Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C22×C3×D32 |
| unambiguous semidirect product | (C3×D3)⋊(C23×C3) |
| Fitting, socle | C22×C33 ⋊ C22 |
| derived | C32 ⋊ C24×C3 |
| Sylow | C33 ⋊ C24 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C3 | D32 |
| commutator, cosocle, hypocenter | C32 | C24×C3 |
| Frattini | C1 | G |
| Fitting, socle | C22×C33 | C22 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C22×C3 | |
642 conjugacy classes of subgroups; only the 208 normal subgroups are shown.
This group has 642 conjugacy classes of subgroups — too many to draw. Only its 208 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 4 generators on 13 points:
(1,2)(3,4)(6,7)(8,9,10)(11,12,13);(5,6,7)(9,10);(3,4);(1,2)108 conjugacy classes — view the full character table.
| maximal subgroup of | D3×(C2×D3)⋊Dic3, #11722902, #11722946, … |
| maximal normal subgroup of | D3×(C2×D3)⋊Dic3, #11722902, #11722946, … |
| maximal quotient of | C2×C3×D3×Dic6, C2×C3×#3795, C2×C3×#3796, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 767);