 78)( 48, 80)( 49, 85)( 50, 87)( 51, 86)( 52, 88)( 53, 93)( 54, 95)( 55, 94)( 56, 96)( 57, 89)( 58, 91)( 59, 90)( 60, 92)( 61, 97)( 62, 99)( 63, 98)( 64,100)( 65,105)( 66,107)( 67,106)( 68,108)( 69,101)( 70,103)( 71,102)( 72,104)(110,111)(113,117)(114,119)(115,118)(116,120)(122,123)(125,129)(126,131)(127,130)(128,132)(134,135)(137,141)(138,143)(139,142)(140,144)(145,181)(146,183)(147,182)(148,184)(149,189)(150,191)(151,190)(152,192)(153,185)(154,187)(155,186)(156,188)(157,193)(158,195)(159,194)(160,196)(161,201)(162,203)(163,202)(164,204)(165,197)(166,199)(167,198)(168,200)(169,205)(170,207)(171,206)(172,208)(173,213)(174,215)(175,214)(176,216)(177,209)(178,211)(179,210)(180,212);;
s1 := ( 1, 37)( 2, 38)( 3, 40)( 4, 39)( 5, 45)( 6, 46)( 7, 48)( 8, 47)( 9, 41)( 10, 42)( 11, 44)( 12, 43)( 13, 53)( 14, 54)( 15, 56)( 16, 55)( 17, 49)( 18, 50)( 19, 52)( 20, 51)( 21, 57)( 22, 58)( 23, 60)( 24, 59)( 25, 69)( 26, 70)( 27, 72)( 28, 71)( 29, 65)( 30, 66)( 31, 68)( 32, 67)( 33, 61)( 34, 62)( 35, 64)( 36, 63)( 75, 76)( 77, 81)( 78, 82)( 79, 84)( 80, 83)( 85, 89)( 86, 90)( 87, 92)( 88, 91)( 95, 96)( 97,105)( 98,106)( 99,108)(100,107)(103,104)(109,145)(110,146)(111,148)(112,147)(113,153)(114,154)(115,156)(116,155)(117,149)(118,150)(119,152)(120,151)(121,161)(122,162)(123,164)(124,163)(125,157)(126,158)(127,160)(128,159)(129,165)(130,166)(131,168)(132,167)(133,177)(134,178)(135,180)(136,179)(137,173)(138,174)(139,176)(140,175)(141,169)(142,170)(143,172)(144,171)(183,184)(185,189)(186,190)(187,192)(188,191)(193,197)(194,198)(195,200)(196,199)(203,204)(205,213)(206,214)(207,216)(208,215)(211,212);;
s2 := ( 1, 16)( 2, 15)( 3, 14)( 4, 13)( 5, 24)( 6, 23)( 7, 22)( 8, 21)( 9, 20)( 10, 19)( 11, 18)( 12, 17)( 25, 28)( 26, 27)( 29, 36)( 30, 35)( 31, 34)( 32, 33)( 37, 52)( 38, 51)( 39, 50)( 40, 49)( 41, 60)( 42, 59)( 43, 58)( 44, 57)( 45, 56)( 46, 55)( 47, 54)( 48, 53)( 61, 64)( 62, 63)( 65, 72)( 66, 71)( 67, 70)( 68, 69)( 73, 88)( 74, 87)( 75, 86)( 76, 85)( 77, 96)( 78, 95)( 79, 94)( 80, 93)( 81, 92)( 82, 91)( 83, 90)( 84, 89)( 97,100)( 98, 99)(101,108)(102,107)(103,106)(104,105)(109,124)(110,123)(111,122)(112,121)(113,132)(114,131)(115,130)(116,129)(117,128)(118,127)(119,126)(120,125)(133,136)(134,135)(137,144)(138,143)(139,142)(140,141)(145,160)(146,159)(147,158)(148,157)(149,168)(150,167)(151,166)(152,165)(153,164)(154,163)(155,162)(156,161)(169,172)(170,171)(173,180)(174,179)(175,178)(176,177)(181,196)(182,195)(183,194)(184,193)(185,204)(186,203)(187,202)(188,201)(189,200)(190,199)(191,198)(192,197)(205,208)(206,207)(209,216)(210,215)(211,214)(212,213);;
s3 := ( 1,109)( 2,110)( 3,111)( 4,112)( 5,117)( 6,118)( 7,119)( 8,120)( 9,113)( 10,114)( 11,115)( 12,116)( 13,133)( 14,134)( 15,135)( 16,136)( 17,141)( 18,142)( 19,143)( 20,144)( 21,137)( 22,138)( 23,139)( 24,140)( 25,121)( 26,122)( 27,123)( 28,124)( 29,129)( 30,130)( 31,131)( 32,132)( 33,125)( 34,126)( 35,127)( 36,128)( 37,145)( 38,146)( 39,147)( 40,148)( 41,153)( 42,154)( 43,155)( 44,156)( 45,149)( 46,150)( 47,151)( 48,152)( 49,169)( 50,170)( 51,171)( 52,172)( 53,177)( 54,178)( 55,179)( 56,180)( 57,173)( 58,174)( 59,175)( 60,176)( 61,157)( 62,158)( 63,159)( 64,160)( 65,165)( 66,166)( 67,167)( 68,168)( 69,161)( 70,162)( 71,163)( 72,164)( 73,181)( 74,182)( 75,183)( 76,184)( 77,189)( 78,190)( 79,191)( 80,192)( 81,185)( 82,186)( 83,187)( 84,188)( 85,205)( 86,206)( 87,207)( 88,208)( 89,213)( 90,214)( 91,215)( 92,216)( 93,209)( 94,210)( 95,211)( 96,212)( 97,193)( 98,194)( 99,195)(100,196)(101,201)(102,202)(103,203)(104,204)(105,197)(106,198)(107,199)(108,200);;
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, s0*s1*s0*s1*s0*s1,
s1*s2*s3*s1*s2*s1*s2*s3*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s3*s2*s1*s3*s0*s2*s3*s2*s3*s1*s2*s3*s2*s3*s1*s0 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(216)!( 2, 3)( 5, 9)( 6, 11)( 7, 10)( 8, 12)( 14, 15)( 17, 21)( 18, 23)( 19, 22)( 20, 24)( 26, 27)( 29, 33)( 30, 35)( 31, 34)( 32, 36)( 37, 73)( 38, 75)( 39, 74)( 40, 76)( 41, 81)( 42, 83)( 43, 82)( 44, 84)( 45, 77)( 46, 79)( 47, 78)( 48, 80)( 49, 85)( 50, 87)( 51, 86)( 52, 88)( 53, 93)( 54, 95)( 55, 94)( 56, 96)( 57, 89)( 58, 91)( 59, 90)( 60, 92)( 61, 97)( 62, 99)( 63, 98)( 64,100)( 65,105)( 66,107)( 67,106)( 68,108)( 69,101)( 70,103)( 71,102)( 72,104)(110,111)(113,117)(114,119)(115,118)(116,120)(122,123)(125,129)(126,131)(127,130)(128,132)(134,135)(137,141)(138,143)(139,142)(140,144)(145,181)(146,183)(147,182)(148,184)(149,189)(150,191)(151,190)(152,192)(153,185)(154,187)(155,186)(156,188)(157,193)(158,195)(159,194)(160,196)(161,201)(162,203)(163,202)(164,204)(165,197)(166,199)(167,198)(168,200)(169,205)(170,207)(171,206)(172,208)(173,213)(174,215)(175,214)(176,216)(177,209)(178,211)(179,210)(180,212);
s1 := Sym(216)!( 1, 37)( 2, 38)( 3, 40)( 4, 39)( 5, 45)( 6, 46)( 7, 48)( 8, 47)( 9, 41)( 10, 42)( 11, 44)( 12, 43)( 13, 53)( 14, 54)( 15, 56)( 16, 55)( 17, 49)( 18, 50)( 19, 52)( 20, 51)( 21, 57)( 22, 58)( 23, 60)( 24, 59)( 25, 69)( 26, 70)( 27, 72)( 28, 71)( 29, 65)( 30, 66)( 31, 68)( 32, 67)( 33, 61)( 34, 62)( 35, 64)( 36, 63)( 75, 76)( 77, 81)( 78, 82)( 79, 84)( 80, 83)( 85, 89)( 86, 90)( 87, 92)( 88, 91)( 95, 96)( 97,105)( 98,106)( 99,108)(100,107)(103,104)(109,145)(110,146)(111,148)(112,147)(113,153)(114,154)(115,156)(116,155)(117,149)(118,150)(119,152)(120,151)(121,161)(122,162)(123,164)(124,163)(125,157)(126,158)(127,160)(128,159)(129,165)(130,166)(131,168)(132,167)(133,177)(134,178)(135,180)(136,179)(137,173)(138,174)(139,176)(140,175)(141,169)(142,170)(143,172)(144,171)(183,184)(185,189)(186,190)(187,192)(188,191)(193,197)(194,198)(195,200)(196,199)(203,204)(205,213)(206,214)(207,216)(208,215)(211,212);
s2 := Sym(216)!( 1, 16)( 2, 15)( 3, 14)( 4, 13)( 5, 24)( 6, 23)( 7, 22)( 8, 21)( 9, 20)( 10, 19)( 11, 18)( 12, 17)( 25, 28)( 26, 27)( 29, 36)( 30, 35)( 31, 34)( 32, 33)( 37, 52)( 38, 51)( 39, 50)( 40, 49)( 41, 60)( 42, 59)( 43, 58)( 44, 57)( 45, 56)( 46, 55)( 47, 54)( 48, 53)( 61, 64)( 62, 63)( 65, 72)( 66, 71)( 67, 70)( 68, 69)( 73, 88)( 74, 87)( 75, 86)( 76, 85)( 77, 96)( 78, 95)( 79, 94)( 80, 93)( 81, 92)( 82, 91)( 83, 90)( 84, 89)( 97,100)( 98, 99)(101,108)(102,107)(103,106)(104,105)(109,124)(110,123)(111,122)(112,121)(113,132)(114,131)(115,130)(116,129)(117,128)(118,127)(119,126)(120,125)(133,136)(134,135)(137,144)(138,143)(139,142)(140,141)(145,160)(146,159)(147,158)(148,157)(149,168)(150,167)(151,166)(152,165)(153,164)(154,163)(155,162)(156,161)(169,172)(170,171)(173,180)(174,179)(175,178)(176,177)(181,196)(182,195)(183,194)(184,193)(185,204)(186,203)(187,202)(188,201)(189,200)(190,199)(191,198)(192,197)(205,208)(206,207)(209,216)(210,215)(211,214)(212,213);
s3 := Sym(216)!( 1,109)( 2,110)( 3,111)( 4,112)( 5,117)( 6,118)( 7,119)( 8,120)( 9,113)( 10,114)( 11,115)( 12,116)( 13,133)( 14,134)( 15,135)( 16,136)( 17,141)( 18,142)( 19,143)( 20,144)( 21,137)( 22,138)( 23,139)( 24,140)( 25,121)( 26,122)( 27,123)( 28,124)( 29,129)( 30,130)( 31,131)( 32,132)( 33,125)( 34,126)( 35,127)( 36,128)( 37,145)( 38,146)( 39,147)( 40,148)( 41,153)( 42,154)( 43,155)( 44,156)( 45,149)( 46,150)( 47,151)( 48,152)( 49,169)( 50,170)( 51,171)( 52,172)( 53,177)( 54,178)( 55,179)( 56,180)( 57,173)( 58,174)( 59,175)( 60,176)( 61,157)( 62,158)( 63,159)( 64,160)( 65,165)( 66,166)( 67,167)( 68,168)( 69,161)( 70,162)( 71,163)( 72,164)( 73,181)( 74,182)( 75,183)( 76,184)( 77,189)( 78,190)( 79,191)( 80,192)( 81,185)( 82,186)( 83,187)( 84,188)( 85,205)( 86,206)( 87,207)( 88,208)( 89,213)( 90,214)( 91,215)( 92,216)( 93,209)( 94,210)( 95,211)( 96,212)( 97,193)( 98,194)( 99,195)(100,196)(101,201)(102,202)(103,203)(104,204)(105,197)(106,198)(107,199)(108,200);
poly := sub<Sym(216)|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,
s0*s1*s0*s1*s0*s1, s1*s2*s3*s1*s2*s1*s2*s3*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s3*s2*s1*s3*s0*s2*s3*s2*s3*s1*s2*s3*s2*s3*s1*s0 >;
References : None.
to this polytope