Polytope of Type {2,24,6,3}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,24,6,3}*1728a
if this polytope has a name.
Group : SmallGroup(1728,15888)
Rank : 5
Schlafli Type : {2,24,6,3}
Number of vertices, edges, etc : 2, 24, 72, 9, 3
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 : {2,12,6,3}*864a
3-fold quotients : {2,24,2,3}*576
4-fold quotients : {2,6,6,3}*432a
6-fold quotients : {2,12,2,3}*288
8-fold quotients : {2,3,6,3}*216
9-fold quotients : {2,8,2,3}*192
12-fold quotients : {2,6,2,3}*144
18-fold quotients : {2,4,2,3}*96
24-fold quotients : {2,3,2,3}*72
36-fold quotients : {2,2,2,3}*48
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)(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);;
s2 := ( 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);;
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 := ( 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);;
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*s4*s3*s4*s3*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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)(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);
s2 := 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);
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)!( 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);
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*s4*s3*s4*s3*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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