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