Polytope of Type {2,6,24}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,24}*1728b
if this polytope has a name.
Group : SmallGroup(1728,15888)
Rank : 4
Schlafli Type : {2,6,24}
Number of vertices, edges, etc : 2, 18, 216, 72
Order of s0s1s2s3 : 24
Order of s0s1s2s3s2s1 : 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 : {2,6,12}*864b
3-fold quotients : {2,6,24}*576a
4-fold quotients : {2,6,6}*432b
6-fold quotients : {2,6,12}*288a
8-fold quotients : {2,6,6}*216
9-fold quotients : {2,2,24}*192, {2,6,8}*192
12-fold quotients : {2,6,6}*144a
18-fold quotients : {2,2,12}*96, {2,6,4}*96a
27-fold quotients : {2,2,8}*64
36-fold quotients : {2,2,6}*48, {2,6,2}*48
54-fold quotients : {2,2,4}*32
72-fold quotients : {2,2,3}*24, {2,3,2}*24
108-fold quotients : {2,2,2}*16
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, 6)( 4, 7)( 5, 8)( 12, 24)( 13, 25)( 14, 26)( 15, 21)( 16, 22)
( 17, 23)( 18, 27)( 19, 28)( 20, 29)( 30, 33)( 31, 34)( 32, 35)( 39, 51)
( 40, 52)( 41, 53)( 42, 48)( 43, 49)( 44, 50)( 45, 54)( 46, 55)( 47, 56)
( 57, 87)( 58, 88)( 59, 89)( 60, 84)( 61, 85)( 62, 86)( 63, 90)( 64, 91)
( 65, 92)( 66,105)( 67,106)( 68,107)( 69,102)( 70,103)( 71,104)( 72,108)
( 73,109)( 74,110)( 75, 96)( 76, 97)( 77, 98)( 78, 93)( 79, 94)( 80, 95)
( 81, 99)( 82,100)( 83,101)(111,168)(112,169)(113,170)(114,165)(115,166)
(116,167)(117,171)(118,172)(119,173)(120,186)(121,187)(122,188)(123,183)
(124,184)(125,185)(126,189)(127,190)(128,191)(129,177)(130,178)(131,179)
(132,174)(133,175)(134,176)(135,180)(136,181)(137,182)(138,195)(139,196)
(140,197)(141,192)(142,193)(143,194)(144,198)(145,199)(146,200)(147,213)
(148,214)(149,215)(150,210)(151,211)(152,212)(153,216)(154,217)(155,218)
(156,204)(157,205)(158,206)(159,201)(160,202)(161,203)(162,207)(163,208)
(164,209);;
s3 := ( 3,120)( 4,122)( 5,121)( 6,124)( 7,123)( 8,125)( 9,128)( 10,127)
( 11,126)( 12,111)( 13,113)( 14,112)( 15,115)( 16,114)( 17,116)( 18,119)
( 19,118)( 20,117)( 21,129)( 22,131)( 23,130)( 24,133)( 25,132)( 26,134)
( 27,137)( 28,136)( 29,135)( 30,147)( 31,149)( 32,148)( 33,151)( 34,150)
( 35,152)( 36,155)( 37,154)( 38,153)( 39,138)( 40,140)( 41,139)( 42,142)
( 43,141)( 44,143)( 45,146)( 46,145)( 47,144)( 48,156)( 49,158)( 50,157)
( 51,160)( 52,159)( 53,161)( 54,164)( 55,163)( 56,162)( 57,201)( 58,203)
( 59,202)( 60,205)( 61,204)( 62,206)( 63,209)( 64,208)( 65,207)( 66,192)
( 67,194)( 68,193)( 69,196)( 70,195)( 71,197)( 72,200)( 73,199)( 74,198)
( 75,210)( 76,212)( 77,211)( 78,214)( 79,213)( 80,215)( 81,218)( 82,217)
( 83,216)( 84,174)( 85,176)( 86,175)( 87,178)( 88,177)( 89,179)( 90,182)
( 91,181)( 92,180)( 93,165)( 94,167)( 95,166)( 96,169)( 97,168)( 98,170)
( 99,173)(100,172)(101,171)(102,183)(103,185)(104,184)(105,187)(106,186)
(107,188)(108,191)(109,190)(110,189);;
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*s1*s0*s1,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s3*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
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, 6)( 4, 7)( 5, 8)( 12, 24)( 13, 25)( 14, 26)( 15, 21)
( 16, 22)( 17, 23)( 18, 27)( 19, 28)( 20, 29)( 30, 33)( 31, 34)( 32, 35)
( 39, 51)( 40, 52)( 41, 53)( 42, 48)( 43, 49)( 44, 50)( 45, 54)( 46, 55)
( 47, 56)( 57, 87)( 58, 88)( 59, 89)( 60, 84)( 61, 85)( 62, 86)( 63, 90)
( 64, 91)( 65, 92)( 66,105)( 67,106)( 68,107)( 69,102)( 70,103)( 71,104)
( 72,108)( 73,109)( 74,110)( 75, 96)( 76, 97)( 77, 98)( 78, 93)( 79, 94)
( 80, 95)( 81, 99)( 82,100)( 83,101)(111,168)(112,169)(113,170)(114,165)
(115,166)(116,167)(117,171)(118,172)(119,173)(120,186)(121,187)(122,188)
(123,183)(124,184)(125,185)(126,189)(127,190)(128,191)(129,177)(130,178)
(131,179)(132,174)(133,175)(134,176)(135,180)(136,181)(137,182)(138,195)
(139,196)(140,197)(141,192)(142,193)(143,194)(144,198)(145,199)(146,200)
(147,213)(148,214)(149,215)(150,210)(151,211)(152,212)(153,216)(154,217)
(155,218)(156,204)(157,205)(158,206)(159,201)(160,202)(161,203)(162,207)
(163,208)(164,209);
s3 := Sym(218)!( 3,120)( 4,122)( 5,121)( 6,124)( 7,123)( 8,125)( 9,128)
( 10,127)( 11,126)( 12,111)( 13,113)( 14,112)( 15,115)( 16,114)( 17,116)
( 18,119)( 19,118)( 20,117)( 21,129)( 22,131)( 23,130)( 24,133)( 25,132)
( 26,134)( 27,137)( 28,136)( 29,135)( 30,147)( 31,149)( 32,148)( 33,151)
( 34,150)( 35,152)( 36,155)( 37,154)( 38,153)( 39,138)( 40,140)( 41,139)
( 42,142)( 43,141)( 44,143)( 45,146)( 46,145)( 47,144)( 48,156)( 49,158)
( 50,157)( 51,160)( 52,159)( 53,161)( 54,164)( 55,163)( 56,162)( 57,201)
( 58,203)( 59,202)( 60,205)( 61,204)( 62,206)( 63,209)( 64,208)( 65,207)
( 66,192)( 67,194)( 68,193)( 69,196)( 70,195)( 71,197)( 72,200)( 73,199)
( 74,198)( 75,210)( 76,212)( 77,211)( 78,214)( 79,213)( 80,215)( 81,218)
( 82,217)( 83,216)( 84,174)( 85,176)( 86,175)( 87,178)( 88,177)( 89,179)
( 90,182)( 91,181)( 92,180)( 93,165)( 94,167)( 95,166)( 96,169)( 97,168)
( 98,170)( 99,173)(100,172)(101,171)(102,183)(103,185)(104,184)(105,187)
(106,186)(107,188)(108,191)(109,190)(110,189);
poly := sub<Sym(218)|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*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s3*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;
to this polytope