Polytope of Type {4,6,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6,6}*1728c
if this polytope has a name.
Group : SmallGroup(1728,47874)
Rank : 4
Schlafli Type : {4,6,6}
Number of vertices, edges, etc : 8, 72, 108, 18
Order of s0s1s2s3 : 6
Order of s0s1s2s3s2s1 : 2
Special Properties :
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}*864i
3-fold quotients : {4,6,6}*576a, {4,6,6}*576b
4-fold quotients : {2,6,6}*432d
6-fold quotients : {4,3,6}*288, {4,6,6}*288d, {4,6,6}*288e, {4,6,6}*288f
9-fold quotients : {4,6,2}*192
12-fold quotients : {4,3,6}*144, {2,6,6}*144a, {2,6,6}*144b, {2,6,6}*144c
18-fold quotients : {4,3,2}*96, {4,6,2}*96b, {4,6,2}*96c
24-fold quotients : {2,3,6}*72, {2,6,3}*72
36-fold quotients : {4,3,2}*48, {2,2,6}*48, {2,6,2}*48
72-fold quotients : {2,2,3}*24, {2,3,2}*24
108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Irregular Quotients (of which this is a minimal cover):
P/N, where N=<s1*s2*s1*s2*s1*s2> of order 2.
12 facets:
6 of {4,3}*48
6 of {4,6}*96
8 vertex figures:
8 of 2-fold non-regular quotient of {6,6}*216d
P/N, where N=<s0*s1*s0*s1> of order 2.
18 facets:
18 of 2-fold non-regular quotient of {4,6}*96
4 vertex figures:
4 of {6,6}*216d
P/N, where N=<s1*s2*s3*s2*s1*s3> of order 3.
6 facets:
6 of {4,6}*96
8 vertex figures:
8 of 3-fold non-regular quotient of {6,6}*216d
P/N, where N=<s0*s1*s0*s1, s2*s1*s2*s3*s2*s3*s2*s1*s3*s2> of order 4.
9 facets:
9 of 2-fold non-regular quotient of {4,6}*96
4 vertex figures:
4 of 2-fold non-regular quotient of {6,6}*216d
P/N, where N=<s0*s1*s0*s1, s1*s2*s1*s2*s1*s2> of order 4.
12 facets:
6 of 2-fold non-regular quotient of {4,3}*48
6 of 2-fold non-regular quotient of {4,6}*96
4 vertex figures:
4 of 2-fold non-regular quotient of {6,6}*216d
P/N, where N=<s0*s1*s0*s1, s1*s2*s3*s2*s1*s3> of order 6.
6 facets:
6 of 2-fold non-regular quotient of {4,6}*96
4 vertex figures:
4 of 3-fold non-regular quotient of {6,6}*216d
Permutation Representation (GAP) :
s0 := ( 1,111)( 2,112)( 3,109)( 4,110)( 5,115)( 6,116)( 7,113)( 8,114)( 9,119)( 10,120)( 11,117)( 12,118)( 13,123)( 14,124)( 15,121)( 16,122)( 17,127)( 18,128)( 19,125)( 20,126)( 21,131)( 22,132)( 23,129)( 24,130)( 25,135)( 26,136)( 27,133)( 28,134)( 29,139)( 30,140)( 31,137)( 32,138)( 33,143)( 34,144)( 35,141)( 36,142)( 37,147)( 38,148)( 39,145)( 40,146)( 41,151)( 42,152)( 43,149)( 44,150)( 45,155)( 46,156)( 47,153)( 48,154)( 49,159)( 50,160)( 51,157)( 52,158)( 53,163)( 54,164)( 55,161)( 56,162)( 57,167)( 58,168)( 59,165)( 60,166)( 61,171)( 62,172)( 63,169)( 64,170)( 65,175)( 66,176)( 67,173)( 68,174)( 69,179)( 70,180)( 71,177)( 72,178)( 73,183)( 74,184)( 75,181)( 76,182)( 77,187)( 78,188)( 79,185)( 80,186)( 81,191)( 82,192)( 83,189)( 84,190)( 85,195)( 86,196)( 87,193)( 88,194)( 89,199)( 90,200)( 91,197)( 92,198)( 93,203)( 94,204)( 95,201)( 96,202)( 97,207)( 98,208)( 99,205)(100,206)(101,211)(102,212)(103,209)(104,210)(105,215)(106,216)(107,213)(108,214);;
s1 := ( 3, 4)( 7, 8)( 11, 12)( 13, 25)( 14, 26)( 15, 28)( 16, 27)( 17, 29)( 18, 30)( 19, 32)( 20, 31)( 21, 33)( 22, 34)( 23, 36)( 24, 35)( 37, 73)( 38, 74)( 39, 76)( 40, 75)( 41, 77)( 42, 78)( 43, 80)( 44, 79)( 45, 81)( 46, 82)( 47, 84)( 48, 83)( 49, 97)( 50, 98)( 51,100)( 52, 99)( 53,101)( 54,102)( 55,104)( 56,103)( 57,105)( 58,106)( 59,108)( 60,107)( 61, 85)( 62, 86)( 63, 88)( 64, 87)( 65, 89)( 66, 90)( 67, 92)( 68, 91)( 69, 93)( 70, 94)( 71, 96)( 72, 95)(111,112)(115,116)(119,120)(121,133)(122,134)(123,136)(124,135)(125,137)(126,138)(127,140)(128,139)(129,141)(130,142)(131,144)(132,143)(145,181)(146,182)(147,184)(148,183)(149,185)(150,186)(151,188)(152,187)(153,189)(154,190)(155,192)(156,191)(157,205)(158,206)(159,208)(160,207)(161,209)(162,210)(163,212)(164,211)(165,213)(166,214)(167,216)(168,215)(169,193)(170,194)(171,196)(172,195)(173,197)(174,198)(175,200)(176,199)(177,201)(178,202)(179,204)(180,203);;
s2 := ( 1, 49)( 2, 52)( 3, 51)( 4, 50)( 5, 57)( 6, 60)( 7, 59)( 8, 58)( 9, 53)( 10, 56)( 11, 55)( 12, 54)( 13, 37)( 14, 40)( 15, 39)( 16, 38)( 17, 45)( 18, 48)( 19, 47)( 20, 46)( 21, 41)( 22, 44)( 23, 43)( 24, 42)( 25, 61)( 26, 64)( 27, 63)( 28, 62)( 29, 69)( 30, 72)( 31, 71)( 32, 70)( 33, 65)( 34, 68)( 35, 67)( 36, 66)( 73, 85)( 74, 88)( 75, 87)( 76, 86)( 77, 93)( 78, 96)( 79, 95)( 80, 94)( 81, 89)( 82, 92)( 83, 91)( 84, 90)( 98,100)(101,105)(102,108)(103,107)(104,106)(109,157)(110,160)(111,159)(112,158)(113,165)(114,168)(115,167)(116,166)(117,161)(118,164)(119,163)(120,162)(121,145)(122,148)(123,147)(124,146)(125,153)(126,156)(127,155)(128,154)(129,149)(130,152)(131,151)(132,150)(133,169)(134,172)(135,171)(136,170)(137,177)(138,180)(139,179)(140,178)(141,173)(142,176)(143,175)(144,174)(181,193)(182,196)(183,195)(184,194)(185,201)(186,204)(187,203)(188,202)(189,197)(190,200)(191,199)(192,198)(206,208)(209,213)(210,216)(211,215)(212,214);;
s3 := ( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 13, 29)( 14, 30)( 15, 31)( 16, 32)( 17, 25)( 18, 26)( 19, 27)( 20, 28)( 21, 33)( 22, 34)( 23, 35)( 24, 36)( 37, 41)( 38, 42)( 39, 43)( 40, 44)( 49, 65)( 50, 66)( 51, 67)( 52, 68)( 53, 61)( 54, 62)( 55, 63)( 56, 64)( 57, 69)( 58, 70)( 59, 71)( 60, 72)( 73, 77)( 74, 78)( 75, 79)( 76, 80)( 85,101)( 86,102)( 87,103)( 88,104)( 89, 97)( 90, 98)( 91, 99)( 92,100)( 93,105)( 94,106)( 95,107)( 96,108)(109,113)(110,114)(111,115)(112,116)(121,137)(122,138)(123,139)(124,140)(125,133)(126,134)(127,135)(128,136)(129,141)(130,142)(131,143)(132,144)(145,149)(146,150)(147,151)(148,152)(157,173)(158,174)(159,175)(160,176)(161,169)(162,170)(163,171)(164,172)(165,177)(166,178)(167,179)(168,180)(181,185)(182,186)(183,187)(184,188)(193,209)(194,210)(195,211)(196,212)(197,205)(198,206)(199,207)(200,208)(201,213)(202,214)(203,215)(204,216);;
poly := Group([s0,s1,s2,s3]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s0*s1*s2*s3*s2*s1*s0*s1*s2*s3*s2*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,
s1*s2*s3*s2*s3*s1*s2*s1*s2*s3*s2*s3*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(216)!( 1,111)( 2,112)( 3,109)( 4,110)( 5,115)( 6,116)( 7,113)( 8,114)( 9,119)( 10,120)( 11,117)( 12,118)( 13,123)( 14,124)( 15,121)( 16,122)( 17,127)( 18,128)( 19,125)( 20,126)( 21,131)( 22,132)( 23,129)( 24,130)( 25,135)( 26,136)( 27,133)( 28,134)( 29,139)( 30,140)( 31,137)( 32,138)( 33,143)( 34,144)( 35,141)( 36,142)( 37,147)( 38,148)( 39,145)( 40,146)( 41,151)( 42,152)( 43,149)( 44,150)( 45,155)( 46,156)( 47,153)( 48,154)( 49,159)( 50,160)( 51,157)( 52,158)( 53,163)( 54,164)( 55,161)( 56,162)( 57,167)( 58,168)( 59,165)( 60,166)( 61,171)( 62,172)( 63,169)( 64,170)( 65,175)( 66,176)( 67,173)( 68,174)( 69,179)( 70,180)( 71,177)( 72,178)( 73,183)( 74,184)( 75,181)( 76,182)( 77,187)( 78,188)( 79,185)( 80,186)( 81,191)( 82,192)( 83,189)( 84,190)( 85,195)( 86,196)( 87,193)( 88,194)( 89,199)( 90,200)( 91,197)( 92,198)( 93,203)( 94,204)( 95,201)( 96,202)( 97,207)( 98,208)( 99,205)(100,206)(101,211)(102,212)(103,209)(104,210)(105,215)(106,216)(107,213)(108,214);
s1 := Sym(216)!( 3, 4)( 7, 8)( 11, 12)( 13, 25)( 14, 26)( 15, 28)( 16, 27)( 17, 29)( 18, 30)( 19, 32)( 20, 31)( 21, 33)( 22, 34)( 23, 36)( 24, 35)( 37, 73)( 38, 74)( 39, 76)( 40, 75)( 41, 77)( 42, 78)( 43, 80)( 44, 79)( 45, 81)( 46, 82)( 47, 84)( 48, 83)( 49, 97)( 50, 98)( 51,100)( 52, 99)( 53,101)( 54,102)( 55,104)( 56,103)( 57,105)( 58,106)( 59,108)( 60,107)( 61, 85)( 62, 86)( 63, 88)( 64, 87)( 65, 89)( 66, 90)( 67, 92)( 68, 91)( 69, 93)( 70, 94)( 71, 96)( 72, 95)(111,112)(115,116)(119,120)(121,133)(122,134)(123,136)(124,135)(125,137)(126,138)(127,140)(128,139)(129,141)(130,142)(131,144)(132,143)(145,181)(146,182)(147,184)(148,183)(149,185)(150,186)(151,188)(152,187)(153,189)(154,190)(155,192)(156,191)(157,205)(158,206)(159,208)(160,207)(161,209)(162,210)(163,212)(164,211)(165,213)(166,214)(167,216)(168,215)(169,193)(170,194)(171,196)(172,195)(173,197)(174,198)(175,200)(176,199)(177,201)(178,202)(179,204)(180,203);
s2 := Sym(216)!( 1, 49)( 2, 52)( 3, 51)( 4, 50)( 5, 57)( 6, 60)( 7, 59)( 8, 58)( 9, 53)( 10, 56)( 11, 55)( 12, 54)( 13, 37)( 14, 40)( 15, 39)( 16, 38)( 17, 45)( 18, 48)( 19, 47)( 20, 46)( 21, 41)( 22, 44)( 23, 43)( 24, 42)( 25, 61)( 26, 64)( 27, 63)( 28, 62)( 29, 69)( 30, 72)( 31, 71)( 32, 70)( 33, 65)( 34, 68)( 35, 67)( 36, 66)( 73, 85)( 74, 88)( 75, 87)( 76, 86)( 77, 93)( 78, 96)( 79, 95)( 80, 94)( 81, 89)( 82, 92)( 83, 91)( 84, 90)( 98,100)(101,105)(102,108)(103,107)(104,106)(109,157)(110,160)(111,159)(112,158)(113,165)(114,168)(115,167)(116,166)(117,161)(118,164)(119,163)(120,162)(121,145)(122,148)(123,147)(124,146)(125,153)(126,156)(127,155)(128,154)(129,149)(130,152)(131,151)(132,150)(133,169)(134,172)(135,171)(136,170)(137,177)(138,180)(139,179)(140,178)(141,173)(142,176)(143,175)(144,174)(181,193)(182,196)(183,195)(184,194)(185,201)(186,204)(187,203)(188,202)(189,197)(190,200)(191,199)(192,198)(206,208)(209,213)(210,216)(211,215)(212,214);
s3 := Sym(216)!( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 13, 29)( 14, 30)( 15, 31)( 16, 32)( 17, 25)( 18, 26)( 19, 27)( 20, 28)( 21, 33)( 22, 34)( 23, 35)( 24, 36)( 37, 41)( 38, 42)( 39, 43)( 40, 44)( 49, 65)( 50, 66)( 51, 67)( 52, 68)( 53, 61)( 54, 62)( 55, 63)( 56, 64)( 57, 69)( 58, 70)( 59, 71)( 60, 72)( 73, 77)( 74, 78)( 75, 79)( 76, 80)( 85,101)( 86,102)( 87,103)( 88,104)( 89, 97)( 90, 98)( 91, 99)( 92,100)( 93,105)( 94,106)( 95,107)( 96,108)(109,113)(110,114)(111,115)(112,116)(121,137)(122,138)(123,139)(124,140)(125,133)(126,134)(127,135)(128,136)(129,141)(130,142)(131,143)(132,144)(145,149)(146,150)(147,151)(148,152)(157,173)(158,174)(159,175)(160,176)(161,169)(162,170)(163,171)(164,172)(165,177)(166,178)(167,179)(168,180)(181,185)(182,186)(183,187)(184,188)(193,209)(194,210)(195,211)(196,212)(197,205)(198,206)(199,207)(200,208)(201,213)(202,214)(203,215)(204,216);
poly := sub<Sym(216)|s0,s1,s2,s3>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2,
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s0*s1*s2*s3*s2*s1*s0*s1*s2*s3*s2*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,
s1*s2*s3*s2*s3*s1*s2*s1*s2*s3*s2*s3*s1*s2 >;
References : None.
to this polytope