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}*384a
if this polytope has a name.
Group : SmallGroup(384,17948)
Rank : 3
Schlafli Type : {6,4}
Number of vertices, edges, etc : 48, 96, 32
Order of s0s1s2 : 6
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Self-Petrie
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Skewing Operation
Facet Of :
{6,4,2} of size 768
Vertex Figure Of :
{2,6,4} of size 768
{3,6,4} of size 1152
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {6,4}*192a
4-fold quotients : {6,4}*96
8-fold quotients : {3,4}*48, {6,4}*48b, {6,4}*48c
16-fold quotients : {3,4}*24, {6,2}*24
32-fold quotients : {3,2}*12
48-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,8}*768e, {6,8}*768g, {12,4}*768b, {6,4}*768a, {12,4}*768c, {6,8}*768m, {6,8}*768n, {6,4}*768b, {6,4}*768c, {12,4}*768g, {12,4}*768h
3-fold covers : {18,4}*1152a, {6,12}*1152b, {6,12}*1152c
5-fold covers : {6,20}*1920a, {30,4}*1920a
Irregular Quotients (of which this is a minimal cover):
P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2> of order 2.
16 facets:
16 of {6}*12
24 vertex figures:
24 of {4}*8
P/N, where N=<s0*s1*s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s0> of order 2.
16 facets:
16 of {6}*12
28 vertex figures:
20 of {4}*8
8 of {2}*4
P/N, where N=<s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2> of order 2.
16 facets:
16 of {6}*12
24 vertex figures:
24 of {4}*8
P/N, where N=<s0*s1*s0*s1*s0*s1> of order 2.
20 facets:
8 of {3}*6
12 of {6}*12
24 vertex figures:
24 of {4}*8
P/N, where N=<s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2> of order 2.
16 facets:
16 of {6}*12
24 vertex figures:
24 of {4}*8
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s0*s2> of order 2.
16 facets:
16 of {6}*12
24 vertex figures:
24 of {4}*8
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1> of order 2.
16 facets:
16 of {6}*12
24 vertex figures:
24 of {4}*8
P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 2.
16 facets:
16 of {6}*12
24 vertex figures:
24 of {4}*8
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2, s0*s1*s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s0> of order 4.
8 facets:
8 of {6}*12
14 vertex figures:
10 of {4}*8
4 of {2}*4
P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2> of order 4.
8 facets:
8 of {6}*12
12 vertex figures:
12 of {4}*8
P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2, s0*s1*s0*s1*s2*s1*s0*s1*s0*s2> of order 4.
8 facets:
8 of {6}*12
12 vertex figures:
12 of {4}*8
P/N, where N=<s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 4.
10 facets:
4 of {3}*6
6 of {6}*12
12 vertex figures:
12 of {4}*8
P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1> of order 4.
8 facets:
8 of {6}*12
12 vertex figures:
12 of {4}*8
P/N, where N=<s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s2, s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2> of order 4.
8 facets:
8 of {6}*12
12 vertex figures:
12 of {4}*8
P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s0*s1*s0*s1*s2*s1*s0*s1*s0*s2> of order 4.
8 facets:
8 of {6}*12
14 vertex figures:
10 of {4}*8
4 of {2}*4
P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1*s2, s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2> of order 4.
8 facets:
8 of {6}*12
12 vertex figures:
12 of {4}*8
P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2, s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2> of order 4.
8 facets:
8 of {6}*12
12 vertex figures:
12 of {4}*8
P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s2, s0*s1*s0*s1*s2*s1*s0*s1*s0*s2> of order 4.
8 facets:
8 of {6}*12
12 vertex figures:
12 of {4}*8
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s0*s2, s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2> of order 4.
8 facets:
8 of {6}*12
12 vertex figures:
12 of {4}*8
P/N, where N=<s0*s1*s2*s1*s0*s2, s1*s0*s1*s2*s1*s0*s2*s1> of order 4.
8 facets:
8 of {6}*12
18 vertex figures:
6 of {4}*8
12 of {2}*4
P/N, where N=<s0*s1*s2*s1*s0*s2, s0*s1*s0*s1*s2*s1*s0*s1*s0*s2> of order 4.
8 facets:
8 of {6}*12
16 vertex figures:
8 of {4}*8
8 of {2}*4
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 4.
8 facets:
8 of {6}*12
12 vertex figures:
12 of {4}*8
P/N, where N=<s0*s1*s0*s1*s0*s1, s0*s2*s1*s0*s1*s0*s1*s2> of order 4.
12 facets:
8 of {3}*6
4 of {6}*12
12 vertex figures:
12 of {4}*8
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2, s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s2> of order 4.
8 facets:
8 of {6}*12
14 vertex figures:
10 of {4}*8
4 of {2}*4
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2, s0*s1*s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s0> of order 4.
8 facets:
8 of {6}*12
14 vertex figures:
10 of {4}*8
4 of {2}*4
P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s2, s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2> of order 4.
8 facets:
8 of {6}*12
12 vertex figures:
12 of {4}*8
P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1, s0*s1*s0*s1*s2*s1*s0*s2*s1*s0> of order 4.
8 facets:
8 of {6}*12
16 vertex figures:
8 of {4}*8
8 of {2}*4
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s0*s2, s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2> of order 4.
8 facets:
8 of {6}*12
14 vertex figures:
10 of {4}*8
4 of {2}*4
P/N, where N=<s0*s1*s0*s1*s0*s1, s0*s1*s0*s2*s1*s0*s2*s1> of order 8.
6 facets:
4 of {3}*6
2 of {6}*12
8 vertex figures:
4 of {4}*8
4 of {2}*4
P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s0*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 8.
4 facets:
4 of {6}*12
6 vertex figures:
6 of {4}*8
P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1, s0*s1*s0*s1*s2*s1*s0*s2*s1*s0, s0*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 8.
4 facets:
4 of {6}*12
8 vertex figures:
4 of {4}*8
4 of {2}*4
P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s1*s0*s1*s2*s1*s0*s1*s2, s0*s1*s0*s1*s2*s1*s0*s1*s0*s2> of order 8.
4 facets:
4 of {6}*12
8 vertex figures:
4 of {4}*8
4 of {2}*4
P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s1*s0*s1*s2*s1*s0*s2*s1*s2, s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2> of order 8.
4 facets:
4 of {6}*12
6 vertex figures:
6 of {4}*8
P/N, where N=<s0*s1*s0*s1*s0*s1, s0*s2*s1*s0*s1*s0*s1*s2, s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s2> of order 8.
6 facets:
4 of {3}*6
2 of {6}*12
6 vertex figures:
6 of {4}*8
P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1, s0*s1*s0*s1*s2*s1*s0*s2*s1*s0, s0*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1> of order 8.
4 facets:
4 of {6}*12
8 vertex figures:
4 of {4}*8
4 of {2}*4
P/N, where N=<s0*s1*s2*s1*s0*s2, s1*s0*s1*s2*s1*s0*s2*s1, s0*s1*s0*s1*s2*s1*s0*s1*s0*s2> of order 8.
4 facets:
4 of {6}*12
10 vertex figures:
2 of {4}*8
8 of {2}*4
P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1*s2, s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2, s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1> of order 8.
4 facets:
4 of {6}*12
6 vertex figures:
6 of {4}*8
P/N, where N=<s0*s1*s0*s1*s0*s1, s0*s2*s1*s0*s1*s0*s1*s2, s0*s1*s0*s2*s1*s0*s2*s1*s2> of order 8.
6 facets:
4 of {3}*6
2 of {6}*12
6 vertex figures:
6 of {4}*8
P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s1*s0*s1*s2*s1*s0*s2*s1*s2, s0*s1*s0*s1*s2*s1*s0*s1*s0*s2> of order 8.
4 facets:
4 of {6}*12
7 vertex figures:
5 of {4}*8
2 of {2}*4
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s2, s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1> of order 8.
4 facets:
4 of {6}*12
6 vertex figures:
6 of {4}*8
Permutation Representation (GAP) :
s0 := ( 3, 4)( 5, 9)( 6,10)( 7,12)( 8,11)(15,16)(17,33)(18,34)(19,36)(20,35)(21,41)(22,42)(23,44)(24,43)(25,37)(26,38)(27,40)(28,39)(29,45)(30,46)(31,48)(32,47)(51,52)(53,57)(54,58)(55,60)(56,59)(63,64)(65,81)(66,82)(67,84)(68,83)(69,89)(70,90)(71,92)(72,91)(73,85)(74,86)(75,88)(76,87)(77,93)(78,94)(79,96)(80,95);;
s1 := ( 1,17)( 2,20)( 3,19)( 4,18)( 5,23)( 6,22)( 7,21)( 8,24)( 9,30)(10,31)(11,32)(12,29)(13,28)(14,25)(15,26)(16,27)(34,36)(37,39)(41,46)(42,47)(43,48)(44,45)(49,65)(50,68)(51,67)(52,66)(53,71)(54,70)(55,69)(56,72)(57,78)(58,79)(59,80)(60,77)(61,76)(62,73)(63,74)(64,75)(82,84)(85,87)(89,94)(90,95)(91,96)(92,93);;
s2 := ( 1,61)( 2,62)( 3,63)( 4,64)( 5,57)( 6,58)( 7,59)( 8,60)( 9,53)(10,54)(11,55)(12,56)(13,49)(14,50)(15,51)(16,52)(17,77)(18,78)(19,79)(20,80)(21,73)(22,74)(23,75)(24,76)(25,69)(26,70)(27,71)(28,72)(29,65)(30,66)(31,67)(32,68)(33,93)(34,94)(35,95)(36,96)(37,89)(38,90)(39,91)(40,92)(41,85)(42,86)(43,87)(44,88)(45,81)(46,82)(47,83)(48,84);;
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,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(96)!( 3, 4)( 5, 9)( 6,10)( 7,12)( 8,11)(15,16)(17,33)(18,34)(19,36)(20,35)(21,41)(22,42)(23,44)(24,43)(25,37)(26,38)(27,40)(28,39)(29,45)(30,46)(31,48)(32,47)(51,52)(53,57)(54,58)(55,60)(56,59)(63,64)(65,81)(66,82)(67,84)(68,83)(69,89)(70,90)(71,92)(72,91)(73,85)(74,86)(75,88)(76,87)(77,93)(78,94)(79,96)(80,95);
s1 := Sym(96)!( 1,17)( 2,20)( 3,19)( 4,18)( 5,23)( 6,22)( 7,21)( 8,24)( 9,30)(10,31)(11,32)(12,29)(13,28)(14,25)(15,26)(16,27)(34,36)(37,39)(41,46)(42,47)(43,48)(44,45)(49,65)(50,68)(51,67)(52,66)(53,71)(54,70)(55,69)(56,72)(57,78)(58,79)(59,80)(60,77)(61,76)(62,73)(63,74)(64,75)(82,84)(85,87)(89,94)(90,95)(91,96)(92,93);
s2 := Sym(96)!( 1,61)( 2,62)( 3,63)( 4,64)( 5,57)( 6,58)( 7,59)( 8,60)( 9,53)(10,54)(11,55)(12,56)(13,49)(14,50)(15,51)(16,52)(17,77)(18,78)(19,79)(20,80)(21,73)(22,74)(23,75)(24,76)(25,69)(26,70)(27,71)(28,72)(29,65)(30,66)(31,67)(32,68)(33,93)(34,94)(35,95)(36,96)(37,89)(38,90)(39,91)(40,92)(41,85)(42,86)(43,87)(44,88)(45,81)(46,82)(47,83)(48,84);
poly := sub<Sym(96)|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,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 >;
References : None.
to this polytope
Twisty Puzzle