Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23×C3 |
| direct product | C3⋊D3×D12 |
| Fitting | C33×C4 ⋊ C22 |
| Sylow | C33 ⋊ C2×D4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D3×C3⋊D3 |
| commutator, socle, cosocle | C2×C33 | C23 |
| Fitting | C33×C4 | C22 |
| hypercenter | C4 | D3×C3⋊D3 |
| hypocenter | C33 | C2×D4 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ C4 | |
388 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 388 conjugacy classes of subgroups — too many to draw. Only its 72 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:
(1,2,3,4)(8,9,10)(11,12,13);(2,4)(5,6,7)(12,13);(6,7)(9,10)54 conjugacy classes — view the full character table.
| maximal subgroup of | #11722706, #11722707, #11722830, … |
| maximal normal subgroup of | #11722706, #11722707, #11722830, … |
| maximal quotient of | C3⋊D3×#204, C3⋊D3×D24, #11723319, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 672);