Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23×C3 |
| direct product | D3×#638 |
| Fitting | C22×C33 ⋊ C22 |
| Sylow | C33 ⋊ C2×D4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D3×C3⋊D3 |
| commutator, socle, cosocle | C2×C33 | C23 |
| Fitting | C22×C33 | C22 |
| hypercenter | C22 | D3×C3⋊D3 |
| hypocenter | C33 | C2×D4 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ C22 | |
388 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 388 conjugacy classes of subgroups — too many to draw. Only its 80 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)(5,6,7)(11,12,13);(1,2)(3,4)(9,10)(12,13);(6,7)(8,9,10)63 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#3804, #11724414, #11724418, … |
| maximal normal subgroup of | D3×#3804, #11724414, #11724418, … |
| maximal quotient of | #11723392, #11723393, #11723414, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 684);