Finite Group DatabasebetaFinite Group Databasebeta| order | 324 |
| isoclinism | isoclinic class of C3⋊(C3⋊D9) |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C33 |
| direct product | C2×C3⋊(C3⋊D9) |
| unambiguous semidirect product | (C2×C3)⋊(C3⋊D9) |
| unambiguous semidirect product | (C2×C9)⋊(C3⋊D3) |
| Fitting | C2×C32×C9 ⋊ C2 |
| derived, Sylow | C32×C9 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C3⋊(C3⋊D9) |
| commutator, cosocle, hypocenter | C32×C9 | C22 |
| Frattini | C3 | C2×C3⋊(C3⋊D3) |
| Fitting | C2×C32×C9 | C2 |
| socle | C2×C33 | D3 |
| LCS | G ▷ C32×C9 | |
| UCS | C1 ◁ C2 | |
250 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 162 |
A minimal faithful permutation representation — 4 generators on 17 points:
(1,2)(4,5)(7,8)(10,12)(11,15)(13,14)(16,17);(1,2)(3,4,5);(6,7,8);(9,17,12,15,14,13,11,10,16)84 conjugacy classes — view the full character table.
| maximal subgroup of | #10619870, #10619872, #10619873, … |
| maximal normal subgroup of | #10619870, #10619872, #10619873, … |
| maximal quotient of | #10619985, C4×C3⋊(C3⋊D9), C4⋊(C3⋊(C3⋊D9)), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(324, 149);