Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22×C3 |
| direct product | C3×#3830 |
| Fitting | C33×D4 ⋊ C2 |
| Sylow | C33 ⋊ #41 |
| 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 | C3×#41 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2×C3 ◁ C3×D4 | |
304 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 304 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 3 generators on 17 points:
(2,6)(3,7)(9,10,11)(15,16,17);(1,7,4,3)(2,8,6,5)(12,13,14);(1,2)(3,5)(4,6)(7,8)(13,14)(16,17)90 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#63783, #11723108, C3×#63927, … |
| maximal normal subgroup of | C3×#63783, #11723108, C3×#63927, … |
| maximal quotient of | C3×#64086, C3×#64087, C3×#64088, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 715);