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