Polytope of Type {4,12,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,12,6}*1728a
if this polytope has a name.
Group : SmallGroup(1728,14771)
Rank : 4
Schlafli Type : {4,12,6}
Number of vertices, edges, etc : 4, 72, 108, 18
Order of s₀s₁s₂s₃ : 12
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 : {2,12,6}*864a, {4,6,6}*864a
   3-fold quotients : {4,12,6}*576b
   4-fold quotients : {2,6,6}*432c
   6-fold quotients : {2,12,6}*288b, {4,6,6}*288b
   8-fold quotients : {2,3,6}*216
   9-fold quotients : {4,12,2}*192a
   12-fold quotients : {2,6,6}*144c
   18-fold quotients : {2,12,2}*96, {4,6,2}*96a
   24-fold quotients : {2,3,6}*72
   27-fold quotients : {4,4,2}*64
   36-fold quotients : {2,6,2}*48
   54-fold quotients : {2,4,2}*32, {4,2,2}*32
   72-fold quotients : {2,3,2}*24
   108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope