Polytope of Type {2,6,24}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,24}*1728c
if this polytope has a name.
Group : SmallGroup(1728,17452)
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}*864c
   3-fold quotients : {2,6,24}*576c
   4-fold quotients : {2,6,6}*432c
   6-fold quotients : {2,6,12}*288c
   8-fold quotients : {2,3,6}*216
   9-fold quotients : {2,6,8}*192
   12-fold quotients : {2,6,6}*144c
   18-fold quotients : {2,6,4}*96a
   24-fold quotients : {2,3,6}*72
   27-fold quotients : {2,2,8}*64
   36-fold quotients : {2,6,2}*48
   54-fold quotients : {2,2,4}*32
   72-fold quotients : {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 := (  6,  9)(  7, 10)(  8, 11)( 12, 21)( 13, 22)( 14, 23)( 15, 27)( 16, 28)
( 17, 29)( 18, 24)( 19, 25)( 20, 26)( 33, 36)( 34, 37)( 35, 38)( 39, 48)
( 40, 49)( 41, 50)( 42, 54)( 43, 55)( 44, 56)( 45, 51)( 46, 52)( 47, 53)
( 60, 63)( 61, 64)( 62, 65)( 66, 75)( 67, 76)( 68, 77)( 69, 81)( 70, 82)
( 71, 83)( 72, 78)( 73, 79)( 74, 80)( 87, 90)( 88, 91)( 89, 92)( 93,102)
( 94,103)( 95,104)( 96,108)( 97,109)( 98,110)( 99,105)(100,106)(101,107)
(114,117)(115,118)(116,119)(120,129)(121,130)(122,131)(123,135)(124,136)
(125,137)(126,132)(127,133)(128,134)(141,144)(142,145)(143,146)(147,156)
(148,157)(149,158)(150,162)(151,163)(152,164)(153,159)(154,160)(155,161)
(168,171)(169,172)(170,173)(174,183)(175,184)(176,185)(177,189)(178,190)
(179,191)(180,186)(181,187)(182,188)(195,198)(196,199)(197,200)(201,210)
(202,211)(203,212)(204,216)(205,217)(206,218)(207,213)(208,214)(209,215);;
s2 := (  3,133)(  4,134)(  5,132)(  6,131)(  7,129)(  8,130)(  9,135)( 10,136)
( 11,137)( 12,124)( 13,125)( 14,123)( 15,122)( 16,120)( 17,121)( 18,126)
( 19,127)( 20,128)( 21,115)( 22,116)( 23,114)( 24,113)( 25,111)( 26,112)
( 27,117)( 28,118)( 29,119)( 30,160)( 31,161)( 32,159)( 33,158)( 34,156)
( 35,157)( 36,162)( 37,163)( 38,164)( 39,151)( 40,152)( 41,150)( 42,149)
( 43,147)( 44,148)( 45,153)( 46,154)( 47,155)( 48,142)( 49,143)( 50,141)
( 51,140)( 52,138)( 53,139)( 54,144)( 55,145)( 56,146)( 57,214)( 58,215)
( 59,213)( 60,212)( 61,210)( 62,211)( 63,216)( 64,217)( 65,218)( 66,205)
( 67,206)( 68,204)( 69,203)( 70,201)( 71,202)( 72,207)( 73,208)( 74,209)
( 75,196)( 76,197)( 77,195)( 78,194)( 79,192)( 80,193)( 81,198)( 82,199)
( 83,200)( 84,187)( 85,188)( 86,186)( 87,185)( 88,183)( 89,184)( 90,189)
( 91,190)( 92,191)( 93,178)( 94,179)( 95,177)( 96,176)( 97,174)( 98,175)
( 99,180)(100,181)(101,182)(102,169)(103,170)(104,168)(105,167)(106,165)
(107,166)(108,171)(109,172)(110,173);;
s3 := (  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)
( 57, 84)( 58, 86)( 59, 85)( 60, 90)( 61, 92)( 62, 91)( 63, 87)( 64, 89)
( 65, 88)( 66, 93)( 67, 95)( 68, 94)( 69, 99)( 70,101)( 71,100)( 72, 96)
( 73, 98)( 74, 97)( 75,102)( 76,104)( 77,103)( 78,108)( 79,110)( 80,109)
( 81,105)( 82,107)( 83,106)(111,165)(112,167)(113,166)(114,171)(115,173)
(116,172)(117,168)(118,170)(119,169)(120,174)(121,176)(122,175)(123,180)
(124,182)(125,181)(126,177)(127,179)(128,178)(129,183)(130,185)(131,184)
(132,189)(133,191)(134,190)(135,186)(136,188)(137,187)(138,192)(139,194)
(140,193)(141,198)(142,200)(143,199)(144,195)(145,197)(146,196)(147,201)
(148,203)(149,202)(150,207)(151,209)(152,208)(153,204)(154,206)(155,205)
(156,210)(157,212)(158,211)(159,216)(160,218)(161,217)(162,213)(163,215)
(164,214);;
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*s2*s1*s2*s3*s2*s3*s1*s2*s3*s2*s1*s2*s3*s2, 
s3*s1*s2*s3*s2*s3*s1*s2*s1*s2*s3*s1*s2*s3*s2*s3*s1*s2*s1*s2, 
s2*s3*s1*s2*s3*s2*s3*s2*s1*s3*s2*s3*s2*s1*s3*s2*s3*s1*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

to this polytope