Finite Group DatabasebetaFinite Group Databasebeta| order | 486 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C3 |
No aliases for this group.
| Fitting, Sylow | #6909 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C3 | C3×(C3⋊D3)⋊C3 |
| commutator, cosocle, hypocenter | H(3,𝔽3) | C2×C32 |
| Frattini | C32 | C32×D3 |
| Fitting | #6909 | C2 |
| hypercenter | C32 | (C3⋊D3)⋊C3 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C3 ◁ C32 | |
88 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 27 |
A minimal faithful permutation representation — 2 generators on 27 points:
(2,7,9)(3,10)(6,14,16)(8,25,11,18,17,20)(12,21)(13,23)(15,26,22,24,19,27);(1,8,26,4,17,24,5,11,27)(2,15,21,7,22,10,9,19,23)(3,20,16,13,18,14,12,25,6)Polycyclic presentation:
2,3^5|d^a=d^2,e^a=e^2f,c^b=cf,d^b=de,e^d=ef34 conjugacy classes — view the full character table.
| maximal subgroup of | (#4174⋊C2)⋊C3, C2×#92281, #13091109, … |
| maximal normal subgroup of | (#4174⋊C2)⋊C3, C2×#92281, #13091113, … |
| maximal quotient of | #11757585, C2×#92281, #13090750, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(486, 126);