Finite Group DatabasebetaFinite Group Databasebeta| order | 324 |
| isoclinism | isoclinic class of (C3⋊D3)⋊C3 |
| automorphism | Aut(G) = C2×D3×((C3⋊D3)⋊C3)⋊C2 (order 1296) · Inn(G) = (C3⋊D3)⋊C3 · Out(G) = C22×D3 |
| homology | Abelianization (H1) = C32×C4 · Schur multiplier (H2) = C32 |
| direct product | C3×(C3⋊Dic3)⋊C3 |
| unambiguous semidirect product | (C3⋊Dic3)⋊C32 |
| unambiguous semidirect product | (C3×C3⋊Dic3)⋊C3 |
| derived | C32 ⋊ C32×C4 |
| Sylow | C3×H(3,𝔽3) ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | (C3⋊D3)⋊C3 |
| commutator, hypocenter | C32 | C32×C4 |
| Frattini | C2×C3 | C32×D3 |
| Fitting, prime radical | C2×C3×H(3,𝔽3) | C2 |
| socle | C2×C32 | C3×D3 |
| cosocle | C2×C32 | C2×C32 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C2×C3 | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,4,3,2)(8,9,11)(10,12,15,14,13,16);(5,6,7)(9,12,13)(11,15,16)60 conjugacy classes — view the full character table.
| maximal subgroup of | #6293⋊C3, C3×#6196, #6289⋊C3, … |
| maximal normal subgroup of | #6293⋊C3, C3×#6196, #6289⋊C3, … |
| maximal quotient of | C3×(C3⋊(C3⋊C8))⋊C3, C2×C3×(C3⋊Dic3)⋊C3, C3×(C3⋊Dic3)⋊C9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(324, 92);