Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of #6290 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23×C3 |
| direct product | C2×#6291 |
| Sylow | C33 ⋊ C2×D4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D3×C3⋊D3 |
| commutator, cosocle | C2×C33 | C23 |
| Frattini | C2 | C2×D3×C3⋊D3 |
| Fitting, socle | C22×C33 | C22 |
| hypocenter | C33 | C2×D4 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C22 | |
452 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 452 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 72 |
A minimal faithful permutation representation — 3 generators on 15 points:
(1,2)(4,5)(7,8)(9,10,11)(12,13,14,15);(3,4,5)(10,11)(13,15);(1,2)(6,7,8)66 conjugacy classes — view the full character table.
| maximal subgroup of | #11722729, #11722932, #11722935, … |
| maximal normal subgroup of | #11722729, #11722932, #11722935, … |
| maximal quotient of | C2×#88790, #11723367, C2×#88794, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 681);