Polytope of Type {2,6,6,12}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,6,12}*1728a
if this polytope has a name.
Group : SmallGroup(1728,30782)
Rank : 5
Schlafli Type : {2,6,6,12}
Number of vertices, edges, etc : 2, 6, 18, 36, 12
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 : {2,3,6,12}*864a, {2,6,6,6}*864a
3-fold quotients : {2,6,2,12}*576
4-fold quotients : {2,3,6,6}*432a, {2,6,6,3}*432a
6-fold quotients : {2,3,2,12}*288, {2,6,2,6}*288
8-fold quotients : {2,3,6,3}*216
9-fold quotients : {2,2,2,12}*192, {2,6,2,4}*192
12-fold quotients : {2,3,2,6}*144, {2,6,2,3}*144
18-fold quotients : {2,3,2,4}*96, {2,2,2,6}*96, {2,6,2,2}*96
24-fold quotients : {2,3,2,3}*72
27-fold quotients : {2,2,2,4}*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)( 6, 9)( 7, 11)( 8, 10)( 13, 14)( 15, 18)( 16, 20)( 17, 19)
( 22, 23)( 24, 27)( 25, 29)( 26, 28)( 31, 32)( 33, 36)( 34, 38)( 35, 37)
( 40, 41)( 42, 45)( 43, 47)( 44, 46)( 49, 50)( 51, 54)( 52, 56)( 53, 55)
( 58, 59)( 60, 63)( 61, 65)( 62, 64)( 67, 68)( 69, 72)( 70, 74)( 71, 73)
( 76, 77)( 78, 81)( 79, 83)( 80, 82)( 85, 86)( 87, 90)( 88, 92)( 89, 91)
( 94, 95)( 96, 99)( 97,101)( 98,100)(103,104)(105,108)(106,110)(107,109)
(112,113)(114,117)(115,119)(116,118)(121,122)(123,126)(124,128)(125,127)
(130,131)(132,135)(133,137)(134,136)(139,140)(141,144)(142,146)(143,145)
(148,149)(150,153)(151,155)(152,154)(157,158)(159,162)(160,164)(161,163)
(166,167)(168,171)(169,173)(170,172)(175,176)(177,180)(178,182)(179,181)
(184,185)(186,189)(187,191)(188,190)(193,194)(195,198)(196,200)(197,199)
(202,203)(204,207)(205,209)(206,208)(211,212)(213,216)(214,218)(215,217);;
s2 := ( 3,114)( 4,116)( 5,115)( 6,111)( 7,113)( 8,112)( 9,117)( 10,119)
( 11,118)( 12,123)( 13,125)( 14,124)( 15,120)( 16,122)( 17,121)( 18,126)
( 19,128)( 20,127)( 21,132)( 22,134)( 23,133)( 24,129)( 25,131)( 26,130)
( 27,135)( 28,137)( 29,136)( 30,141)( 31,143)( 32,142)( 33,138)( 34,140)
( 35,139)( 36,144)( 37,146)( 38,145)( 39,150)( 40,152)( 41,151)( 42,147)
( 43,149)( 44,148)( 45,153)( 46,155)( 47,154)( 48,159)( 49,161)( 50,160)
( 51,156)( 52,158)( 53,157)( 54,162)( 55,164)( 56,163)( 57,168)( 58,170)
( 59,169)( 60,165)( 61,167)( 62,166)( 63,171)( 64,173)( 65,172)( 66,177)
( 67,179)( 68,178)( 69,174)( 70,176)( 71,175)( 72,180)( 73,182)( 74,181)
( 75,186)( 76,188)( 77,187)( 78,183)( 79,185)( 80,184)( 81,189)( 82,191)
( 83,190)( 84,195)( 85,197)( 86,196)( 87,192)( 88,194)( 89,193)( 90,198)
( 91,200)( 92,199)( 93,204)( 94,206)( 95,205)( 96,201)( 97,203)( 98,202)
( 99,207)(100,209)(101,208)(102,213)(103,215)(104,214)(105,210)(106,212)
(107,211)(108,216)(109,218)(110,217);;
s3 := ( 3, 12)( 4, 14)( 5, 13)( 6, 16)( 7, 15)( 8, 17)( 9, 20)( 10, 19)
( 11, 18)( 22, 23)( 24, 25)( 27, 29)( 30, 39)( 31, 41)( 32, 40)( 33, 43)
( 34, 42)( 35, 44)( 36, 47)( 37, 46)( 38, 45)( 49, 50)( 51, 52)( 54, 56)
( 57, 93)( 58, 95)( 59, 94)( 60, 97)( 61, 96)( 62, 98)( 63,101)( 64,100)
( 65, 99)( 66, 84)( 67, 86)( 68, 85)( 69, 88)( 70, 87)( 71, 89)( 72, 92)
( 73, 91)( 74, 90)( 75,102)( 76,104)( 77,103)( 78,106)( 79,105)( 80,107)
( 81,110)( 82,109)( 83,108)(111,120)(112,122)(113,121)(114,124)(115,123)
(116,125)(117,128)(118,127)(119,126)(130,131)(132,133)(135,137)(138,147)
(139,149)(140,148)(141,151)(142,150)(143,152)(144,155)(145,154)(146,153)
(157,158)(159,160)(162,164)(165,201)(166,203)(167,202)(168,205)(169,204)
(170,206)(171,209)(172,208)(173,207)(174,192)(175,194)(176,193)(177,196)
(178,195)(179,197)(180,200)(181,199)(182,198)(183,210)(184,212)(185,211)
(186,214)(187,213)(188,215)(189,218)(190,217)(191,216);;
s4 := ( 3, 57)( 4, 59)( 5, 58)( 6, 60)( 7, 62)( 8, 61)( 9, 63)( 10, 65)
( 11, 64)( 12, 75)( 13, 77)( 14, 76)( 15, 78)( 16, 80)( 17, 79)( 18, 81)
( 19, 83)( 20, 82)( 21, 66)( 22, 68)( 23, 67)( 24, 69)( 25, 71)( 26, 70)
( 27, 72)( 28, 74)( 29, 73)( 30, 84)( 31, 86)( 32, 85)( 33, 87)( 34, 89)
( 35, 88)( 36, 90)( 37, 92)( 38, 91)( 39,102)( 40,104)( 41,103)( 42,105)
( 43,107)( 44,106)( 45,108)( 46,110)( 47,109)( 48, 93)( 49, 95)( 50, 94)
( 51, 96)( 52, 98)( 53, 97)( 54, 99)( 55,101)( 56,100)(111,165)(112,167)
(113,166)(114,168)(115,170)(116,169)(117,171)(118,173)(119,172)(120,183)
(121,185)(122,184)(123,186)(124,188)(125,187)(126,189)(127,191)(128,190)
(129,174)(130,176)(131,175)(132,177)(133,179)(134,178)(135,180)(136,182)
(137,181)(138,192)(139,194)(140,193)(141,195)(142,197)(143,196)(144,198)
(145,200)(146,199)(147,210)(148,212)(149,211)(150,213)(151,215)(152,214)
(153,216)(154,218)(155,217)(156,201)(157,203)(158,202)(159,204)(160,206)
(161,205)(162,207)(163,209)(164,208);;
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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(218)!(1,2);
s1 := Sym(218)!( 4, 5)( 6, 9)( 7, 11)( 8, 10)( 13, 14)( 15, 18)( 16, 20)
( 17, 19)( 22, 23)( 24, 27)( 25, 29)( 26, 28)( 31, 32)( 33, 36)( 34, 38)
( 35, 37)( 40, 41)( 42, 45)( 43, 47)( 44, 46)( 49, 50)( 51, 54)( 52, 56)
( 53, 55)( 58, 59)( 60, 63)( 61, 65)( 62, 64)( 67, 68)( 69, 72)( 70, 74)
( 71, 73)( 76, 77)( 78, 81)( 79, 83)( 80, 82)( 85, 86)( 87, 90)( 88, 92)
( 89, 91)( 94, 95)( 96, 99)( 97,101)( 98,100)(103,104)(105,108)(106,110)
(107,109)(112,113)(114,117)(115,119)(116,118)(121,122)(123,126)(124,128)
(125,127)(130,131)(132,135)(133,137)(134,136)(139,140)(141,144)(142,146)
(143,145)(148,149)(150,153)(151,155)(152,154)(157,158)(159,162)(160,164)
(161,163)(166,167)(168,171)(169,173)(170,172)(175,176)(177,180)(178,182)
(179,181)(184,185)(186,189)(187,191)(188,190)(193,194)(195,198)(196,200)
(197,199)(202,203)(204,207)(205,209)(206,208)(211,212)(213,216)(214,218)
(215,217);
s2 := Sym(218)!( 3,114)( 4,116)( 5,115)( 6,111)( 7,113)( 8,112)( 9,117)
( 10,119)( 11,118)( 12,123)( 13,125)( 14,124)( 15,120)( 16,122)( 17,121)
( 18,126)( 19,128)( 20,127)( 21,132)( 22,134)( 23,133)( 24,129)( 25,131)
( 26,130)( 27,135)( 28,137)( 29,136)( 30,141)( 31,143)( 32,142)( 33,138)
( 34,140)( 35,139)( 36,144)( 37,146)( 38,145)( 39,150)( 40,152)( 41,151)
( 42,147)( 43,149)( 44,148)( 45,153)( 46,155)( 47,154)( 48,159)( 49,161)
( 50,160)( 51,156)( 52,158)( 53,157)( 54,162)( 55,164)( 56,163)( 57,168)
( 58,170)( 59,169)( 60,165)( 61,167)( 62,166)( 63,171)( 64,173)( 65,172)
( 66,177)( 67,179)( 68,178)( 69,174)( 70,176)( 71,175)( 72,180)( 73,182)
( 74,181)( 75,186)( 76,188)( 77,187)( 78,183)( 79,185)( 80,184)( 81,189)
( 82,191)( 83,190)( 84,195)( 85,197)( 86,196)( 87,192)( 88,194)( 89,193)
( 90,198)( 91,200)( 92,199)( 93,204)( 94,206)( 95,205)( 96,201)( 97,203)
( 98,202)( 99,207)(100,209)(101,208)(102,213)(103,215)(104,214)(105,210)
(106,212)(107,211)(108,216)(109,218)(110,217);
s3 := Sym(218)!( 3, 12)( 4, 14)( 5, 13)( 6, 16)( 7, 15)( 8, 17)( 9, 20)
( 10, 19)( 11, 18)( 22, 23)( 24, 25)( 27, 29)( 30, 39)( 31, 41)( 32, 40)
( 33, 43)( 34, 42)( 35, 44)( 36, 47)( 37, 46)( 38, 45)( 49, 50)( 51, 52)
( 54, 56)( 57, 93)( 58, 95)( 59, 94)( 60, 97)( 61, 96)( 62, 98)( 63,101)
( 64,100)( 65, 99)( 66, 84)( 67, 86)( 68, 85)( 69, 88)( 70, 87)( 71, 89)
( 72, 92)( 73, 91)( 74, 90)( 75,102)( 76,104)( 77,103)( 78,106)( 79,105)
( 80,107)( 81,110)( 82,109)( 83,108)(111,120)(112,122)(113,121)(114,124)
(115,123)(116,125)(117,128)(118,127)(119,126)(130,131)(132,133)(135,137)
(138,147)(139,149)(140,148)(141,151)(142,150)(143,152)(144,155)(145,154)
(146,153)(157,158)(159,160)(162,164)(165,201)(166,203)(167,202)(168,205)
(169,204)(170,206)(171,209)(172,208)(173,207)(174,192)(175,194)(176,193)
(177,196)(178,195)(179,197)(180,200)(181,199)(182,198)(183,210)(184,212)
(185,211)(186,214)(187,213)(188,215)(189,218)(190,217)(191,216);
s4 := Sym(218)!( 3, 57)( 4, 59)( 5, 58)( 6, 60)( 7, 62)( 8, 61)( 9, 63)
( 10, 65)( 11, 64)( 12, 75)( 13, 77)( 14, 76)( 15, 78)( 16, 80)( 17, 79)
( 18, 81)( 19, 83)( 20, 82)( 21, 66)( 22, 68)( 23, 67)( 24, 69)( 25, 71)
( 26, 70)( 27, 72)( 28, 74)( 29, 73)( 30, 84)( 31, 86)( 32, 85)( 33, 87)
( 34, 89)( 35, 88)( 36, 90)( 37, 92)( 38, 91)( 39,102)( 40,104)( 41,103)
( 42,105)( 43,107)( 44,106)( 45,108)( 46,110)( 47,109)( 48, 93)( 49, 95)
( 50, 94)( 51, 96)( 52, 98)( 53, 97)( 54, 99)( 55,101)( 56,100)(111,165)
(112,167)(113,166)(114,168)(115,170)(116,169)(117,171)(118,173)(119,172)
(120,183)(121,185)(122,184)(123,186)(124,188)(125,187)(126,189)(127,191)
(128,190)(129,174)(130,176)(131,175)(132,177)(133,179)(134,178)(135,180)
(136,182)(137,181)(138,192)(139,194)(140,193)(141,195)(142,197)(143,196)
(144,198)(145,200)(146,199)(147,210)(148,212)(149,211)(150,213)(151,215)
(152,214)(153,216)(154,218)(155,217)(156,201)(157,203)(158,202)(159,204)
(160,206)(161,205)(162,207)(163,209)(164,208);
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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;
to this polytope