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