Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of #625 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C2×C3×#625 |
| Sylow | C33 ⋊ C2×D4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C3 | D32 |
| commutator, cosocle | C2×C32 | C23×C3 |
| Frattini | C2 | C2×C3×D32 |
| Fitting, socle | C22×C33 | C22 |
| hypocenter | C32 | C2×C3×D4 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C22×C3 | |
306 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 306 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) = 15 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 15 points:
(1,2)(4,5)(6,7,8)(9,10,11)(12,13)(14,15);(3,4,5)(7,8)(13,15);(1,2)90 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#63739, #11722773, Dic3×#625, … |
| maximal normal subgroup of | C3×#63739, #11722773, Dic3×#625, … |
| maximal quotient of | C2×C3×#3713, C3×#63822, C3×#63823, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 655);