Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of (C3⋊S4)⋊C3 |
| automorphism | Aut(G) = C2×((C3⋊S4)⋊C3)⋊C2 (order 864) · Inn(G) = (C3⋊S4)⋊C3 · Out(G) = C22 |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22×C3 |
| direct product | C2×(C3⋊S4)⋊C3 |
| unambiguous semidirect product | (C2×C3⋊S4)⋊C3 |
| Fitting | C23×C32 ⋊ D3 |
| derived | C3×A4 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | (C3⋊S4)⋊C3 |
| commutator, cosocle, hypocenter | C3×A4 | C22×C3 |
| Frattini | C3 | C2×C3×S4 |
| Fitting | C23×C32 | D3 |
| socle | C23×C3 | C3×D3 |
| LCS | G ▷ C3×A4 | |
| UCS | C1 ◁ C2 | |
134 conjugacy classes of subgroups; only the 22 normal subgroups are shown.
This group has 134 conjugacy classes of subgroups — too many to draw. Only its 22 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 18 |
A minimal faithful permutation representation — 2 generators on 15 points:
(3,4,6,5)(8,13,15,10,11,14)(9,12);(1,2)(4,5,6)(7,8,10)(9,11,14)(12,15,13)38 conjugacy classes — view the full character table.
| maximal subgroup of | ((C3⋊S4)⋊C3)⋊C4, #11722272, #11722273, … |
| maximal normal subgroup of | ((C3⋊S4)⋊C3)⋊C4, #11722272, #11722273, … |
| maximal quotient of | #64259⋊C3, C4×(C3⋊S4)⋊C3, C4⋊((C3⋊S4)⋊C3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 535);