Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of D3×C3⋊D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C3 |
| direct product | C4×D3×C3⋊D3 |
| Fitting | C33×C4 ⋊ C22 |
| derived, Sylow | C33 ⋊ C22×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | D3×C3⋊D3 |
| commutator, hypocenter | C33 | C22×C4 |
| Frattini | C2 | C2×D3×C3⋊D3 |
| Fitting | C33×C4 | C22 |
| socle, cosocle | C2×C33 | C23 |
| prime radical | C2×D3×C3⋊D3 | C2 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C4 | |
388 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 388 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 72 |
A minimal faithful permutation representation — 3 generators on 13 points:
(5,6,7)(9,10)(12,13);(6,7)(11,12,13);(1,2,3,4)(8,9,10)72 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#3710, D3×#3712, #11722818, … |
| maximal normal subgroup of | D3×#3710, D3×#3712, #11722818, … |
| maximal quotient of | D3×C8×C3⋊D3, D3×#3743, C3⋊D3×ΓL(1,𝔽25), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 670);