Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C22×C3 |
| direct product | C32×#237 |
| Fitting | C33×D4 ⋊ C2 |
| Sylow | C33 ⋊ #41 |
| 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 | C32×#41 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C32 ◁ C32×D4 | |
304 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 304 conjugacy classes of subgroups — too many to draw. Only its 138 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:
(4,5,6)(7,8,9)(11,15)(12,16);(1,2,3)(10,16,13,12)(11,17,15,14);(8,9)(10,11)(12,14)(13,15)(16,17)135 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#63925, C3×#63929, C3×#63933, … |
| maximal normal subgroup of | C3×#63925, C3×#63929, C3×#63933, … |
| maximal quotient of | C32×#877, C32×#878, C32×#879, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 705);