Finite Group DatabasebetaFinite Group Databasebeta| order | 486 |
| isoclinism | isoclinic class of #92318 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C3 |
No aliases for this group.
| Fitting, Sylow | #6903 ⋊ C2 |
| derived | C3×C27 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×C27 | C2×C3 |
| Frattini | C9 | C3×C3⋊D3 |
| Fitting | #6903 | C2 |
| socle | C3 | C3×C3⋊D9 |
| LCS | G ▷ C3×C27 | |
| UCS | C1 | |
62 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 81 |
A minimal faithful permutation representation — 3 generators on 81 points:
(2,6)(3,31,34,11,12,33)(4,14)(5,17)(7,9)(8,50,53,23,24,52)(10,29)(13,61,63,36,35,65)(15,16)(18,56,58,25,42,54)(19,38)(20,30)(21,71,70,46,47,73)(22,49)(26,75,76,48,55,74)(27,45)(28,44)(32,68,62,39,60,37)(40,41)(43,72,79,64,69,80)(51,81,78,66,77,57)(59,67);(1,3,11)(2,8,25)(4,13,32)(5,12,33)(6,18,23)(7,21,48)(9,26,46)(10,24,54)(14,39,36)(15,37,61)(16,35,62)(17,34,31)(19,43,51)(20,42,52)(22,47,74)(27,57,72)(28,55,73)(29,58,50)(30,53,56)(38,66,64)(40,68,65)(41,63,60)(44,70,75)(45,69,78)(49,76,71)(59,79,77)(67,81,80);(1,38,20,40,49,44,4,2,45,5,67,29,15,9,7,16,10,59,17,27,6,14,28,22,41,30,19)(3,66,42,68,76,70,13,8,69,12,81,58,37,26,21,35,24,79,34,57,18,39,55,47,63,53,43)(11,64,52,65,71,75,32,25,78,33,80,50,61,46,48,62,54,77,31,72,23,36,73,74,60,56,51)Polycyclic presentation:
2,3^5|c^3=e^2f,e^3=f^2,c^a=c^2e,d^a=d^2,e^a=e^2f,f^a=f^2,d^b=df54 conjugacy classes — view the full character table.
| maximal subgroup of | #92318⋊C2, C2×#92318, #13090542, … |
| maximal normal subgroup of | #92318⋊C2, C2×#92318, C3×#92318, … |
| maximal quotient of | #11757622, C2×#92318, #13090867, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(486, 163);