Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C23×C3 |
| direct product | C32×D3×D4 |
| Fitting | C33×D4 ⋊ C2 |
| Sylow | C33 ⋊ C2×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32 | C22×D3 |
| commutator, cosocle | C2×C3 | C23×C32 |
| Frattini | C2 | C22×C32×D3 |
| Fitting | C33×D4 | C2 |
| socle | C2×C33 | C23 |
| hypercenter | C32×D4 | D3 |
| hypocenter | C3 | C2×C32×D4 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C32 ◁ C32×D4 | |
388 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 388 conjugacy classes of subgroups — too many to draw. Only its 150 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 3 generators on 13 points:
(2,4)(8,9,10)(11,12,13);(1,2)(3,4)(5,6,7);(12,13)135 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D3×#213, C3×D3×#214, C3×#63932, … |
| maximal normal subgroup of | C3×D3×#213, C3×D3×#214, C3×#63932, … |
| maximal quotient of | C32×#878, C32×D3×C22⋊C4, C32×#67⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 704);