Polytope of Type {2,3,6,24}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,3,6,24}*1728a
if this polytope has a name.
Group : SmallGroup(1728,15888)
Rank : 5
Schlafli Type : {2,3,6,24}
Number of vertices, edges, etc : 2, 3, 9, 72, 24
Order of s0s1s2s3s4 : 24
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
3-fold quotients : {2,3,2,24}*576
4-fold quotients : {2,3,6,6}*432a
6-fold quotients : {2,3,2,12}*288
8-fold quotients : {2,3,6,3}*216
9-fold quotients : {2,3,2,8}*192
12-fold quotients : {2,3,2,6}*144
18-fold quotients : {2,3,2,4}*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)( 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, 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);;
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)(111,174)(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);;
s4 := ( 3,111)( 4,113)( 5,112)( 6,114)( 7,116)( 8,115)( 9,117)( 10,119)
( 11,118)( 12,129)( 13,131)( 14,130)( 15,132)( 16,134)( 17,133)( 18,135)
( 19,137)( 20,136)( 21,120)( 22,122)( 23,121)( 24,123)( 25,125)( 26,124)
( 27,126)( 28,128)( 29,127)( 30,138)( 31,140)( 32,139)( 33,141)( 34,143)
( 35,142)( 36,144)( 37,146)( 38,145)( 39,156)( 40,158)( 41,157)( 42,159)
( 43,161)( 44,160)( 45,162)( 46,164)( 47,163)( 48,147)( 49,149)( 50,148)
( 51,150)( 52,152)( 53,151)( 54,153)( 55,155)( 56,154)( 57,192)( 58,194)
( 59,193)( 60,195)( 61,197)( 62,196)( 63,198)( 64,200)( 65,199)( 66,210)
( 67,212)( 68,211)( 69,213)( 70,215)( 71,214)( 72,216)( 73,218)( 74,217)
( 75,201)( 76,203)( 77,202)( 78,204)( 79,206)( 80,205)( 81,207)( 82,209)
( 83,208)( 84,165)( 85,167)( 86,166)( 87,168)( 88,170)( 89,169)( 90,171)
( 91,173)( 92,172)( 93,183)( 94,185)( 95,184)( 96,186)( 97,188)( 98,187)
( 99,189)(100,191)(101,190)(102,174)(103,176)(104,175)(105,177)(106,179)
(107,178)(108,180)(109,182)(110,181);;
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, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2,
s4*s2*s3*s2*s3*s4*s2*s3*s2*s3, s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*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, 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);
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)(111,174)(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);
s4 := Sym(218)!( 3,111)( 4,113)( 5,112)( 6,114)( 7,116)( 8,115)( 9,117)
( 10,119)( 11,118)( 12,129)( 13,131)( 14,130)( 15,132)( 16,134)( 17,133)
( 18,135)( 19,137)( 20,136)( 21,120)( 22,122)( 23,121)( 24,123)( 25,125)
( 26,124)( 27,126)( 28,128)( 29,127)( 30,138)( 31,140)( 32,139)( 33,141)
( 34,143)( 35,142)( 36,144)( 37,146)( 38,145)( 39,156)( 40,158)( 41,157)
( 42,159)( 43,161)( 44,160)( 45,162)( 46,164)( 47,163)( 48,147)( 49,149)
( 50,148)( 51,150)( 52,152)( 53,151)( 54,153)( 55,155)( 56,154)( 57,192)
( 58,194)( 59,193)( 60,195)( 61,197)( 62,196)( 63,198)( 64,200)( 65,199)
( 66,210)( 67,212)( 68,211)( 69,213)( 70,215)( 71,214)( 72,216)( 73,218)
( 74,217)( 75,201)( 76,203)( 77,202)( 78,204)( 79,206)( 80,205)( 81,207)
( 82,209)( 83,208)( 84,165)( 85,167)( 86,166)( 87,168)( 88,170)( 89,169)
( 90,171)( 91,173)( 92,172)( 93,183)( 94,185)( 95,184)( 96,186)( 97,188)
( 98,187)( 99,189)(100,191)(101,190)(102,174)(103,176)(104,175)(105,177)
(106,179)(107,178)(108,180)(109,182)(110,181);
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,
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s4*s2*s3*s2*s3*s4*s2*s3*s2*s3,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*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