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