Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of #6189 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
| direct product | C2×#6195 |
| Sylow | C3×C9 ⋊ C2×D4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D3×D9 |
| commutator, cosocle | C2×C3×C9 | C23 |
| Frattini | C2×C3 | C2×D32 |
| Fitting | C22×C3×C9 | C22 |
| socle | C22×C32 | C2×D3 |
| hypocenter | C3×C9 | C2×D4 |
| LCS | G ▷ C2×C3×C9 ▷ C3×C9 | |
| UCS | C1 ◁ C22 | |
194 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 194 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2)(3,4,5)(6,7,8,9)(11,13)(12,16)(14,15)(17,18);(4,5)(7,9)(10,15,17,16,11,13,12,18,14);(1,2)66 conjugacy classes — view the full character table.
| maximal subgroup of | #11720954, #11720956, #11720957, … |
| maximal normal subgroup of | #11720954, #11720956, #11720957, … |
| maximal quotient of | C2×#88422, #11720885, C2×#88424, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 312);