Overview
- Group
- SmallGroup(1944,2344)
- Rank
- 4
- Schläfli Type
- {6,6,3}
- Vertices, edges, …
- 54, 162, 81, 3
- Order of s0s1s2s3
- 6
- Order of s0s1s2s3s2s1
- 6
- Also known as
- if this polytope has a name.
Special Properties
- Universal
- Orientable
- Flat
Quotients maximal quotients in bold
3-fold
9-fold
27-fold
54-fold
81-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=<(s0*s1*s2*s1)^2> of order 3
3 facets
- 3 of 3-fold non-regular quotient of {6,6}*648a
18 vertex figures
- 18 of {6,3}*36
P/N, where N=<(s0*s1)^2> of order 3
3 facets
- 3 of 3-fold non-regular quotient of {6,6}*648a
18 vertex figures
- 18 of {6,3}*36
P/N, where N=<((s1*s0)^2*s1*s2)^2> of order 3
3 facets
- 3 of 3-fold non-regular quotient of {6,6}*648a
18 vertex figures
- 18 of {6,3}*36
P/N, where N=<s0*s1*s2*s1*s0*(s2*s1)^2*s2> of order 3
3 facets
- 3 of 3-fold non-regular quotient of {6,6}*648a
18 vertex figures
- 18 of {6,3}*36
P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1> of order 3
3 facets
- 3 of 3-fold non-regular quotient of {6,6}*648a
30 vertex figures
P/N, where N=<(s0*s1*s2*s1)^2, s1*s0*(s2*s1)^2*s0*s2*s1*s2> of order 9
3 facets
- 3 of 9-fold non-regular quotient of {6,6}*648a
6 vertex figures
- 6 of {6,3}*36
P/N, where N=<(s0*s1*s2*s1)^2, s1*s0*s1*s2*s1*s0*s2*s1> of order 9
3 facets
- 3 of 9-fold non-regular quotient of {6,6}*648a
10 vertex figures
P/N, where N=<(s1*s2)^2, s0*s1*s2*s1*s0*s2> of order 9
3 facets
- 3 of 9-fold non-regular quotient of {6,6}*648a
14 vertex figures
Representations
Permutation Representation (GAP)
s0 := ( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27)(31,34)(32,35)(33,36)(40,43)(41,44)(42,45)(49,52)(50,53)(51,54)(58,61)(59,62)(60,63)(67,70)(68,71)(69,72)(76,79)(77,80)(78,81);; s1 := ( 2, 3)( 4, 6)( 7, 8)(10,27)(11,26)(12,25)(13,20)(14,19)(15,21)(16,22)(17,24)(18,23)(29,30)(31,33)(34,35)(37,54)(38,53)(39,52)(40,47)(41,46)(42,48)(43,49)(44,51)(45,50)(56,57)(58,60)(61,62)(64,81)(65,80)(66,79)(67,74)(68,73)(69,75)(70,76)(71,78)(72,77);; s2 := ( 1,10)( 2,12)( 3,11)( 4,16)( 5,18)( 6,17)( 7,13)( 8,15)( 9,14)(20,21)(22,25)(23,27)(24,26)(28,64)(29,66)(30,65)(31,70)(32,72)(33,71)(34,67)(35,69)(36,68)(37,55)(38,57)(39,56)(40,61)(41,63)(42,62)(43,58)(44,60)(45,59)(46,73)(47,75)(48,74)(49,79)(50,81)(51,80)(52,76)(53,78)(54,77);; s3 := ( 1,28)( 2,30)( 3,29)( 4,34)( 5,36)( 6,35)( 7,31)( 8,33)( 9,32)(10,46)(11,48)(12,47)(13,52)(14,54)(15,53)(16,49)(17,51)(18,50)(19,37)(20,39)(21,38)(22,43)(23,45)(24,44)(25,40)(26,42)(27,41)(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);; 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, s2*s3*s2*s3*s2*s3,
s1*s2*s3*s1*s2*s1*s2*s3*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma)
s0 := Sym(81)!( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27)(31,34)(32,35)(33,36)(40,43)(41,44)(42,45)(49,52)(50,53)(51,54)(58,61)(59,62)(60,63)(67,70)(68,71)(69,72)(76,79)(77,80)(78,81); s1 := Sym(81)!( 2, 3)( 4, 6)( 7, 8)(10,27)(11,26)(12,25)(13,20)(14,19)(15,21)(16,22)(17,24)(18,23)(29,30)(31,33)(34,35)(37,54)(38,53)(39,52)(40,47)(41,46)(42,48)(43,49)(44,51)(45,50)(56,57)(58,60)(61,62)(64,81)(65,80)(66,79)(67,74)(68,73)(69,75)(70,76)(71,78)(72,77); s2 := Sym(81)!( 1,10)( 2,12)( 3,11)( 4,16)( 5,18)( 6,17)( 7,13)( 8,15)( 9,14)(20,21)(22,25)(23,27)(24,26)(28,64)(29,66)(30,65)(31,70)(32,72)(33,71)(34,67)(35,69)(36,68)(37,55)(38,57)(39,56)(40,61)(41,63)(42,62)(43,58)(44,60)(45,59)(46,73)(47,75)(48,74)(49,79)(50,81)(51,80)(52,76)(53,78)(54,77); s3 := Sym(81)!( 1,28)( 2,30)( 3,29)( 4,34)( 5,36)( 6,35)( 7,31)( 8,33)( 9,32)(10,46)(11,48)(12,47)(13,52)(14,54)(15,53)(16,49)(17,51)(18,50)(19,37)(20,39)(21,38)(22,43)(23,45)(24,44)(25,40)(26,42)(27,41)(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); 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, s2*s3*s2*s3*s2*s3, s1*s2*s3*s1*s2*s1*s2*s3*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1 >;
References
None.
to this polytope.