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 | C2×#6289 |
| derived, Sylow | C33 ⋊ C22×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D3×C3⋊D3 |
| commutator, hypocenter | C33 | C22×C4 |
| Frattini | C2 | C2×D3×C3⋊D3 |
| Fitting, socle | C22×C33 | C22 |
| prime radical | C22×C3⋊(C3⋊D3) | C2 |
| cosocle | C2×C33 | C23 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C22 | |
452 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 452 conjugacy classes of subgroups — too many to draw. Only its 100 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)(12,14,15,13);(3,4,5)(10,11)(12,13,15,14);(1,2)(6,7,8)72 conjugacy classes — view the full character table.
| maximal subgroup of | #11722726, #11722900, #11722906, … |
| maximal normal subgroup of | #11722726, #11722900, #11722906, … |
| maximal quotient of | C2×#88785, #11723356, #11723357, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 679);