Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of #625 |
| automorphism | Aut(G) = C25×D32 (order 1152) · Inn(G) = D32 · Out(G) = C25 |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | C3×Dic3⋊Dic3 |
| unambiguous semidirect product | Dic3⋊(C3×Dic3) |
| Sylow | C33 ⋊ C4⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C3 | D32 |
| commutator | C2×C32 | C2×C3×C4 |
| Frattini | C22 | C3×D32 |
| Fitting, socle, prime radical | C22×C33 | C22 |
| cosocle | C22×C32 | C22×C3 |
| hypocenter | C32 | C3×C4⋊C4 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C22×C3 | |
162 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 162 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 17 points:
(1,4,3,2)(6,8)(9,10,11)(13,14)(15,16,17);(5,8,7,6)(12,13,14)(16,17)90 conjugacy classes — view the full character table.
| maximal subgroup of | D3×Dic3⋊Dic3, #11722908, #11722911, … |
| maximal normal subgroup of | D3×Dic3⋊Dic3, #11722908, #11722911, … |
| maximal quotient of | C3×Dic3⋊(C3⋊C8), C3×#63573, C3×#63575, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 428);