Polytope of Type {2,4,6,6,3}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,4,6,6,3}*1728c
if this polytope has a name.
Group : SmallGroup(1728,46116)
Rank : 6
Schlafli Type : {2,4,6,6,3}
Number of vertices, edges, etc : 2, 4, 12, 18, 9, 3
Order of s0s1s2s3s4s5 : 6
Order of s0s1s2s3s4s5s4s3s2s1 : 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 : {2,4,3,6,3}*864
3-fold quotients : {2,4,6,2,3}*576b
6-fold quotients : {2,4,3,2,3}*288
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)( 7, 11)( 8, 13)( 9, 12)( 10, 14)( 16, 17)( 19, 23)( 20, 25)
( 21, 24)( 22, 26)( 28, 29)( 31, 35)( 32, 37)( 33, 36)( 34, 38)( 39, 75)
( 40, 77)( 41, 76)( 42, 78)( 43, 83)( 44, 85)( 45, 84)( 46, 86)( 47, 79)
( 48, 81)( 49, 80)( 50, 82)( 51, 87)( 52, 89)( 53, 88)( 54, 90)( 55, 95)
( 56, 97)( 57, 96)( 58, 98)( 59, 91)( 60, 93)( 61, 92)( 62, 94)( 63, 99)
( 64,101)( 65,100)( 66,102)( 67,107)( 68,109)( 69,108)( 70,110)( 71,103)
( 72,105)( 73,104)( 74,106)(112,113)(115,119)(116,121)(117,120)(118,122)
(124,125)(127,131)(128,133)(129,132)(130,134)(136,137)(139,143)(140,145)
(141,144)(142,146)(147,183)(148,185)(149,184)(150,186)(151,191)(152,193)
(153,192)(154,194)(155,187)(156,189)(157,188)(158,190)(159,195)(160,197)
(161,196)(162,198)(163,203)(164,205)(165,204)(166,206)(167,199)(168,201)
(169,200)(170,202)(171,207)(172,209)(173,208)(174,210)(175,215)(176,217)
(177,216)(178,218)(179,211)(180,213)(181,212)(182,214);;
s3 := ( 3,183)( 4,186)( 5,185)( 6,184)( 7,191)( 8,194)( 9,193)( 10,192)
( 11,187)( 12,190)( 13,189)( 14,188)( 15,203)( 16,206)( 17,205)( 18,204)
( 19,199)( 20,202)( 21,201)( 22,200)( 23,195)( 24,198)( 25,197)( 26,196)
( 27,211)( 28,214)( 29,213)( 30,212)( 31,207)( 32,210)( 33,209)( 34,208)
( 35,215)( 36,218)( 37,217)( 38,216)( 39,147)( 40,150)( 41,149)( 42,148)
( 43,155)( 44,158)( 45,157)( 46,156)( 47,151)( 48,154)( 49,153)( 50,152)
( 51,167)( 52,170)( 53,169)( 54,168)( 55,163)( 56,166)( 57,165)( 58,164)
( 59,159)( 60,162)( 61,161)( 62,160)( 63,175)( 64,178)( 65,177)( 66,176)
( 67,171)( 68,174)( 69,173)( 70,172)( 71,179)( 72,182)( 73,181)( 74,180)
( 75,111)( 76,114)( 77,113)( 78,112)( 79,119)( 80,122)( 81,121)( 82,120)
( 83,115)( 84,118)( 85,117)( 86,116)( 87,131)( 88,134)( 89,133)( 90,132)
( 91,127)( 92,130)( 93,129)( 94,128)( 95,123)( 96,126)( 97,125)( 98,124)
( 99,139)(100,142)(101,141)(102,140)(103,135)(104,138)(105,137)(106,136)
(107,143)(108,146)(109,145)(110,144);;
s4 := ( 3, 15)( 4, 16)( 5, 17)( 6, 18)( 7, 23)( 8, 24)( 9, 25)( 10, 26)
( 11, 19)( 12, 20)( 13, 21)( 14, 22)( 31, 35)( 32, 36)( 33, 37)( 34, 38)
( 39, 51)( 40, 52)( 41, 53)( 42, 54)( 43, 59)( 44, 60)( 45, 61)( 46, 62)
( 47, 55)( 48, 56)( 49, 57)( 50, 58)( 67, 71)( 68, 72)( 69, 73)( 70, 74)
( 75, 87)( 76, 88)( 77, 89)( 78, 90)( 79, 95)( 80, 96)( 81, 97)( 82, 98)
( 83, 91)( 84, 92)( 85, 93)( 86, 94)(103,107)(104,108)(105,109)(106,110)
(111,123)(112,124)(113,125)(114,126)(115,131)(116,132)(117,133)(118,134)
(119,127)(120,128)(121,129)(122,130)(139,143)(140,144)(141,145)(142,146)
(147,159)(148,160)(149,161)(150,162)(151,167)(152,168)(153,169)(154,170)
(155,163)(156,164)(157,165)(158,166)(175,179)(176,180)(177,181)(178,182)
(183,195)(184,196)(185,197)(186,198)(187,203)(188,204)(189,205)(190,206)
(191,199)(192,200)(193,201)(194,202)(211,215)(212,216)(213,217)(214,218);;
s5 := ( 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,s5]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4","s5");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;; s4 := F.5;; s5 := F.6;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s5*s5,
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, s0*s5*s0*s5, s1*s5*s1*s5,
s2*s5*s2*s5, s3*s5*s3*s5, s4*s5*s4*s5*s4*s5,
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s1*s2*s3*s1*s2*s3,
s4*s2*s3*s4*s3*s4*s2*s3*s4*s3, s5*s3*s4*s3*s4*s5*s3*s4*s3*s4 ];;
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)( 7, 11)( 8, 13)( 9, 12)( 10, 14)( 16, 17)( 19, 23)
( 20, 25)( 21, 24)( 22, 26)( 28, 29)( 31, 35)( 32, 37)( 33, 36)( 34, 38)
( 39, 75)( 40, 77)( 41, 76)( 42, 78)( 43, 83)( 44, 85)( 45, 84)( 46, 86)
( 47, 79)( 48, 81)( 49, 80)( 50, 82)( 51, 87)( 52, 89)( 53, 88)( 54, 90)
( 55, 95)( 56, 97)( 57, 96)( 58, 98)( 59, 91)( 60, 93)( 61, 92)( 62, 94)
( 63, 99)( 64,101)( 65,100)( 66,102)( 67,107)( 68,109)( 69,108)( 70,110)
( 71,103)( 72,105)( 73,104)( 74,106)(112,113)(115,119)(116,121)(117,120)
(118,122)(124,125)(127,131)(128,133)(129,132)(130,134)(136,137)(139,143)
(140,145)(141,144)(142,146)(147,183)(148,185)(149,184)(150,186)(151,191)
(152,193)(153,192)(154,194)(155,187)(156,189)(157,188)(158,190)(159,195)
(160,197)(161,196)(162,198)(163,203)(164,205)(165,204)(166,206)(167,199)
(168,201)(169,200)(170,202)(171,207)(172,209)(173,208)(174,210)(175,215)
(176,217)(177,216)(178,218)(179,211)(180,213)(181,212)(182,214);
s3 := Sym(218)!( 3,183)( 4,186)( 5,185)( 6,184)( 7,191)( 8,194)( 9,193)
( 10,192)( 11,187)( 12,190)( 13,189)( 14,188)( 15,203)( 16,206)( 17,205)
( 18,204)( 19,199)( 20,202)( 21,201)( 22,200)( 23,195)( 24,198)( 25,197)
( 26,196)( 27,211)( 28,214)( 29,213)( 30,212)( 31,207)( 32,210)( 33,209)
( 34,208)( 35,215)( 36,218)( 37,217)( 38,216)( 39,147)( 40,150)( 41,149)
( 42,148)( 43,155)( 44,158)( 45,157)( 46,156)( 47,151)( 48,154)( 49,153)
( 50,152)( 51,167)( 52,170)( 53,169)( 54,168)( 55,163)( 56,166)( 57,165)
( 58,164)( 59,159)( 60,162)( 61,161)( 62,160)( 63,175)( 64,178)( 65,177)
( 66,176)( 67,171)( 68,174)( 69,173)( 70,172)( 71,179)( 72,182)( 73,181)
( 74,180)( 75,111)( 76,114)( 77,113)( 78,112)( 79,119)( 80,122)( 81,121)
( 82,120)( 83,115)( 84,118)( 85,117)( 86,116)( 87,131)( 88,134)( 89,133)
( 90,132)( 91,127)( 92,130)( 93,129)( 94,128)( 95,123)( 96,126)( 97,125)
( 98,124)( 99,139)(100,142)(101,141)(102,140)(103,135)(104,138)(105,137)
(106,136)(107,143)(108,146)(109,145)(110,144);
s4 := Sym(218)!( 3, 15)( 4, 16)( 5, 17)( 6, 18)( 7, 23)( 8, 24)( 9, 25)
( 10, 26)( 11, 19)( 12, 20)( 13, 21)( 14, 22)( 31, 35)( 32, 36)( 33, 37)
( 34, 38)( 39, 51)( 40, 52)( 41, 53)( 42, 54)( 43, 59)( 44, 60)( 45, 61)
( 46, 62)( 47, 55)( 48, 56)( 49, 57)( 50, 58)( 67, 71)( 68, 72)( 69, 73)
( 70, 74)( 75, 87)( 76, 88)( 77, 89)( 78, 90)( 79, 95)( 80, 96)( 81, 97)
( 82, 98)( 83, 91)( 84, 92)( 85, 93)( 86, 94)(103,107)(104,108)(105,109)
(106,110)(111,123)(112,124)(113,125)(114,126)(115,131)(116,132)(117,133)
(118,134)(119,127)(120,128)(121,129)(122,130)(139,143)(140,144)(141,145)
(142,146)(147,159)(148,160)(149,161)(150,162)(151,167)(152,168)(153,169)
(154,170)(155,163)(156,164)(157,165)(158,166)(175,179)(176,180)(177,181)
(178,182)(183,195)(184,196)(185,197)(186,198)(187,203)(188,204)(189,205)
(190,206)(191,199)(192,200)(193,201)(194,202)(211,215)(212,216)(213,217)
(214,218);
s5 := 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,s5>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4,s5> := Group< s0,s1,s2,s3,s4,s5 | s0*s0, s1*s1, s2*s2,
s3*s3, s4*s4, s5*s5, 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, s0*s5*s0*s5,
s1*s5*s1*s5, s2*s5*s2*s5, s3*s5*s3*s5,
s4*s5*s4*s5*s4*s5, s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s1*s2*s3*s1*s2*s3, s4*s2*s3*s4*s3*s4*s2*s3*s4*s3,
s5*s3*s4*s3*s4*s5*s3*s4*s3*s4 >;
to this polytope