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}*1728e
if this polytope has a name.
Group : SmallGroup(1728,46116)
Rank : 5
Schlafli Type : {2,6,6,4}
Number of vertices, edges, etc : 2, 18, 54, 36, 4
Order of s0s1s2s3s4 : 6
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
Degenerate
Universal
Non-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 : {2,6,3,4}*864
3-fold quotients : {2,6,6,4}*576e
6-fold quotients : {2,6,3,4}*288
9-fold quotients : {2,2,6,4}*192c
18-fold quotients : {2,2,3,4}*96
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 7, 11)( 8, 12)( 9, 13)( 10, 14)( 15, 27)( 16, 28)( 17, 29)( 18, 30)
( 19, 35)( 20, 36)( 21, 37)( 22, 38)( 23, 31)( 24, 32)( 25, 33)( 26, 34)
( 43, 47)( 44, 48)( 45, 49)( 46, 50)( 51, 63)( 52, 64)( 53, 65)( 54, 66)
( 55, 71)( 56, 72)( 57, 73)( 58, 74)( 59, 67)( 60, 68)( 61, 69)( 62, 70)
( 79, 83)( 80, 84)( 81, 85)( 82, 86)( 87, 99)( 88,100)( 89,101)( 90,102)
( 91,107)( 92,108)( 93,109)( 94,110)( 95,103)( 96,104)( 97,105)( 98,106)
(115,119)(116,120)(117,121)(118,122)(123,135)(124,136)(125,137)(126,138)
(127,143)(128,144)(129,145)(130,146)(131,139)(132,140)(133,141)(134,142)
(151,155)(152,156)(153,157)(154,158)(159,171)(160,172)(161,173)(162,174)
(163,179)(164,180)(165,181)(166,182)(167,175)(168,176)(169,177)(170,178)
(187,191)(188,192)(189,193)(190,194)(195,207)(196,208)(197,209)(198,210)
(199,215)(200,216)(201,217)(202,218)(203,211)(204,212)(205,213)(206,214);;
s2 := ( 3, 15)( 4, 17)( 5, 16)( 6, 18)( 7, 19)( 8, 21)( 9, 20)( 10, 22)
( 11, 23)( 12, 25)( 13, 24)( 14, 26)( 28, 29)( 32, 33)( 36, 37)( 39, 87)
( 40, 89)( 41, 88)( 42, 90)( 43, 91)( 44, 93)( 45, 92)( 46, 94)( 47, 95)
( 48, 97)( 49, 96)( 50, 98)( 51, 75)( 52, 77)( 53, 76)( 54, 78)( 55, 79)
( 56, 81)( 57, 80)( 58, 82)( 59, 83)( 60, 85)( 61, 84)( 62, 86)( 63, 99)
( 64,101)( 65,100)( 66,102)( 67,103)( 68,105)( 69,104)( 70,106)( 71,107)
( 72,109)( 73,108)( 74,110)(111,123)(112,125)(113,124)(114,126)(115,127)
(116,129)(117,128)(118,130)(119,131)(120,133)(121,132)(122,134)(136,137)
(140,141)(144,145)(147,195)(148,197)(149,196)(150,198)(151,199)(152,201)
(153,200)(154,202)(155,203)(156,205)(157,204)(158,206)(159,183)(160,185)
(161,184)(162,186)(163,187)(164,189)(165,188)(166,190)(167,191)(168,193)
(169,192)(170,194)(171,207)(172,209)(173,208)(174,210)(175,211)(176,213)
(177,212)(178,214)(179,215)(180,217)(181,216)(182,218);;
s3 := ( 3,147)( 4,148)( 5,150)( 6,149)( 7,151)( 8,152)( 9,154)( 10,153)
( 11,155)( 12,156)( 13,158)( 14,157)( 15,179)( 16,180)( 17,182)( 18,181)
( 19,171)( 20,172)( 21,174)( 22,173)( 23,175)( 24,176)( 25,178)( 26,177)
( 27,163)( 28,164)( 29,166)( 30,165)( 31,167)( 32,168)( 33,170)( 34,169)
( 35,159)( 36,160)( 37,162)( 38,161)( 39,111)( 40,112)( 41,114)( 42,113)
( 43,115)( 44,116)( 45,118)( 46,117)( 47,119)( 48,120)( 49,122)( 50,121)
( 51,143)( 52,144)( 53,146)( 54,145)( 55,135)( 56,136)( 57,138)( 58,137)
( 59,139)( 60,140)( 61,142)( 62,141)( 63,127)( 64,128)( 65,130)( 66,129)
( 67,131)( 68,132)( 69,134)( 70,133)( 71,123)( 72,124)( 73,126)( 74,125)
( 75,183)( 76,184)( 77,186)( 78,185)( 79,187)( 80,188)( 81,190)( 82,189)
( 83,191)( 84,192)( 85,194)( 86,193)( 87,215)( 88,216)( 89,218)( 90,217)
( 91,207)( 92,208)( 93,210)( 94,209)( 95,211)( 96,212)( 97,214)( 98,213)
( 99,199)(100,200)(101,202)(102,201)(103,203)(104,204)(105,206)(106,205)
(107,195)(108,196)(109,198)(110,197);;
s4 := ( 3, 6)( 4, 5)( 7, 10)( 8, 9)( 11, 14)( 12, 13)( 15, 18)( 16, 17)
( 19, 22)( 20, 21)( 23, 26)( 24, 25)( 27, 30)( 28, 29)( 31, 34)( 32, 33)
( 35, 38)( 36, 37)( 39, 42)( 40, 41)( 43, 46)( 44, 45)( 47, 50)( 48, 49)
( 51, 54)( 52, 53)( 55, 58)( 56, 57)( 59, 62)( 60, 61)( 63, 66)( 64, 65)
( 67, 70)( 68, 69)( 71, 74)( 72, 73)( 75, 78)( 76, 77)( 79, 82)( 80, 81)
( 83, 86)( 84, 85)( 87, 90)( 88, 89)( 91, 94)( 92, 93)( 95, 98)( 96, 97)
( 99,102)(100,101)(103,106)(104,105)(107,110)(108,109)(111,114)(112,113)
(115,118)(116,117)(119,122)(120,121)(123,126)(124,125)(127,130)(128,129)
(131,134)(132,133)(135,138)(136,137)(139,142)(140,141)(143,146)(144,145)
(147,150)(148,149)(151,154)(152,153)(155,158)(156,157)(159,162)(160,161)
(163,166)(164,165)(167,170)(168,169)(171,174)(172,173)(175,178)(176,177)
(179,182)(180,181)(183,186)(184,185)(187,190)(188,189)(191,194)(192,193)
(195,198)(196,197)(199,202)(200,201)(203,206)(204,205)(207,210)(208,209)
(211,214)(212,213)(215,218)(216,217);;
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*s4, s4*s3*s2*s4*s3*s4*s3*s2*s3,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2*s3*s1*s2*s1*s3*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(218)!(1,2);
s1 := Sym(218)!( 7, 11)( 8, 12)( 9, 13)( 10, 14)( 15, 27)( 16, 28)( 17, 29)
( 18, 30)( 19, 35)( 20, 36)( 21, 37)( 22, 38)( 23, 31)( 24, 32)( 25, 33)
( 26, 34)( 43, 47)( 44, 48)( 45, 49)( 46, 50)( 51, 63)( 52, 64)( 53, 65)
( 54, 66)( 55, 71)( 56, 72)( 57, 73)( 58, 74)( 59, 67)( 60, 68)( 61, 69)
( 62, 70)( 79, 83)( 80, 84)( 81, 85)( 82, 86)( 87, 99)( 88,100)( 89,101)
( 90,102)( 91,107)( 92,108)( 93,109)( 94,110)( 95,103)( 96,104)( 97,105)
( 98,106)(115,119)(116,120)(117,121)(118,122)(123,135)(124,136)(125,137)
(126,138)(127,143)(128,144)(129,145)(130,146)(131,139)(132,140)(133,141)
(134,142)(151,155)(152,156)(153,157)(154,158)(159,171)(160,172)(161,173)
(162,174)(163,179)(164,180)(165,181)(166,182)(167,175)(168,176)(169,177)
(170,178)(187,191)(188,192)(189,193)(190,194)(195,207)(196,208)(197,209)
(198,210)(199,215)(200,216)(201,217)(202,218)(203,211)(204,212)(205,213)
(206,214);
s2 := Sym(218)!( 3, 15)( 4, 17)( 5, 16)( 6, 18)( 7, 19)( 8, 21)( 9, 20)
( 10, 22)( 11, 23)( 12, 25)( 13, 24)( 14, 26)( 28, 29)( 32, 33)( 36, 37)
( 39, 87)( 40, 89)( 41, 88)( 42, 90)( 43, 91)( 44, 93)( 45, 92)( 46, 94)
( 47, 95)( 48, 97)( 49, 96)( 50, 98)( 51, 75)( 52, 77)( 53, 76)( 54, 78)
( 55, 79)( 56, 81)( 57, 80)( 58, 82)( 59, 83)( 60, 85)( 61, 84)( 62, 86)
( 63, 99)( 64,101)( 65,100)( 66,102)( 67,103)( 68,105)( 69,104)( 70,106)
( 71,107)( 72,109)( 73,108)( 74,110)(111,123)(112,125)(113,124)(114,126)
(115,127)(116,129)(117,128)(118,130)(119,131)(120,133)(121,132)(122,134)
(136,137)(140,141)(144,145)(147,195)(148,197)(149,196)(150,198)(151,199)
(152,201)(153,200)(154,202)(155,203)(156,205)(157,204)(158,206)(159,183)
(160,185)(161,184)(162,186)(163,187)(164,189)(165,188)(166,190)(167,191)
(168,193)(169,192)(170,194)(171,207)(172,209)(173,208)(174,210)(175,211)
(176,213)(177,212)(178,214)(179,215)(180,217)(181,216)(182,218);
s3 := Sym(218)!( 3,147)( 4,148)( 5,150)( 6,149)( 7,151)( 8,152)( 9,154)
( 10,153)( 11,155)( 12,156)( 13,158)( 14,157)( 15,179)( 16,180)( 17,182)
( 18,181)( 19,171)( 20,172)( 21,174)( 22,173)( 23,175)( 24,176)( 25,178)
( 26,177)( 27,163)( 28,164)( 29,166)( 30,165)( 31,167)( 32,168)( 33,170)
( 34,169)( 35,159)( 36,160)( 37,162)( 38,161)( 39,111)( 40,112)( 41,114)
( 42,113)( 43,115)( 44,116)( 45,118)( 46,117)( 47,119)( 48,120)( 49,122)
( 50,121)( 51,143)( 52,144)( 53,146)( 54,145)( 55,135)( 56,136)( 57,138)
( 58,137)( 59,139)( 60,140)( 61,142)( 62,141)( 63,127)( 64,128)( 65,130)
( 66,129)( 67,131)( 68,132)( 69,134)( 70,133)( 71,123)( 72,124)( 73,126)
( 74,125)( 75,183)( 76,184)( 77,186)( 78,185)( 79,187)( 80,188)( 81,190)
( 82,189)( 83,191)( 84,192)( 85,194)( 86,193)( 87,215)( 88,216)( 89,218)
( 90,217)( 91,207)( 92,208)( 93,210)( 94,209)( 95,211)( 96,212)( 97,214)
( 98,213)( 99,199)(100,200)(101,202)(102,201)(103,203)(104,204)(105,206)
(106,205)(107,195)(108,196)(109,198)(110,197);
s4 := Sym(218)!( 3, 6)( 4, 5)( 7, 10)( 8, 9)( 11, 14)( 12, 13)( 15, 18)
( 16, 17)( 19, 22)( 20, 21)( 23, 26)( 24, 25)( 27, 30)( 28, 29)( 31, 34)
( 32, 33)( 35, 38)( 36, 37)( 39, 42)( 40, 41)( 43, 46)( 44, 45)( 47, 50)
( 48, 49)( 51, 54)( 52, 53)( 55, 58)( 56, 57)( 59, 62)( 60, 61)( 63, 66)
( 64, 65)( 67, 70)( 68, 69)( 71, 74)( 72, 73)( 75, 78)( 76, 77)( 79, 82)
( 80, 81)( 83, 86)( 84, 85)( 87, 90)( 88, 89)( 91, 94)( 92, 93)( 95, 98)
( 96, 97)( 99,102)(100,101)(103,106)(104,105)(107,110)(108,109)(111,114)
(112,113)(115,118)(116,117)(119,122)(120,121)(123,126)(124,125)(127,130)
(128,129)(131,134)(132,133)(135,138)(136,137)(139,142)(140,141)(143,146)
(144,145)(147,150)(148,149)(151,154)(152,153)(155,158)(156,157)(159,162)
(160,161)(163,166)(164,165)(167,170)(168,169)(171,174)(172,173)(175,178)
(176,177)(179,182)(180,181)(183,186)(184,185)(187,190)(188,189)(191,194)
(192,193)(195,198)(196,197)(199,202)(200,201)(203,206)(204,205)(207,210)
(208,209)(211,214)(212,213)(215,218)(216,217);
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*s4,
s4*s3*s2*s4*s3*s4*s3*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2*s3*s1*s2*s1*s3*s2 >;
to this polytope