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