Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of D3×C3⋊S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22×C3 |
| direct product | D3×C3⋊S4 |
| unambiguous semidirect product | (C3×D3)⋊S4 |
| unambiguous semidirect product | (D3×A4)⋊D3 |
| Fitting, socle | C22×C32 ⋊ C2×D3 |
| derived | C32×A4 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C32×A4 | C22 |
| Fitting, socle | C22×C32 | C2×D3 |
| LCS | G ▷ C32×A4 | |
| UCS | C1 | |
207 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 207 conjugacy classes of subgroups — too many to draw. Only its 28 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 10 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 3 generators on 10 points:
(1,2,4,3)(5,6,7)(9,10);(6,7)(8,9,10);(2,3,4)27 conjugacy classes — view the full character table.
| maximal subgroup of | D32×S4, C2×D3×C3⋊S4, #10619745⋊C2, … |
| maximal normal subgroup of | D32×S4, C2×D3×C3⋊S4, D3×(C3⋊S4)⋊C3, … |
| maximal quotient of | #11724191, #11724192, #11724193, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 747);