Polytope of Type {2,3,12,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,3,12,6}*1728a
if this polytope has a name.
Group : SmallGroup(1728,46116)
Rank : 5
Schlafli Type : {2,3,12,6}
Number of vertices, edges, etc : 2, 6, 36, 72, 6
Order of s0s1s2s3s4 : 6
Order of s0s1s2s3s4s3s2s1 : 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) :
3-fold quotients : {2,3,4,6}*576
4-fold quotients : {2,3,6,6}*432a
8-fold quotients : {2,3,6,3}*216
9-fold quotients : {2,3,4,2}*192
12-fold quotients : {2,3,2,6}*144
18-fold quotients : {2,3,4,2}*96
24-fold quotients : {2,3,2,3}*72
36-fold quotients : {2,3,2,2}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 4, 5)( 7, 11)( 8, 13)( 9, 12)( 10, 14)( 16, 17)( 19, 23)( 20, 25)
( 21, 24)( 22, 26)( 28, 29)( 31, 35)( 32, 37)( 33, 36)( 34, 38)( 39, 75)
( 40, 77)( 41, 76)( 42, 78)( 43, 83)( 44, 85)( 45, 84)( 46, 86)( 47, 79)
( 48, 81)( 49, 80)( 50, 82)( 51, 87)( 52, 89)( 53, 88)( 54, 90)( 55, 95)
( 56, 97)( 57, 96)( 58, 98)( 59, 91)( 60, 93)( 61, 92)( 62, 94)( 63, 99)
( 64,101)( 65,100)( 66,102)( 67,107)( 68,109)( 69,108)( 70,110)( 71,103)
( 72,105)( 73,104)( 74,106)(112,113)(115,119)(116,121)(117,120)(118,122)
(124,125)(127,131)(128,133)(129,132)(130,134)(136,137)(139,143)(140,145)
(141,144)(142,146)(147,183)(148,185)(149,184)(150,186)(151,191)(152,193)
(153,192)(154,194)(155,187)(156,189)(157,188)(158,190)(159,195)(160,197)
(161,196)(162,198)(163,203)(164,205)(165,204)(166,206)(167,199)(168,201)
(169,200)(170,202)(171,207)(172,209)(173,208)(174,210)(175,215)(176,217)
(177,216)(178,218)(179,211)(180,213)(181,212)(182,214);;
s2 := ( 3, 39)( 4, 40)( 5, 42)( 6, 41)( 7, 47)( 8, 48)( 9, 50)( 10, 49)
( 11, 43)( 12, 44)( 13, 46)( 14, 45)( 15, 55)( 16, 56)( 17, 58)( 18, 57)
( 19, 51)( 20, 52)( 21, 54)( 22, 53)( 23, 59)( 24, 60)( 25, 62)( 26, 61)
( 27, 71)( 28, 72)( 29, 74)( 30, 73)( 31, 67)( 32, 68)( 33, 70)( 34, 69)
( 35, 63)( 36, 64)( 37, 66)( 38, 65)( 77, 78)( 79, 83)( 80, 84)( 81, 86)
( 82, 85)( 87, 91)( 88, 92)( 89, 94)( 90, 93)( 97, 98)( 99,107)(100,108)
(101,110)(102,109)(105,106)(111,147)(112,148)(113,150)(114,149)(115,155)
(116,156)(117,158)(118,157)(119,151)(120,152)(121,154)(122,153)(123,163)
(124,164)(125,166)(126,165)(127,159)(128,160)(129,162)(130,161)(131,167)
(132,168)(133,170)(134,169)(135,179)(136,180)(137,182)(138,181)(139,175)
(140,176)(141,178)(142,177)(143,171)(144,172)(145,174)(146,173)(185,186)
(187,191)(188,192)(189,194)(190,193)(195,199)(196,200)(197,202)(198,201)
(205,206)(207,215)(208,216)(209,218)(210,217)(213,214);;
s3 := ( 3, 18)( 4, 17)( 5, 16)( 6, 15)( 7, 26)( 8, 25)( 9, 24)( 10, 23)
( 11, 22)( 12, 21)( 13, 20)( 14, 19)( 27, 30)( 28, 29)( 31, 38)( 32, 37)
( 33, 36)( 34, 35)( 39, 54)( 40, 53)( 41, 52)( 42, 51)( 43, 62)( 44, 61)
( 45, 60)( 46, 59)( 47, 58)( 48, 57)( 49, 56)( 50, 55)( 63, 66)( 64, 65)
( 67, 74)( 68, 73)( 69, 72)( 70, 71)( 75, 90)( 76, 89)( 77, 88)( 78, 87)
( 79, 98)( 80, 97)( 81, 96)( 82, 95)( 83, 94)( 84, 93)( 85, 92)( 86, 91)
( 99,102)(100,101)(103,110)(104,109)(105,108)(106,107)(111,126)(112,125)
(113,124)(114,123)(115,134)(116,133)(117,132)(118,131)(119,130)(120,129)
(121,128)(122,127)(135,138)(136,137)(139,146)(140,145)(141,144)(142,143)
(147,162)(148,161)(149,160)(150,159)(151,170)(152,169)(153,168)(154,167)
(155,166)(156,165)(157,164)(158,163)(171,174)(172,173)(175,182)(176,181)
(177,180)(178,179)(183,198)(184,197)(185,196)(186,195)(187,206)(188,205)
(189,204)(190,203)(191,202)(192,201)(193,200)(194,199)(207,210)(208,209)
(211,218)(212,217)(213,216)(214,215);;
s4 := ( 3,111)( 4,112)( 5,113)( 6,114)( 7,119)( 8,120)( 9,121)( 10,122)
( 11,115)( 12,116)( 13,117)( 14,118)( 15,135)( 16,136)( 17,137)( 18,138)
( 19,143)( 20,144)( 21,145)( 22,146)( 23,139)( 24,140)( 25,141)( 26,142)
( 27,123)( 28,124)( 29,125)( 30,126)( 31,131)( 32,132)( 33,133)( 34,134)
( 35,127)( 36,128)( 37,129)( 38,130)( 39,147)( 40,148)( 41,149)( 42,150)
( 43,155)( 44,156)( 45,157)( 46,158)( 47,151)( 48,152)( 49,153)( 50,154)
( 51,171)( 52,172)( 53,173)( 54,174)( 55,179)( 56,180)( 57,181)( 58,182)
( 59,175)( 60,176)( 61,177)( 62,178)( 63,159)( 64,160)( 65,161)( 66,162)
( 67,167)( 68,168)( 69,169)( 70,170)( 71,163)( 72,164)( 73,165)( 74,166)
( 75,183)( 76,184)( 77,185)( 78,186)( 79,191)( 80,192)( 81,193)( 82,194)
( 83,187)( 84,188)( 85,189)( 86,190)( 87,207)( 88,208)( 89,209)( 90,210)
( 91,215)( 92,216)( 93,217)( 94,218)( 95,211)( 96,212)( 97,213)( 98,214)
( 99,195)(100,196)(101,197)(102,198)(103,203)(104,204)(105,205)(106,206)
(107,199)(108,200)(109,201)(110,202);;
poly := Group([s0,s1,s2,s3,s4]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;; s4 := F.5;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s1*s0*s1,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4,
s1*s2*s1*s2*s1*s2, s4*s2*s3*s2*s3*s4*s2*s3*s2*s3,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4,
s4*s1*s3*s4*s3*s2*s4*s3*s4*s3*s1*s2*s4*s3*s4*s3*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(218)!(1,2);
s1 := Sym(218)!( 4, 5)( 7, 11)( 8, 13)( 9, 12)( 10, 14)( 16, 17)( 19, 23)
( 20, 25)( 21, 24)( 22, 26)( 28, 29)( 31, 35)( 32, 37)( 33, 36)( 34, 38)
( 39, 75)( 40, 77)( 41, 76)( 42, 78)( 43, 83)( 44, 85)( 45, 84)( 46, 86)
( 47, 79)( 48, 81)( 49, 80)( 50, 82)( 51, 87)( 52, 89)( 53, 88)( 54, 90)
( 55, 95)( 56, 97)( 57, 96)( 58, 98)( 59, 91)( 60, 93)( 61, 92)( 62, 94)
( 63, 99)( 64,101)( 65,100)( 66,102)( 67,107)( 68,109)( 69,108)( 70,110)
( 71,103)( 72,105)( 73,104)( 74,106)(112,113)(115,119)(116,121)(117,120)
(118,122)(124,125)(127,131)(128,133)(129,132)(130,134)(136,137)(139,143)
(140,145)(141,144)(142,146)(147,183)(148,185)(149,184)(150,186)(151,191)
(152,193)(153,192)(154,194)(155,187)(156,189)(157,188)(158,190)(159,195)
(160,197)(161,196)(162,198)(163,203)(164,205)(165,204)(166,206)(167,199)
(168,201)(169,200)(170,202)(171,207)(172,209)(173,208)(174,210)(175,215)
(176,217)(177,216)(178,218)(179,211)(180,213)(181,212)(182,214);
s2 := Sym(218)!( 3, 39)( 4, 40)( 5, 42)( 6, 41)( 7, 47)( 8, 48)( 9, 50)
( 10, 49)( 11, 43)( 12, 44)( 13, 46)( 14, 45)( 15, 55)( 16, 56)( 17, 58)
( 18, 57)( 19, 51)( 20, 52)( 21, 54)( 22, 53)( 23, 59)( 24, 60)( 25, 62)
( 26, 61)( 27, 71)( 28, 72)( 29, 74)( 30, 73)( 31, 67)( 32, 68)( 33, 70)
( 34, 69)( 35, 63)( 36, 64)( 37, 66)( 38, 65)( 77, 78)( 79, 83)( 80, 84)
( 81, 86)( 82, 85)( 87, 91)( 88, 92)( 89, 94)( 90, 93)( 97, 98)( 99,107)
(100,108)(101,110)(102,109)(105,106)(111,147)(112,148)(113,150)(114,149)
(115,155)(116,156)(117,158)(118,157)(119,151)(120,152)(121,154)(122,153)
(123,163)(124,164)(125,166)(126,165)(127,159)(128,160)(129,162)(130,161)
(131,167)(132,168)(133,170)(134,169)(135,179)(136,180)(137,182)(138,181)
(139,175)(140,176)(141,178)(142,177)(143,171)(144,172)(145,174)(146,173)
(185,186)(187,191)(188,192)(189,194)(190,193)(195,199)(196,200)(197,202)
(198,201)(205,206)(207,215)(208,216)(209,218)(210,217)(213,214);
s3 := Sym(218)!( 3, 18)( 4, 17)( 5, 16)( 6, 15)( 7, 26)( 8, 25)( 9, 24)
( 10, 23)( 11, 22)( 12, 21)( 13, 20)( 14, 19)( 27, 30)( 28, 29)( 31, 38)
( 32, 37)( 33, 36)( 34, 35)( 39, 54)( 40, 53)( 41, 52)( 42, 51)( 43, 62)
( 44, 61)( 45, 60)( 46, 59)( 47, 58)( 48, 57)( 49, 56)( 50, 55)( 63, 66)
( 64, 65)( 67, 74)( 68, 73)( 69, 72)( 70, 71)( 75, 90)( 76, 89)( 77, 88)
( 78, 87)( 79, 98)( 80, 97)( 81, 96)( 82, 95)( 83, 94)( 84, 93)( 85, 92)
( 86, 91)( 99,102)(100,101)(103,110)(104,109)(105,108)(106,107)(111,126)
(112,125)(113,124)(114,123)(115,134)(116,133)(117,132)(118,131)(119,130)
(120,129)(121,128)(122,127)(135,138)(136,137)(139,146)(140,145)(141,144)
(142,143)(147,162)(148,161)(149,160)(150,159)(151,170)(152,169)(153,168)
(154,167)(155,166)(156,165)(157,164)(158,163)(171,174)(172,173)(175,182)
(176,181)(177,180)(178,179)(183,198)(184,197)(185,196)(186,195)(187,206)
(188,205)(189,204)(190,203)(191,202)(192,201)(193,200)(194,199)(207,210)
(208,209)(211,218)(212,217)(213,216)(214,215);
s4 := Sym(218)!( 3,111)( 4,112)( 5,113)( 6,114)( 7,119)( 8,120)( 9,121)
( 10,122)( 11,115)( 12,116)( 13,117)( 14,118)( 15,135)( 16,136)( 17,137)
( 18,138)( 19,143)( 20,144)( 21,145)( 22,146)( 23,139)( 24,140)( 25,141)
( 26,142)( 27,123)( 28,124)( 29,125)( 30,126)( 31,131)( 32,132)( 33,133)
( 34,134)( 35,127)( 36,128)( 37,129)( 38,130)( 39,147)( 40,148)( 41,149)
( 42,150)( 43,155)( 44,156)( 45,157)( 46,158)( 47,151)( 48,152)( 49,153)
( 50,154)( 51,171)( 52,172)( 53,173)( 54,174)( 55,179)( 56,180)( 57,181)
( 58,182)( 59,175)( 60,176)( 61,177)( 62,178)( 63,159)( 64,160)( 65,161)
( 66,162)( 67,167)( 68,168)( 69,169)( 70,170)( 71,163)( 72,164)( 73,165)
( 74,166)( 75,183)( 76,184)( 77,185)( 78,186)( 79,191)( 80,192)( 81,193)
( 82,194)( 83,187)( 84,188)( 85,189)( 86,190)( 87,207)( 88,208)( 89,209)
( 90,210)( 91,215)( 92,216)( 93,217)( 94,218)( 95,211)( 96,212)( 97,213)
( 98,214)( 99,195)(100,196)(101,197)(102,198)(103,203)(104,204)(105,205)
(106,206)(107,199)(108,200)(109,201)(110,202);
poly := sub<Sym(218)|s0,s1,s2,s3,s4>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2,
s3*s3, s4*s4, s0*s1*s0*s1, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4,
s1*s4*s1*s4, s2*s4*s2*s4, s1*s2*s1*s2*s1*s2,
s4*s2*s3*s2*s3*s4*s2*s3*s2*s3, s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4,
s4*s1*s3*s4*s3*s2*s4*s3*s4*s3*s1*s2*s4*s3*s4*s3*s1*s2 >;
to this polytope