Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of (C3⋊D3)⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42 · Schur multiplier (H2) = C3×C4 |
| direct product | C4×(C3⋊Dic3)⋊C3 |
| unambiguous semidirect product | (C4×C3⋊Dic3)⋊C3 |
| derived | C32 ⋊ C3×C42 |
| Sylow | H(3,𝔽3) ⋊ C42 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | (C3⋊D3)⋊C3 |
| commutator, hypocenter | C32 | C3×C42 |
| Frattini, socle | C22×C3 | C2×C3×D3 |
| Fitting | C2×C4×H(3,𝔽3) | C2 |
| prime radical | C22×H(3,𝔽3) | C22 |
| cosocle | C22×C32 | C22×C3 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C2×C4 | |
121 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 2 generators on 17 points:
(2,7,9,4,5,8)(3,6)(14,15,16,17);(1,2,4)(3,5,8)(6,9,7)(10,11,12,13)80 conjugacy classes — view the full character table.
| maximal subgroup of | #11720310, #11720311, #11720312, … |
| maximal normal subgroup of | #11720310, #11720311, #11720312, … |
| maximal quotient of | C4×(C3⋊(C3⋊C8))⋊C3, #63630⋊C3, C8×(C3⋊Dic3)⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 138);