Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23×C3 |
| direct product | C3×D4×C3⋊D3 |
| Fitting | C33×D4 ⋊ C2 |
| Sylow | C33 ⋊ C2×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | C22×C3⋊D3 |
| commutator, cosocle | C2×C32 | C23×C3 |
| Frattini | C2 | C22×C3×C3⋊D3 |
| Fitting | C33×D4 | C2 |
| socle | C2×C33 | C23 |
| hypercenter | C3×D4 | C3⋊D3 |
| hypocenter | C32 | C2×C3×D4 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2×C3 ◁ C3×D4 | |
388 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 388 conjugacy classes of subgroups — too many to draw. Only its 98 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:
(5,6,7)(9,10)(12,13);(2,4)(11,12,13);(1,2)(3,4)(8,9,10)90 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#63782, #11723107, C3×#63926, … |
| maximal normal subgroup of | C3×#63782, #11723107, C3×#63926, … |
| maximal quotient of | C3×#64087, C3×C22⋊C4×C3⋊D3, C3×#64090, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 714);