Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of #1113 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×#6294 |
| derived, Sylow | C33 ⋊ C22×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #1113 |
| commutator, hypocenter | C33 | C22×C4 |
| Frattini | C2 | C2×#1113 |
| Fitting, socle | C22×C33 | C22 |
| prime radical | C22×C3×C3⋊D3 | C2 |
| cosocle | C2×C33 | C23 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C22 | |
290 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 290 conjugacy classes of subgroups — too many to draw. Only its 79 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:
(4,5)(6,7,8)(9,10)(12,13,14,15);(3,4,5)(7,8)(10,11);(1,2)60 conjugacy classes — view the full character table.
| maximal subgroup of | #11722769, #11722791, Dic3×#624, … |
| maximal normal subgroup of | #11722769, #11722791, Dic3×#624, … |
| maximal quotient of | C2×#88808, #11723656, #11723657, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 692);