Finite Group DatabasebetaFinite Group Databasebeta| order | 324 |
| isoclinism | isoclinic class of C3⋊((C3⋊D3)⋊C3) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C32 |
| direct product | C2×C3⋊((C3⋊D3)⋊C3) |
| unambiguous semidirect product | (C2×C3)⋊((C3⋊D3)⋊C3) |
| Fitting | C2×C3×H(3,𝔽3) ⋊ C2 |
| derived | C33 ⋊ C22×C3 |
| Sylow | C3×H(3,𝔽3) ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C3⋊((C3⋊D3)⋊C3) |
| commutator, cosocle, hypocenter | C33 | C22×C3 |
| Frattini | C3 | C2×C3×C3⋊D3 |
| Fitting | C2×C3×H(3,𝔽3) | C2 |
| socle | C2×C32 | C3×D3 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C2 | |
160 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 54 |
A minimal faithful permutation representation — 3 generators on 14 points:
(4,5)(7,12,14,9,10,13)(8,11);(1,2)(3,4,5);(6,7,9)(8,10,13)(11,14,12)42 conjugacy classes — view the full character table.
| maximal subgroup of | #6289⋊C3, #6292⋊C3, #6291⋊C3, … |
| maximal normal subgroup of | #6289⋊C3, #6292⋊C3, #6291⋊C3, … |
| maximal quotient of | #10619956, C4×C3⋊((C3⋊D3)⋊C3), C4⋊(C3⋊((C3⋊D3)⋊C3)), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(324, 144);