Polytope of Type {6,4}
Play with this polytope as a twisty puzzle
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,4}*1296a
if this polytope has a name.
Group : SmallGroup(1296,2909)
Rank : 3
Schlafli Type : {6,4}
Number of vertices, edges, etc : 162, 324, 108
Order of s0s1s2 : 12
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Skewing Operation
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {6,4}*432a, {6,4}*432b
6-fold quotients : {6,4}*216
9-fold quotients : {6,4}*144
18-fold quotients : {6,4}*72
27-fold quotients : {6,4}*48a
54-fold quotients : {6,2}*24
81-fold quotients : {2,4}*16
108-fold quotients : {3,2}*12
162-fold quotients : {2,2}*8
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*s0*s1*s2*s1*s2> of order 2.
54 facets:
54 of {6}*12
81 vertex figures:
81 of {4}*8
P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 2.
57 facets:
51 of {6}*12
6 of {3}*6
81 vertex figures:
81 of {4}*8
P/N, where N=<s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 2.
54 facets:
54 of {6}*12
90 vertex figures:
72 of {4}*8
18 of {2}*4
P/N, where N=<s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0> of order 3.
36 facets:
36 of {6}*12
54 vertex figures:
54 of {4}*8
P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1> of order 3.
36 facets:
36 of {6}*12
54 vertex figures:
54 of {4}*8
P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1> of order 3.
36 facets:
36 of {6}*12
54 vertex figures:
54 of {4}*8
P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2> of order 3.
36 facets:
36 of {6}*12
54 vertex figures:
54 of {4}*8
P/N, where N=<s0*s2*s1*s0*s1*s2> of order 3.
42 facets:
33 of {6}*12
9 of {2}*4
54 vertex figures:
54 of {4}*8
P/N, where N=<s0*s1*s0*s1*s0*s1, s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 4.
30 facets:
6 of {3}*6
24 of {6}*12
45 vertex figures:
36 of {4}*8
9 of {2}*4
P/N, where N=<s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2, s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1> of order 4.
27 facets:
27 of {6}*12
45 vertex figures:
36 of {4}*8
9 of {2}*4
P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1, s0*s1*s0*s1*s2*s1*s0*s2*s1*s0> of order 6.
18 facets:
18 of {6}*12
36 vertex figures:
18 of {4}*8
18 of {2}*4
P/N, where N=<s1*s2*s1*s2, s0*s1*s2*s1*s0*s2> of order 6.
18 facets:
18 of {6}*12
36 vertex figures:
18 of {2}*4
18 of {4}*8
P/N, where N=<s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2, s0*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 6.
21 facets:
15 of {6}*12
6 of {3}*6
27 vertex figures:
27 of {4}*8
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1, s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1> of order 9.
12 facets:
12 of {6}*12
18 vertex figures:
18 of {4}*8
P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s2*s1*s0, s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1> of order 9.
16 facets:
10 of {6}*12
6 of {2}*4
18 vertex figures:
18 of {4}*8
P/N, where N=<s0*s1*s0*s1*s0*s1, s0*s1*s0*s2*s1*s0*s2*s1> of order 12.
11 facets:
4 of {3}*6
7 of {6}*12
18 vertex figures:
9 of {4}*8
9 of {2}*4
Permutation Representation (GAP) :
s0 := ( 4, 7)( 5, 8)( 6, 9)(10,19)(11,20)(12,21)(13,25)(14,26)(15,27)(16,22)(17,23)(18,24)(28,55)(29,56)(30,57)(31,61)(32,62)(33,63)(34,58)(35,59)(36,60)(37,73)(38,74)(39,75)(40,79)(41,80)(42,81)(43,76)(44,77)(45,78)(46,64)(47,65)(48,66)(49,70)(50,71)(51,72)(52,67)(53,68)(54,69);;
s1 := ( 1,31)( 2,33)( 3,32)( 4,28)( 5,30)( 6,29)( 7,34)( 8,36)( 9,35)(10,41)(11,40)(12,42)(13,38)(14,37)(15,39)(16,44)(17,43)(18,45)(19,51)(20,50)(21,49)(22,48)(23,47)(24,46)(25,54)(26,53)(27,52)(55,58)(56,60)(57,59)(62,63)(64,68)(65,67)(66,69)(70,71)(73,78)(74,77)(75,76)(79,81);;
s2 := ( 1, 2)( 4, 5)( 7, 8)(10,29)(11,28)(12,30)(13,32)(14,31)(15,33)(16,35)(17,34)(18,36)(19,56)(20,55)(21,57)(22,59)(23,58)(24,60)(25,62)(26,61)(27,63)(37,39)(40,42)(43,45)(46,64)(47,66)(48,65)(49,67)(50,69)(51,68)(52,70)(53,72)(54,71)(73,75)(76,78)(79,81);;
poly := Group([s0,s1,s2]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2");;
s0 := F.1;; s1 := F.2;; s2 := F.3;;
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s0*s2*s1*s2*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s2*s1*s2*s1*s0*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(81)!( 4, 7)( 5, 8)( 6, 9)(10,19)(11,20)(12,21)(13,25)(14,26)(15,27)(16,22)(17,23)(18,24)(28,55)(29,56)(30,57)(31,61)(32,62)(33,63)(34,58)(35,59)(36,60)(37,73)(38,74)(39,75)(40,79)(41,80)(42,81)(43,76)(44,77)(45,78)(46,64)(47,65)(48,66)(49,70)(50,71)(51,72)(52,67)(53,68)(54,69);
s1 := Sym(81)!( 1,31)( 2,33)( 3,32)( 4,28)( 5,30)( 6,29)( 7,34)( 8,36)( 9,35)(10,41)(11,40)(12,42)(13,38)(14,37)(15,39)(16,44)(17,43)(18,45)(19,51)(20,50)(21,49)(22,48)(23,47)(24,46)(25,54)(26,53)(27,52)(55,58)(56,60)(57,59)(62,63)(64,68)(65,67)(66,69)(70,71)(73,78)(74,77)(75,76)(79,81);
s2 := Sym(81)!( 1, 2)( 4, 5)( 7, 8)(10,29)(11,28)(12,30)(13,32)(14,31)(15,33)(16,35)(17,34)(18,36)(19,56)(20,55)(21,57)(22,59)(23,58)(24,60)(25,62)(26,61)(27,63)(37,39)(40,42)(43,45)(46,64)(47,66)(48,65)(49,67)(50,69)(51,68)(52,70)(53,72)(54,71)(73,75)(76,78)(79,81);
poly := sub<Sym(81)|s0,s1,s2>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2,
s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s0*s2*s1*s2*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s2*s1*s2*s1*s0*s1*s2*s0*s1*s2*s0*s1 >;
References : None.
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