Polytope of Type {2,6,6,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,6,4}*1728c
if this polytope has a name.
Group : SmallGroup(1728,30882)
Rank : 5
Schlafli Type : {2,6,6,4}
Number of vertices, edges, etc : 2, 18, 54, 36, 4
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 : {2,3,6,4}*864a, {2,6,6,2}*864c
3-fold quotients : {2,6,6,4}*576c
4-fold quotients : {2,3,6,2}*432
6-fold quotients : {2,3,6,4}*288, {2,6,6,2}*288c
9-fold quotients : {2,6,2,4}*192
12-fold quotients : {2,3,6,2}*144
18-fold quotients : {2,3,2,4}*96, {2,6,2,2}*96
27-fold quotients : {2,2,2,4}*64
36-fold quotients : {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 := ( 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)( 57,187)( 58,188)
( 59,186)( 60,185)( 61,183)( 62,184)( 63,189)( 64,190)( 65,191)( 66,178)
( 67,179)( 68,177)( 69,176)( 70,174)( 71,175)( 72,180)( 73,181)( 74,182)
( 75,169)( 76,170)( 77,168)( 78,167)( 79,165)( 80,166)( 81,171)( 82,172)
( 83,173)( 84,214)( 85,215)( 86,213)( 87,212)( 88,210)( 89,211)( 90,216)
( 91,217)( 92,218)( 93,205)( 94,206)( 95,204)( 96,203)( 97,201)( 98,202)
( 99,207)(100,208)(101,209)(102,196)(103,197)(104,195)(105,194)(106,192)
(107,193)(108,198)(109,199)(110,200);;
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)(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)(165,192)(166,194)(167,193)(168,198)(169,200)
(170,199)(171,195)(172,197)(173,196)(174,201)(175,203)(176,202)(177,207)
(178,209)(179,208)(180,204)(181,206)(182,205)(183,210)(184,212)(185,211)
(186,216)(187,218)(188,217)(189,213)(190,215)(191,214);;
s4 := ( 3, 57)( 4, 58)( 5, 59)( 6, 60)( 7, 61)( 8, 62)( 9, 63)( 10, 64)
( 11, 65)( 12, 66)( 13, 67)( 14, 68)( 15, 69)( 16, 70)( 17, 71)( 18, 72)
( 19, 73)( 20, 74)( 21, 75)( 22, 76)( 23, 77)( 24, 78)( 25, 79)( 26, 80)
( 27, 81)( 28, 82)( 29, 83)( 30, 84)( 31, 85)( 32, 86)( 33, 87)( 34, 88)
( 35, 89)( 36, 90)( 37, 91)( 38, 92)( 39, 93)( 40, 94)( 41, 95)( 42, 96)
( 43, 97)( 44, 98)( 45, 99)( 46,100)( 47,101)( 48,102)( 49,103)( 50,104)
( 51,105)( 52,106)( 53,107)( 54,108)( 55,109)( 56,110)(111,165)(112,166)
(113,167)(114,168)(115,169)(116,170)(117,171)(118,172)(119,173)(120,174)
(121,175)(122,176)(123,177)(124,178)(125,179)(126,180)(127,181)(128,182)
(129,183)(130,184)(131,185)(132,186)(133,187)(134,188)(135,189)(136,190)
(137,191)(138,192)(139,193)(140,194)(141,195)(142,196)(143,197)(144,198)
(145,199)(146,200)(147,201)(148,202)(149,203)(150,204)(151,205)(152,206)
(153,207)(154,208)(155,209)(156,210)(157,211)(158,212)(159,213)(160,214)
(161,215)(162,216)(163,217)(164,218);;
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,
s2*s3*s4*s3*s2*s3*s4*s3, s3*s4*s3*s4*s3*s4*s3*s4,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2 ];;
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)( 57,187)
( 58,188)( 59,186)( 60,185)( 61,183)( 62,184)( 63,189)( 64,190)( 65,191)
( 66,178)( 67,179)( 68,177)( 69,176)( 70,174)( 71,175)( 72,180)( 73,181)
( 74,182)( 75,169)( 76,170)( 77,168)( 78,167)( 79,165)( 80,166)( 81,171)
( 82,172)( 83,173)( 84,214)( 85,215)( 86,213)( 87,212)( 88,210)( 89,211)
( 90,216)( 91,217)( 92,218)( 93,205)( 94,206)( 95,204)( 96,203)( 97,201)
( 98,202)( 99,207)(100,208)(101,209)(102,196)(103,197)(104,195)(105,194)
(106,192)(107,193)(108,198)(109,199)(110,200);
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)(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)(165,192)(166,194)(167,193)(168,198)
(169,200)(170,199)(171,195)(172,197)(173,196)(174,201)(175,203)(176,202)
(177,207)(178,209)(179,208)(180,204)(181,206)(182,205)(183,210)(184,212)
(185,211)(186,216)(187,218)(188,217)(189,213)(190,215)(191,214);
s4 := Sym(218)!( 3, 57)( 4, 58)( 5, 59)( 6, 60)( 7, 61)( 8, 62)( 9, 63)
( 10, 64)( 11, 65)( 12, 66)( 13, 67)( 14, 68)( 15, 69)( 16, 70)( 17, 71)
( 18, 72)( 19, 73)( 20, 74)( 21, 75)( 22, 76)( 23, 77)( 24, 78)( 25, 79)
( 26, 80)( 27, 81)( 28, 82)( 29, 83)( 30, 84)( 31, 85)( 32, 86)( 33, 87)
( 34, 88)( 35, 89)( 36, 90)( 37, 91)( 38, 92)( 39, 93)( 40, 94)( 41, 95)
( 42, 96)( 43, 97)( 44, 98)( 45, 99)( 46,100)( 47,101)( 48,102)( 49,103)
( 50,104)( 51,105)( 52,106)( 53,107)( 54,108)( 55,109)( 56,110)(111,165)
(112,166)(113,167)(114,168)(115,169)(116,170)(117,171)(118,172)(119,173)
(120,174)(121,175)(122,176)(123,177)(124,178)(125,179)(126,180)(127,181)
(128,182)(129,183)(130,184)(131,185)(132,186)(133,187)(134,188)(135,189)
(136,190)(137,191)(138,192)(139,193)(140,194)(141,195)(142,196)(143,197)
(144,198)(145,199)(146,200)(147,201)(148,202)(149,203)(150,204)(151,205)
(152,206)(153,207)(154,208)(155,209)(156,210)(157,211)(158,212)(159,213)
(160,214)(161,215)(162,216)(163,217)(164,218);
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, s2*s3*s4*s3*s2*s3*s4*s3,
s3*s4*s3*s4*s3*s4*s3*s4, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2 >;
to this polytope