Polytope of Type {2,12,6,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,12,6,6}*1728a
if this polytope has a name.
Group : SmallGroup(1728,30782)
Rank : 5
Schlafli Type : {2,12,6,6}
Number of vertices, edges, etc : 2, 12, 36, 18, 6
Order of s0s1s2s3s4 : 12
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) :
2-fold quotients : {2,12,6,3}*864a, {2,6,6,6}*864a
3-fold quotients : {2,12,2,6}*576
4-fold quotients : {2,3,6,6}*432a, {2,6,6,3}*432a
6-fold quotients : {2,12,2,3}*288, {2,6,2,6}*288
8-fold quotients : {2,3,6,3}*216
9-fold quotients : {2,12,2,2}*192, {2,4,2,6}*192
12-fold quotients : {2,3,2,6}*144, {2,6,2,3}*144
18-fold quotients : {2,4,2,3}*96, {2,2,2,6}*96, {2,6,2,2}*96
24-fold quotients : {2,3,2,3}*72
27-fold quotients : {2,4,2,2}*64
36-fold quotients : {2,2,2,3}*48, {2,3,2,2}*48
54-fold quotients : {2,2,2,2}*32
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 4, 5)( 7, 8)( 10, 11)( 12, 21)( 13, 23)( 14, 22)( 15, 24)( 16, 26)
( 17, 25)( 18, 27)( 19, 29)( 20, 28)( 31, 32)( 34, 35)( 37, 38)( 39, 48)
( 40, 50)( 41, 49)( 42, 51)( 43, 53)( 44, 52)( 45, 54)( 46, 56)( 47, 55)
( 57, 84)( 58, 86)( 59, 85)( 60, 87)( 61, 89)( 62, 88)( 63, 90)( 64, 92)
( 65, 91)( 66,102)( 67,104)( 68,103)( 69,105)( 70,107)( 71,106)( 72,108)
( 73,110)( 74,109)( 75, 93)( 76, 95)( 77, 94)( 78, 96)( 79, 98)( 80, 97)
( 81, 99)( 82,101)( 83,100)(112,113)(115,116)(118,119)(120,129)(121,131)
(122,130)(123,132)(124,134)(125,133)(126,135)(127,137)(128,136)(139,140)
(142,143)(145,146)(147,156)(148,158)(149,157)(150,159)(151,161)(152,160)
(153,162)(154,164)(155,163)(165,192)(166,194)(167,193)(168,195)(169,197)
(170,196)(171,198)(172,200)(173,199)(174,210)(175,212)(176,211)(177,213)
(178,215)(179,214)(180,216)(181,218)(182,217)(183,201)(184,203)(185,202)
(186,204)(187,206)(188,205)(189,207)(190,209)(191,208);;
s2 := ( 3, 66)( 4, 68)( 5, 67)( 6, 70)( 7, 69)( 8, 71)( 9, 74)( 10, 73)
( 11, 72)( 12, 57)( 13, 59)( 14, 58)( 15, 61)( 16, 60)( 17, 62)( 18, 65)
( 19, 64)( 20, 63)( 21, 75)( 22, 77)( 23, 76)( 24, 79)( 25, 78)( 26, 80)
( 27, 83)( 28, 82)( 29, 81)( 30, 93)( 31, 95)( 32, 94)( 33, 97)( 34, 96)
( 35, 98)( 36,101)( 37,100)( 38, 99)( 39, 84)( 40, 86)( 41, 85)( 42, 88)
( 43, 87)( 44, 89)( 45, 92)( 46, 91)( 47, 90)( 48,102)( 49,104)( 50,103)
( 51,106)( 52,105)( 53,107)( 54,110)( 55,109)( 56,108)(111,174)(112,176)
(113,175)(114,178)(115,177)(116,179)(117,182)(118,181)(119,180)(120,165)
(121,167)(122,166)(123,169)(124,168)(125,170)(126,173)(127,172)(128,171)
(129,183)(130,185)(131,184)(132,187)(133,186)(134,188)(135,191)(136,190)
(137,189)(138,201)(139,203)(140,202)(141,205)(142,204)(143,206)(144,209)
(145,208)(146,207)(147,192)(148,194)(149,193)(150,196)(151,195)(152,197)
(153,200)(154,199)(155,198)(156,210)(157,212)(158,211)(159,214)(160,213)
(161,215)(162,218)(163,217)(164,216);;
s3 := ( 3, 6)( 4, 8)( 5, 7)( 10, 11)( 12, 15)( 13, 17)( 14, 16)( 19, 20)
( 21, 24)( 22, 26)( 23, 25)( 28, 29)( 30, 33)( 31, 35)( 32, 34)( 37, 38)
( 39, 42)( 40, 44)( 41, 43)( 46, 47)( 48, 51)( 49, 53)( 50, 52)( 55, 56)
( 57, 60)( 58, 62)( 59, 61)( 64, 65)( 66, 69)( 67, 71)( 68, 70)( 73, 74)
( 75, 78)( 76, 80)( 77, 79)( 82, 83)( 84, 87)( 85, 89)( 86, 88)( 91, 92)
( 93, 96)( 94, 98)( 95, 97)(100,101)(102,105)(103,107)(104,106)(109,110)
(111,114)(112,116)(113,115)(118,119)(120,123)(121,125)(122,124)(127,128)
(129,132)(130,134)(131,133)(136,137)(138,141)(139,143)(140,142)(145,146)
(147,150)(148,152)(149,151)(154,155)(156,159)(157,161)(158,160)(163,164)
(165,168)(166,170)(167,169)(172,173)(174,177)(175,179)(176,178)(181,182)
(183,186)(184,188)(185,187)(190,191)(192,195)(193,197)(194,196)(199,200)
(201,204)(202,206)(203,205)(208,209)(210,213)(211,215)(212,214)(217,218);;
s4 := ( 3,111)( 4,113)( 5,112)( 6,117)( 7,119)( 8,118)( 9,114)( 10,116)
( 11,115)( 12,120)( 13,122)( 14,121)( 15,126)( 16,128)( 17,127)( 18,123)
( 19,125)( 20,124)( 21,129)( 22,131)( 23,130)( 24,135)( 25,137)( 26,136)
( 27,132)( 28,134)( 29,133)( 30,138)( 31,140)( 32,139)( 33,144)( 34,146)
( 35,145)( 36,141)( 37,143)( 38,142)( 39,147)( 40,149)( 41,148)( 42,153)
( 43,155)( 44,154)( 45,150)( 46,152)( 47,151)( 48,156)( 49,158)( 50,157)
( 51,162)( 52,164)( 53,163)( 54,159)( 55,161)( 56,160)( 57,165)( 58,167)
( 59,166)( 60,171)( 61,173)( 62,172)( 63,168)( 64,170)( 65,169)( 66,174)
( 67,176)( 68,175)( 69,180)( 70,182)( 71,181)( 72,177)( 73,179)( 74,178)
( 75,183)( 76,185)( 77,184)( 78,189)( 79,191)( 80,190)( 81,186)( 82,188)
( 83,187)( 84,192)( 85,194)( 86,193)( 87,198)( 88,200)( 89,199)( 90,195)
( 91,197)( 92,196)( 93,201)( 94,203)( 95,202)( 96,207)( 97,209)( 98,208)
( 99,204)(100,206)(101,205)(102,210)(103,212)(104,211)(105,216)(106,218)
(107,217)(108,213)(109,215)(110,214);;
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,
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s4*s2*s3*s2*s3*s4*s2*s3*s2*s3,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4,
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)!(1,2);
s1 := Sym(218)!( 4, 5)( 7, 8)( 10, 11)( 12, 21)( 13, 23)( 14, 22)( 15, 24)
( 16, 26)( 17, 25)( 18, 27)( 19, 29)( 20, 28)( 31, 32)( 34, 35)( 37, 38)
( 39, 48)( 40, 50)( 41, 49)( 42, 51)( 43, 53)( 44, 52)( 45, 54)( 46, 56)
( 47, 55)( 57, 84)( 58, 86)( 59, 85)( 60, 87)( 61, 89)( 62, 88)( 63, 90)
( 64, 92)( 65, 91)( 66,102)( 67,104)( 68,103)( 69,105)( 70,107)( 71,106)
( 72,108)( 73,110)( 74,109)( 75, 93)( 76, 95)( 77, 94)( 78, 96)( 79, 98)
( 80, 97)( 81, 99)( 82,101)( 83,100)(112,113)(115,116)(118,119)(120,129)
(121,131)(122,130)(123,132)(124,134)(125,133)(126,135)(127,137)(128,136)
(139,140)(142,143)(145,146)(147,156)(148,158)(149,157)(150,159)(151,161)
(152,160)(153,162)(154,164)(155,163)(165,192)(166,194)(167,193)(168,195)
(169,197)(170,196)(171,198)(172,200)(173,199)(174,210)(175,212)(176,211)
(177,213)(178,215)(179,214)(180,216)(181,218)(182,217)(183,201)(184,203)
(185,202)(186,204)(187,206)(188,205)(189,207)(190,209)(191,208);
s2 := Sym(218)!( 3, 66)( 4, 68)( 5, 67)( 6, 70)( 7, 69)( 8, 71)( 9, 74)
( 10, 73)( 11, 72)( 12, 57)( 13, 59)( 14, 58)( 15, 61)( 16, 60)( 17, 62)
( 18, 65)( 19, 64)( 20, 63)( 21, 75)( 22, 77)( 23, 76)( 24, 79)( 25, 78)
( 26, 80)( 27, 83)( 28, 82)( 29, 81)( 30, 93)( 31, 95)( 32, 94)( 33, 97)
( 34, 96)( 35, 98)( 36,101)( 37,100)( 38, 99)( 39, 84)( 40, 86)( 41, 85)
( 42, 88)( 43, 87)( 44, 89)( 45, 92)( 46, 91)( 47, 90)( 48,102)( 49,104)
( 50,103)( 51,106)( 52,105)( 53,107)( 54,110)( 55,109)( 56,108)(111,174)
(112,176)(113,175)(114,178)(115,177)(116,179)(117,182)(118,181)(119,180)
(120,165)(121,167)(122,166)(123,169)(124,168)(125,170)(126,173)(127,172)
(128,171)(129,183)(130,185)(131,184)(132,187)(133,186)(134,188)(135,191)
(136,190)(137,189)(138,201)(139,203)(140,202)(141,205)(142,204)(143,206)
(144,209)(145,208)(146,207)(147,192)(148,194)(149,193)(150,196)(151,195)
(152,197)(153,200)(154,199)(155,198)(156,210)(157,212)(158,211)(159,214)
(160,213)(161,215)(162,218)(163,217)(164,216);
s3 := Sym(218)!( 3, 6)( 4, 8)( 5, 7)( 10, 11)( 12, 15)( 13, 17)( 14, 16)
( 19, 20)( 21, 24)( 22, 26)( 23, 25)( 28, 29)( 30, 33)( 31, 35)( 32, 34)
( 37, 38)( 39, 42)( 40, 44)( 41, 43)( 46, 47)( 48, 51)( 49, 53)( 50, 52)
( 55, 56)( 57, 60)( 58, 62)( 59, 61)( 64, 65)( 66, 69)( 67, 71)( 68, 70)
( 73, 74)( 75, 78)( 76, 80)( 77, 79)( 82, 83)( 84, 87)( 85, 89)( 86, 88)
( 91, 92)( 93, 96)( 94, 98)( 95, 97)(100,101)(102,105)(103,107)(104,106)
(109,110)(111,114)(112,116)(113,115)(118,119)(120,123)(121,125)(122,124)
(127,128)(129,132)(130,134)(131,133)(136,137)(138,141)(139,143)(140,142)
(145,146)(147,150)(148,152)(149,151)(154,155)(156,159)(157,161)(158,160)
(163,164)(165,168)(166,170)(167,169)(172,173)(174,177)(175,179)(176,178)
(181,182)(183,186)(184,188)(185,187)(190,191)(192,195)(193,197)(194,196)
(199,200)(201,204)(202,206)(203,205)(208,209)(210,213)(211,215)(212,214)
(217,218);
s4 := Sym(218)!( 3,111)( 4,113)( 5,112)( 6,117)( 7,119)( 8,118)( 9,114)
( 10,116)( 11,115)( 12,120)( 13,122)( 14,121)( 15,126)( 16,128)( 17,127)
( 18,123)( 19,125)( 20,124)( 21,129)( 22,131)( 23,130)( 24,135)( 25,137)
( 26,136)( 27,132)( 28,134)( 29,133)( 30,138)( 31,140)( 32,139)( 33,144)
( 34,146)( 35,145)( 36,141)( 37,143)( 38,142)( 39,147)( 40,149)( 41,148)
( 42,153)( 43,155)( 44,154)( 45,150)( 46,152)( 47,151)( 48,156)( 49,158)
( 50,157)( 51,162)( 52,164)( 53,163)( 54,159)( 55,161)( 56,160)( 57,165)
( 58,167)( 59,166)( 60,171)( 61,173)( 62,172)( 63,168)( 64,170)( 65,169)
( 66,174)( 67,176)( 68,175)( 69,180)( 70,182)( 71,181)( 72,177)( 73,179)
( 74,178)( 75,183)( 76,185)( 77,184)( 78,189)( 79,191)( 80,190)( 81,186)
( 82,188)( 83,187)( 84,192)( 85,194)( 86,193)( 87,198)( 88,200)( 89,199)
( 90,195)( 91,197)( 92,196)( 93,201)( 94,203)( 95,202)( 96,207)( 97,209)
( 98,208)( 99,204)(100,206)(101,205)(102,210)(103,212)(104,211)(105,216)
(106,218)(107,217)(108,213)(109,215)(110,214);
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, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2,
s4*s2*s3*s2*s3*s4*s2*s3*s2*s3, s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4,
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