Overview
- Group
- SmallGroup(1296,3528)
- Rank
- 4
- Schläfli Type
- {3,6,4}
- Vertices, edges, …
- 3, 81, 108, 36
- Order of s0s1s2s3
- 12
- Order of s0s1s2s3s2s1
- 6
- Also known as
- if this polytope has a name.
Special Properties
- Orientable
- Flat
Quotients maximal quotients in bold
3-fold
9-fold
18-fold
27-fold
54-fold
Covers minimal covers in bold
None in this atlas.
Irregular Quotients of which this is a minimal cover
Click an entry to reveal its facets and vertex figures.
P/N, where N=<(s2*s3)^2> of order 2
18 facets
- 18 of {3,6}*36
3 vertex figures
- 3 of 2-fold non-regular quotient of {6,4}*432b
P/N, where N=<s0*s1*s3*s2*s1*s0*(s3*s2*s1)^2*s3*s2> of order 3
12 facets
- 12 of {3,6}*36
3 vertex figures
- 3 of 3-fold non-regular quotient of {6,4}*432b
P/N, where N=<(s2*s1*s2*s3)^2> of order 3
12 facets
- 12 of {3,6}*36
3 vertex figures
- 3 of 3-fold non-regular quotient of {6,4}*432b
P/N, where N=<s1*s2*s1*s3*(s2*s1)^2*s3*s2> of order 3
12 facets
- 12 of {3,6}*36
3 vertex figures
- 3 of 3-fold non-regular quotient of {6,4}*432b
P/N, where N=<s2*s1*s3*s2*s1*s2*s3*s2> of order 3
18 facets
3 vertex figures
- 3 of 3-fold non-regular quotient of {6,4}*432b
P/N, where N=<(s2*s3)^2, s1*s0*s2*s1*s2*s3*s2*s1*s0*s3*s2*s1> of order 6
6 facets
- 6 of {3,6}*36
3 vertex figures
- 3 of 6-fold non-regular quotient of {6,4}*432b
P/N, where N=<(s2*s3)^2, (s2*s1*s2*s3)^2> of order 6
6 facets
- 6 of {3,6}*36
3 vertex figures
- 3 of 6-fold non-regular quotient of {6,4}*432b
P/N, where N=<s1*s3*s2*s1*s2*s3, s1*s2*s1*s3*(s2*s1)^2*s3*s2> of order 9
6 facets
3 vertex figures
- 3 of 9-fold non-regular quotient of {6,4}*432b
P/N, where N=<(s2*s1*s2*s3)^2, s2*s1*s3*s2*s1*s2*s3*s2> of order 9
8 facets
3 vertex figures
- 3 of 9-fold non-regular quotient of {6,4}*432b
Representations
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)(28,55)(29,57)(30,56)(31,61)(32,63)(33,62)(34,58)(35,60)(36,59)(37,73)(38,75)(39,74)(40,79)(41,81)(42,80)(43,76)(44,78)(45,77)(46,64)(47,66)(48,65)(49,70)(50,72)(51,71)(52,67)(53,69)(54,68);; s1 := ( 1,32)( 2,31)( 3,33)( 4,29)( 5,28)( 6,30)( 7,35)( 8,34)( 9,36)(10,50)(11,49)(12,51)(13,47)(14,46)(15,48)(16,53)(17,52)(18,54)(19,41)(20,40)(21,42)(22,38)(23,37)(24,39)(25,44)(26,43)(27,45)(55,59)(56,58)(57,60)(61,62)(64,77)(65,76)(66,78)(67,74)(68,73)(69,75)(70,80)(71,79)(72,81);; s2 := ( 4, 7)( 5, 8)( 6, 9)(10,28)(11,29)(12,30)(13,34)(14,35)(15,36)(16,31)(17,32)(18,33)(19,55)(20,56)(21,57)(22,61)(23,62)(24,63)(25,58)(26,59)(27,60)(40,43)(41,44)(42,45)(46,64)(47,65)(48,66)(49,70)(50,71)(51,72)(52,67)(53,68)(54,69)(76,79)(77,80)(78,81);; s3 := (10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(37,46)(38,47)(39,48)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54)(64,73)(65,74)(66,75)(67,76)(68,77)(69,78)(70,79)(71,80)(72,81);; 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, s0*s1*s0*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1,
s2*s0*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s0*s1*s2*s3*s2*s1 ];;
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)(28,55)(29,57)(30,56)(31,61)(32,63)(33,62)(34,58)(35,60)(36,59)(37,73)(38,75)(39,74)(40,79)(41,81)(42,80)(43,76)(44,78)(45,77)(46,64)(47,66)(48,65)(49,70)(50,72)(51,71)(52,67)(53,69)(54,68); s1 := Sym(81)!( 1,32)( 2,31)( 3,33)( 4,29)( 5,28)( 6,30)( 7,35)( 8,34)( 9,36)(10,50)(11,49)(12,51)(13,47)(14,46)(15,48)(16,53)(17,52)(18,54)(19,41)(20,40)(21,42)(22,38)(23,37)(24,39)(25,44)(26,43)(27,45)(55,59)(56,58)(57,60)(61,62)(64,77)(65,76)(66,78)(67,74)(68,73)(69,75)(70,80)(71,79)(72,81); s2 := Sym(81)!( 4, 7)( 5, 8)( 6, 9)(10,28)(11,29)(12,30)(13,34)(14,35)(15,36)(16,31)(17,32)(18,33)(19,55)(20,56)(21,57)(22,61)(23,62)(24,63)(25,58)(26,59)(27,60)(40,43)(41,44)(42,45)(46,64)(47,65)(48,66)(49,70)(50,71)(51,72)(52,67)(53,68)(54,69)(76,79)(77,80)(78,81); s3 := Sym(81)!(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(37,46)(38,47)(39,48)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54)(64,73)(65,74)(66,75)(67,76)(68,77)(69,78)(70,79)(71,80)(72,81); 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, s0*s1*s0*s1*s0*s1, s2*s3*s2*s3*s2*s3*s2*s3, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s2*s0*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s0*s1*s2*s3*s2*s1 >;
References
None.
to this polytope.