Polytope of Type {6,6,4,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,4,6}*1728c
if this polytope has a name.
Group : SmallGroup(1728,47409)
Rank : 5
Schlafli Type : {6,6,4,6}
Number of vertices, edges, etc : 6, 18, 12, 12, 6
Order of s0s1s2s3s4 : 12
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
Universal
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 : {3,6,4,6}*864, {6,6,2,6}*864c
3-fold quotients : {6,2,4,6}*576a, {6,6,4,2}*576c
4-fold quotients : {3,6,2,6}*432, {6,6,2,3}*432c
6-fold quotients : {3,2,4,6}*288a, {3,6,4,2}*288, {6,2,2,6}*288, {6,6,2,2}*288c
8-fold quotients : {3,6,2,3}*216
9-fold quotients : {2,2,4,6}*192a, {6,2,4,2}*192
12-fold quotients : {3,2,2,6}*144, {3,6,2,2}*144, {6,2,2,3}*144
18-fold quotients : {3,2,4,2}*96, {2,2,2,6}*96, {6,2,2,2}*96
24-fold quotients : {3,2,2,3}*72
27-fold quotients : {2,2,4,2}*64
36-fold quotients : {2,2,2,3}*48, {3,2,2,2}*48
54-fold quotients : {2,2,2,2}*32
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 2, 3)( 5, 6)( 8, 9)( 10, 19)( 11, 21)( 12, 20)( 13, 22)( 14, 24)
( 15, 23)( 16, 25)( 17, 27)( 18, 26)( 29, 30)( 32, 33)( 35, 36)( 37, 46)
( 38, 48)( 39, 47)( 40, 49)( 41, 51)( 42, 50)( 43, 52)( 44, 54)( 45, 53)
( 56, 57)( 59, 60)( 62, 63)( 64, 73)( 65, 75)( 66, 74)( 67, 76)( 68, 78)
( 69, 77)( 70, 79)( 71, 81)( 72, 80)( 83, 84)( 86, 87)( 89, 90)( 91,100)
( 92,102)( 93,101)( 94,103)( 95,105)( 96,104)( 97,106)( 98,108)( 99,107)
(110,111)(113,114)(116,117)(118,127)(119,129)(120,128)(121,130)(122,132)
(123,131)(124,133)(125,135)(126,134)(137,138)(140,141)(143,144)(145,154)
(146,156)(147,155)(148,157)(149,159)(150,158)(151,160)(152,162)(153,161)
(164,165)(167,168)(170,171)(172,181)(173,183)(174,182)(175,184)(176,186)
(177,185)(178,187)(179,189)(180,188)(191,192)(194,195)(197,198)(199,208)
(200,210)(201,209)(202,211)(203,213)(204,212)(205,214)(206,216)(207,215);;
s1 := ( 1,119)( 2,118)( 3,120)( 4,122)( 5,121)( 6,123)( 7,125)( 8,124)
( 9,126)( 10,110)( 11,109)( 12,111)( 13,113)( 14,112)( 15,114)( 16,116)
( 17,115)( 18,117)( 19,128)( 20,127)( 21,129)( 22,131)( 23,130)( 24,132)
( 25,134)( 26,133)( 27,135)( 28,146)( 29,145)( 30,147)( 31,149)( 32,148)
( 33,150)( 34,152)( 35,151)( 36,153)( 37,137)( 38,136)( 39,138)( 40,140)
( 41,139)( 42,141)( 43,143)( 44,142)( 45,144)( 46,155)( 47,154)( 48,156)
( 49,158)( 50,157)( 51,159)( 52,161)( 53,160)( 54,162)( 55,173)( 56,172)
( 57,174)( 58,176)( 59,175)( 60,177)( 61,179)( 62,178)( 63,180)( 64,164)
( 65,163)( 66,165)( 67,167)( 68,166)( 69,168)( 70,170)( 71,169)( 72,171)
( 73,182)( 74,181)( 75,183)( 76,185)( 77,184)( 78,186)( 79,188)( 80,187)
( 81,189)( 82,200)( 83,199)( 84,201)( 85,203)( 86,202)( 87,204)( 88,206)
( 89,205)( 90,207)( 91,191)( 92,190)( 93,192)( 94,194)( 95,193)( 96,195)
( 97,197)( 98,196)( 99,198)(100,209)(101,208)(102,210)(103,212)(104,211)
(105,213)(106,215)(107,214)(108,216);;
s2 := ( 10, 19)( 11, 20)( 12, 21)( 13, 22)( 14, 23)( 15, 24)( 16, 25)( 17, 26)
( 18, 27)( 37, 46)( 38, 47)( 39, 48)( 40, 49)( 41, 50)( 42, 51)( 43, 52)
( 44, 53)( 45, 54)( 55, 82)( 56, 83)( 57, 84)( 58, 85)( 59, 86)( 60, 87)
( 61, 88)( 62, 89)( 63, 90)( 64,100)( 65,101)( 66,102)( 67,103)( 68,104)
( 69,105)( 70,106)( 71,107)( 72,108)( 73, 91)( 74, 92)( 75, 93)( 76, 94)
( 77, 95)( 78, 96)( 79, 97)( 80, 98)( 81, 99)(118,127)(119,128)(120,129)
(121,130)(122,131)(123,132)(124,133)(125,134)(126,135)(145,154)(146,155)
(147,156)(148,157)(149,158)(150,159)(151,160)(152,161)(153,162)(163,190)
(164,191)(165,192)(166,193)(167,194)(168,195)(169,196)(170,197)(171,198)
(172,208)(173,209)(174,210)(175,211)(176,212)(177,213)(178,214)(179,215)
(180,216)(181,199)(182,200)(183,201)(184,202)(185,203)(186,204)(187,205)
(188,206)(189,207);;
s3 := ( 1, 55)( 2, 56)( 3, 57)( 4, 61)( 5, 62)( 6, 63)( 7, 58)( 8, 59)
( 9, 60)( 10, 64)( 11, 65)( 12, 66)( 13, 70)( 14, 71)( 15, 72)( 16, 67)
( 17, 68)( 18, 69)( 19, 73)( 20, 74)( 21, 75)( 22, 79)( 23, 80)( 24, 81)
( 25, 76)( 26, 77)( 27, 78)( 28, 82)( 29, 83)( 30, 84)( 31, 88)( 32, 89)
( 33, 90)( 34, 85)( 35, 86)( 36, 87)( 37, 91)( 38, 92)( 39, 93)( 40, 97)
( 41, 98)( 42, 99)( 43, 94)( 44, 95)( 45, 96)( 46,100)( 47,101)( 48,102)
( 49,106)( 50,107)( 51,108)( 52,103)( 53,104)( 54,105)(109,163)(110,164)
(111,165)(112,169)(113,170)(114,171)(115,166)(116,167)(117,168)(118,172)
(119,173)(120,174)(121,178)(122,179)(123,180)(124,175)(125,176)(126,177)
(127,181)(128,182)(129,183)(130,187)(131,188)(132,189)(133,184)(134,185)
(135,186)(136,190)(137,191)(138,192)(139,196)(140,197)(141,198)(142,193)
(143,194)(144,195)(145,199)(146,200)(147,201)(148,205)(149,206)(150,207)
(151,202)(152,203)(153,204)(154,208)(155,209)(156,210)(157,214)(158,215)
(159,216)(160,211)(161,212)(162,213);;
s4 := ( 1, 4)( 2, 5)( 3, 6)( 10, 13)( 11, 14)( 12, 15)( 19, 22)( 20, 23)
( 21, 24)( 28, 31)( 29, 32)( 30, 33)( 37, 40)( 38, 41)( 39, 42)( 46, 49)
( 47, 50)( 48, 51)( 55, 58)( 56, 59)( 57, 60)( 64, 67)( 65, 68)( 66, 69)
( 73, 76)( 74, 77)( 75, 78)( 82, 85)( 83, 86)( 84, 87)( 91, 94)( 92, 95)
( 93, 96)(100,103)(101,104)(102,105)(109,112)(110,113)(111,114)(118,121)
(119,122)(120,123)(127,130)(128,131)(129,132)(136,139)(137,140)(138,141)
(145,148)(146,149)(147,150)(154,157)(155,158)(156,159)(163,166)(164,167)
(165,168)(172,175)(173,176)(174,177)(181,184)(182,185)(183,186)(190,193)
(191,194)(192,195)(199,202)(200,203)(201,204)(208,211)(209,212)(210,213);;
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, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3, s2*s3*s4*s3*s2*s3*s4*s3,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(216)!( 2, 3)( 5, 6)( 8, 9)( 10, 19)( 11, 21)( 12, 20)( 13, 22)
( 14, 24)( 15, 23)( 16, 25)( 17, 27)( 18, 26)( 29, 30)( 32, 33)( 35, 36)
( 37, 46)( 38, 48)( 39, 47)( 40, 49)( 41, 51)( 42, 50)( 43, 52)( 44, 54)
( 45, 53)( 56, 57)( 59, 60)( 62, 63)( 64, 73)( 65, 75)( 66, 74)( 67, 76)
( 68, 78)( 69, 77)( 70, 79)( 71, 81)( 72, 80)( 83, 84)( 86, 87)( 89, 90)
( 91,100)( 92,102)( 93,101)( 94,103)( 95,105)( 96,104)( 97,106)( 98,108)
( 99,107)(110,111)(113,114)(116,117)(118,127)(119,129)(120,128)(121,130)
(122,132)(123,131)(124,133)(125,135)(126,134)(137,138)(140,141)(143,144)
(145,154)(146,156)(147,155)(148,157)(149,159)(150,158)(151,160)(152,162)
(153,161)(164,165)(167,168)(170,171)(172,181)(173,183)(174,182)(175,184)
(176,186)(177,185)(178,187)(179,189)(180,188)(191,192)(194,195)(197,198)
(199,208)(200,210)(201,209)(202,211)(203,213)(204,212)(205,214)(206,216)
(207,215);
s1 := Sym(216)!( 1,119)( 2,118)( 3,120)( 4,122)( 5,121)( 6,123)( 7,125)
( 8,124)( 9,126)( 10,110)( 11,109)( 12,111)( 13,113)( 14,112)( 15,114)
( 16,116)( 17,115)( 18,117)( 19,128)( 20,127)( 21,129)( 22,131)( 23,130)
( 24,132)( 25,134)( 26,133)( 27,135)( 28,146)( 29,145)( 30,147)( 31,149)
( 32,148)( 33,150)( 34,152)( 35,151)( 36,153)( 37,137)( 38,136)( 39,138)
( 40,140)( 41,139)( 42,141)( 43,143)( 44,142)( 45,144)( 46,155)( 47,154)
( 48,156)( 49,158)( 50,157)( 51,159)( 52,161)( 53,160)( 54,162)( 55,173)
( 56,172)( 57,174)( 58,176)( 59,175)( 60,177)( 61,179)( 62,178)( 63,180)
( 64,164)( 65,163)( 66,165)( 67,167)( 68,166)( 69,168)( 70,170)( 71,169)
( 72,171)( 73,182)( 74,181)( 75,183)( 76,185)( 77,184)( 78,186)( 79,188)
( 80,187)( 81,189)( 82,200)( 83,199)( 84,201)( 85,203)( 86,202)( 87,204)
( 88,206)( 89,205)( 90,207)( 91,191)( 92,190)( 93,192)( 94,194)( 95,193)
( 96,195)( 97,197)( 98,196)( 99,198)(100,209)(101,208)(102,210)(103,212)
(104,211)(105,213)(106,215)(107,214)(108,216);
s2 := Sym(216)!( 10, 19)( 11, 20)( 12, 21)( 13, 22)( 14, 23)( 15, 24)( 16, 25)
( 17, 26)( 18, 27)( 37, 46)( 38, 47)( 39, 48)( 40, 49)( 41, 50)( 42, 51)
( 43, 52)( 44, 53)( 45, 54)( 55, 82)( 56, 83)( 57, 84)( 58, 85)( 59, 86)
( 60, 87)( 61, 88)( 62, 89)( 63, 90)( 64,100)( 65,101)( 66,102)( 67,103)
( 68,104)( 69,105)( 70,106)( 71,107)( 72,108)( 73, 91)( 74, 92)( 75, 93)
( 76, 94)( 77, 95)( 78, 96)( 79, 97)( 80, 98)( 81, 99)(118,127)(119,128)
(120,129)(121,130)(122,131)(123,132)(124,133)(125,134)(126,135)(145,154)
(146,155)(147,156)(148,157)(149,158)(150,159)(151,160)(152,161)(153,162)
(163,190)(164,191)(165,192)(166,193)(167,194)(168,195)(169,196)(170,197)
(171,198)(172,208)(173,209)(174,210)(175,211)(176,212)(177,213)(178,214)
(179,215)(180,216)(181,199)(182,200)(183,201)(184,202)(185,203)(186,204)
(187,205)(188,206)(189,207);
s3 := Sym(216)!( 1, 55)( 2, 56)( 3, 57)( 4, 61)( 5, 62)( 6, 63)( 7, 58)
( 8, 59)( 9, 60)( 10, 64)( 11, 65)( 12, 66)( 13, 70)( 14, 71)( 15, 72)
( 16, 67)( 17, 68)( 18, 69)( 19, 73)( 20, 74)( 21, 75)( 22, 79)( 23, 80)
( 24, 81)( 25, 76)( 26, 77)( 27, 78)( 28, 82)( 29, 83)( 30, 84)( 31, 88)
( 32, 89)( 33, 90)( 34, 85)( 35, 86)( 36, 87)( 37, 91)( 38, 92)( 39, 93)
( 40, 97)( 41, 98)( 42, 99)( 43, 94)( 44, 95)( 45, 96)( 46,100)( 47,101)
( 48,102)( 49,106)( 50,107)( 51,108)( 52,103)( 53,104)( 54,105)(109,163)
(110,164)(111,165)(112,169)(113,170)(114,171)(115,166)(116,167)(117,168)
(118,172)(119,173)(120,174)(121,178)(122,179)(123,180)(124,175)(125,176)
(126,177)(127,181)(128,182)(129,183)(130,187)(131,188)(132,189)(133,184)
(134,185)(135,186)(136,190)(137,191)(138,192)(139,196)(140,197)(141,198)
(142,193)(143,194)(144,195)(145,199)(146,200)(147,201)(148,205)(149,206)
(150,207)(151,202)(152,203)(153,204)(154,208)(155,209)(156,210)(157,214)
(158,215)(159,216)(160,211)(161,212)(162,213);
s4 := Sym(216)!( 1, 4)( 2, 5)( 3, 6)( 10, 13)( 11, 14)( 12, 15)( 19, 22)
( 20, 23)( 21, 24)( 28, 31)( 29, 32)( 30, 33)( 37, 40)( 38, 41)( 39, 42)
( 46, 49)( 47, 50)( 48, 51)( 55, 58)( 56, 59)( 57, 60)( 64, 67)( 65, 68)
( 66, 69)( 73, 76)( 74, 77)( 75, 78)( 82, 85)( 83, 86)( 84, 87)( 91, 94)
( 92, 95)( 93, 96)(100,103)(101,104)(102,105)(109,112)(110,113)(111,114)
(118,121)(119,122)(120,123)(127,130)(128,131)(129,132)(136,139)(137,140)
(138,141)(145,148)(146,149)(147,150)(154,157)(155,158)(156,159)(163,166)
(164,167)(165,168)(172,175)(173,176)(174,177)(181,184)(182,185)(183,186)
(190,193)(191,194)(192,195)(199,202)(200,203)(201,204)(208,211)(209,212)
(210,213);
poly := sub<Sym(216)|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, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3, s2*s3*s4*s3*s2*s3*s4*s3,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;
References : None.
to this polytope