Polytope of Type {6,12,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,12,6}*1728e
if this polytope has a name.
Group : SmallGroup(1728,47874)
Rank : 4
Schlafli Type : {6,12,6}
Number of vertices, edges, etc : 6, 72, 72, 12
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) :
   3-fold quotients : {6,4,6}*576a, {2,12,6}*576a
   4-fold quotients : {6,6,6}*432b
   6-fold quotients : {6,4,3}*288, {2,12,6}*288d
   9-fold quotients : {2,4,6}*192
   12-fold quotients : {2,6,6}*144a, {6,2,6}*144, {6,6,2}*144a
   18-fold quotients : {2,4,3}*96, {2,4,6}*96b, {2,4,6}*96c
   24-fold quotients : {3,2,6}*72, {6,2,3}*72
   36-fold quotients : {2,4,3}*48, {2,2,6}*48, {2,6,2}*48, {6,2,2}*48
   48-fold quotients : {3,2,3}*36
   72-fold quotients : {2,2,3}*24, {2,3,2}*24, {3,2,2}*24
   108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := ( 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);;
s1 := (  1, 15)(  2, 16)(  3, 13)(  4, 14)(  5, 23)(  6, 24)(  7, 21)(  8, 22)
(  9, 19)( 10, 20)( 11, 17)( 12, 18)( 25, 27)( 26, 28)( 29, 35)( 30, 36)
( 31, 33)( 32, 34)( 37, 51)( 38, 52)( 39, 49)( 40, 50)( 41, 59)( 42, 60)
( 43, 57)( 44, 58)( 45, 55)( 46, 56)( 47, 53)( 48, 54)( 61, 63)( 62, 64)
( 65, 71)( 66, 72)( 67, 69)( 68, 70)( 73, 87)( 74, 88)( 75, 85)( 76, 86)
( 77, 95)( 78, 96)( 79, 93)( 80, 94)( 81, 91)( 82, 92)( 83, 89)( 84, 90)
( 97, 99)( 98,100)(101,107)(102,108)(103,105)(104,106)(109,123)(110,124)
(111,121)(112,122)(113,131)(114,132)(115,129)(116,130)(117,127)(118,128)
(119,125)(120,126)(133,135)(134,136)(137,143)(138,144)(139,141)(140,142)
(145,159)(146,160)(147,157)(148,158)(149,167)(150,168)(151,165)(152,166)
(153,163)(154,164)(155,161)(156,162)(169,171)(170,172)(173,179)(174,180)
(175,177)(176,178)(181,195)(182,196)(183,193)(184,194)(185,203)(186,204)
(187,201)(188,202)(189,199)(190,200)(191,197)(192,198)(205,207)(206,208)
(209,215)(210,216)(211,213)(212,214);;
s2 := (  1,  5)(  2,  6)(  3,  8)(  4,  7)( 11, 12)( 13, 17)( 14, 18)( 15, 20)
( 16, 19)( 23, 24)( 25, 29)( 26, 30)( 27, 32)( 28, 31)( 35, 36)( 37, 77)
( 38, 78)( 39, 80)( 40, 79)( 41, 73)( 42, 74)( 43, 76)( 44, 75)( 45, 81)
( 46, 82)( 47, 84)( 48, 83)( 49, 89)( 50, 90)( 51, 92)( 52, 91)( 53, 85)
( 54, 86)( 55, 88)( 56, 87)( 57, 93)( 58, 94)( 59, 96)( 60, 95)( 61,101)
( 62,102)( 63,104)( 64,103)( 65, 97)( 66, 98)( 67,100)( 68, 99)( 69,105)
( 70,106)( 71,108)( 72,107)(109,113)(110,114)(111,116)(112,115)(119,120)
(121,125)(122,126)(123,128)(124,127)(131,132)(133,137)(134,138)(135,140)
(136,139)(143,144)(145,185)(146,186)(147,188)(148,187)(149,181)(150,182)
(151,184)(152,183)(153,189)(154,190)(155,192)(156,191)(157,197)(158,198)
(159,200)(160,199)(161,193)(162,194)(163,196)(164,195)(165,201)(166,202)
(167,204)(168,203)(169,209)(170,210)(171,212)(172,211)(173,205)(174,206)
(175,208)(176,207)(177,213)(178,214)(179,216)(180,215);;
s3 := (  1,145)(  2,148)(  3,147)(  4,146)(  5,149)(  6,152)(  7,151)(  8,150)
(  9,153)( 10,156)( 11,155)( 12,154)( 13,157)( 14,160)( 15,159)( 16,158)
( 17,161)( 18,164)( 19,163)( 20,162)( 21,165)( 22,168)( 23,167)( 24,166)
( 25,169)( 26,172)( 27,171)( 28,170)( 29,173)( 30,176)( 31,175)( 32,174)
( 33,177)( 34,180)( 35,179)( 36,178)( 37,109)( 38,112)( 39,111)( 40,110)
( 41,113)( 42,116)( 43,115)( 44,114)( 45,117)( 46,120)( 47,119)( 48,118)
( 49,121)( 50,124)( 51,123)( 52,122)( 53,125)( 54,128)( 55,127)( 56,126)
( 57,129)( 58,132)( 59,131)( 60,130)( 61,133)( 62,136)( 63,135)( 64,134)
( 65,137)( 66,140)( 67,139)( 68,138)( 69,141)( 70,144)( 71,143)( 72,142)
( 73,181)( 74,184)( 75,183)( 76,182)( 77,185)( 78,188)( 79,187)( 80,186)
( 81,189)( 82,192)( 83,191)( 84,190)( 85,193)( 86,196)( 87,195)( 88,194)
( 89,197)( 90,200)( 91,199)( 92,198)( 93,201)( 94,204)( 95,203)( 96,202)
( 97,205)( 98,208)( 99,207)(100,206)(101,209)(102,212)(103,211)(104,210)
(105,213)(106,216)(107,215)(108,214);;
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*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s1*s2, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s1*s2*s3*s2*s3*s1*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := 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);
s1 := Sym(216)!(  1, 15)(  2, 16)(  3, 13)(  4, 14)(  5, 23)(  6, 24)(  7, 21)
(  8, 22)(  9, 19)( 10, 20)( 11, 17)( 12, 18)( 25, 27)( 26, 28)( 29, 35)
( 30, 36)( 31, 33)( 32, 34)( 37, 51)( 38, 52)( 39, 49)( 40, 50)( 41, 59)
( 42, 60)( 43, 57)( 44, 58)( 45, 55)( 46, 56)( 47, 53)( 48, 54)( 61, 63)
( 62, 64)( 65, 71)( 66, 72)( 67, 69)( 68, 70)( 73, 87)( 74, 88)( 75, 85)
( 76, 86)( 77, 95)( 78, 96)( 79, 93)( 80, 94)( 81, 91)( 82, 92)( 83, 89)
( 84, 90)( 97, 99)( 98,100)(101,107)(102,108)(103,105)(104,106)(109,123)
(110,124)(111,121)(112,122)(113,131)(114,132)(115,129)(116,130)(117,127)
(118,128)(119,125)(120,126)(133,135)(134,136)(137,143)(138,144)(139,141)
(140,142)(145,159)(146,160)(147,157)(148,158)(149,167)(150,168)(151,165)
(152,166)(153,163)(154,164)(155,161)(156,162)(169,171)(170,172)(173,179)
(174,180)(175,177)(176,178)(181,195)(182,196)(183,193)(184,194)(185,203)
(186,204)(187,201)(188,202)(189,199)(190,200)(191,197)(192,198)(205,207)
(206,208)(209,215)(210,216)(211,213)(212,214);
s2 := Sym(216)!(  1,  5)(  2,  6)(  3,  8)(  4,  7)( 11, 12)( 13, 17)( 14, 18)
( 15, 20)( 16, 19)( 23, 24)( 25, 29)( 26, 30)( 27, 32)( 28, 31)( 35, 36)
( 37, 77)( 38, 78)( 39, 80)( 40, 79)( 41, 73)( 42, 74)( 43, 76)( 44, 75)
( 45, 81)( 46, 82)( 47, 84)( 48, 83)( 49, 89)( 50, 90)( 51, 92)( 52, 91)
( 53, 85)( 54, 86)( 55, 88)( 56, 87)( 57, 93)( 58, 94)( 59, 96)( 60, 95)
( 61,101)( 62,102)( 63,104)( 64,103)( 65, 97)( 66, 98)( 67,100)( 68, 99)
( 69,105)( 70,106)( 71,108)( 72,107)(109,113)(110,114)(111,116)(112,115)
(119,120)(121,125)(122,126)(123,128)(124,127)(131,132)(133,137)(134,138)
(135,140)(136,139)(143,144)(145,185)(146,186)(147,188)(148,187)(149,181)
(150,182)(151,184)(152,183)(153,189)(154,190)(155,192)(156,191)(157,197)
(158,198)(159,200)(160,199)(161,193)(162,194)(163,196)(164,195)(165,201)
(166,202)(167,204)(168,203)(169,209)(170,210)(171,212)(172,211)(173,205)
(174,206)(175,208)(176,207)(177,213)(178,214)(179,216)(180,215);
s3 := Sym(216)!(  1,145)(  2,148)(  3,147)(  4,146)(  5,149)(  6,152)(  7,151)
(  8,150)(  9,153)( 10,156)( 11,155)( 12,154)( 13,157)( 14,160)( 15,159)
( 16,158)( 17,161)( 18,164)( 19,163)( 20,162)( 21,165)( 22,168)( 23,167)
( 24,166)( 25,169)( 26,172)( 27,171)( 28,170)( 29,173)( 30,176)( 31,175)
( 32,174)( 33,177)( 34,180)( 35,179)( 36,178)( 37,109)( 38,112)( 39,111)
( 40,110)( 41,113)( 42,116)( 43,115)( 44,114)( 45,117)( 46,120)( 47,119)
( 48,118)( 49,121)( 50,124)( 51,123)( 52,122)( 53,125)( 54,128)( 55,127)
( 56,126)( 57,129)( 58,132)( 59,131)( 60,130)( 61,133)( 62,136)( 63,135)
( 64,134)( 65,137)( 66,140)( 67,139)( 68,138)( 69,141)( 70,144)( 71,143)
( 72,142)( 73,181)( 74,184)( 75,183)( 76,182)( 77,185)( 78,188)( 79,187)
( 80,186)( 81,189)( 82,192)( 83,191)( 84,190)( 85,193)( 86,196)( 87,195)
( 88,194)( 89,197)( 90,200)( 91,199)( 92,198)( 93,201)( 94,204)( 95,203)
( 96,202)( 97,205)( 98,208)( 99,207)(100,206)(101,209)(102,212)(103,211)
(104,210)(105,213)(106,216)(107,215)(108,214);
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*s2*s1*s0*s1*s2*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s1*s2, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s1*s2*s3*s2*s3*s1*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2 >;

References : None.
to this polytope