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# Polytope of Type {6,6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,6}*1944h
if this polytope has a name.
Group : SmallGroup(1944,2344)
Rank : 4
Schlafli Type : {6,6,6}
Number of vertices, edges, etc : 27, 81, 81, 6
Order of s0s1s2s3 : 18
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Non-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 : {6,6,2}*648a, {6,6,6}*648b
9-fold quotients : {6,6,2}*216
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,19)(11,21)(12,20)(13,25)(14,27)(15,26)
(16,22)(17,24)(18,23)(29,30)(31,34)(32,36)(33,35)(37,46)(38,48)(39,47)(40,52)
(41,54)(42,53)(43,49)(44,51)(45,50)(56,57)(58,61)(59,63)(60,62)(64,73)(65,75)
(66,74)(67,79)(68,81)(69,80)(70,76)(71,78)(72,77);;
s1 := ( 1,10)( 2,12)( 3,11)( 4,13)( 5,15)( 6,14)( 7,16)( 8,18)( 9,17)(20,21)
(23,24)(26,27)(28,64)(29,66)(30,65)(31,67)(32,69)(33,68)(34,70)(35,72)(36,71)
(37,55)(38,57)(39,56)(40,58)(41,60)(42,59)(43,61)(44,63)(45,62)(46,73)(47,75)
(48,74)(49,76)(50,78)(51,77)(52,79)(53,81)(54,80);;
s2 := ( 1,28)( 2,29)( 3,30)( 4,35)( 5,36)( 6,34)( 7,33)( 8,31)( 9,32)(10,41)
(11,42)(12,40)(13,39)(14,37)(15,38)(16,43)(17,44)(18,45)(19,54)(20,52)(21,53)
(22,49)(23,50)(24,51)(25,47)(26,48)(27,46)(58,62)(59,63)(60,61)(64,68)(65,69)
(66,67)(73,81)(74,79)(75,80);;
s3 := (28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,61)(35,62)(36,63)(37,64)
(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(46,73)(47,74)(48,75)
(49,76)(50,77)(51,78)(52,79)(53,80)(54,81);;
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*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1,
s1*s2*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0 ];;
poly := F / rels;;

Permutation Representation (Magma) :
s0 := Sym(81)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)(10,19)(11,21)(12,20)(13,25)(14,27)
(15,26)(16,22)(17,24)(18,23)(29,30)(31,34)(32,36)(33,35)(37,46)(38,48)(39,47)
(40,52)(41,54)(42,53)(43,49)(44,51)(45,50)(56,57)(58,61)(59,63)(60,62)(64,73)
(65,75)(66,74)(67,79)(68,81)(69,80)(70,76)(71,78)(72,77);
s1 := Sym(81)!( 1,10)( 2,12)( 3,11)( 4,13)( 5,15)( 6,14)( 7,16)( 8,18)( 9,17)
(20,21)(23,24)(26,27)(28,64)(29,66)(30,65)(31,67)(32,69)(33,68)(34,70)(35,72)
(36,71)(37,55)(38,57)(39,56)(40,58)(41,60)(42,59)(43,61)(44,63)(45,62)(46,73)
(47,75)(48,74)(49,76)(50,78)(51,77)(52,79)(53,81)(54,80);
s2 := Sym(81)!( 1,28)( 2,29)( 3,30)( 4,35)( 5,36)( 6,34)( 7,33)( 8,31)( 9,32)
(10,41)(11,42)(12,40)(13,39)(14,37)(15,38)(16,43)(17,44)(18,45)(19,54)(20,52)
(21,53)(22,49)(23,50)(24,51)(25,47)(26,48)(27,46)(58,62)(59,63)(60,61)(64,68)
(65,69)(66,67)(73,81)(74,79)(75,80);
s3 := Sym(81)!(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,61)(35,62)(36,63)
(37,64)(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(44,71)(45,72)(46,73)(47,74)
(48,75)(49,76)(50,77)(51,78)(52,79)(53,80)(54,81);
poly := sub<Sym(81)|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*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1,
s1*s2*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0 >;

References : None.
to this polytope