Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of D32 |
| automorphism | Aut(G) = C22×#64241 (order 1152) · Inn(G) = D32 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C2 |
| direct product | C3×#3710 |
| derived | C32 ⋊ C2×C3×C8 |
| Sylow | C33 ⋊ C2×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C4 | D32 |
| commutator, hypocenter | C32 | C2×C3×C8 |
| Frattini | C4 | C3×D32 |
| Fitting | C33×C4 | C22 |
| socle | C2×C33 | C2×C4 |
| prime radical | C2×C3×C3⋊D3 | C4 |
| cosocle | C32×C4 | C22×C3 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C3×C4 | |
126 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 17 points:
(1,5,7,2,4,8,3,6)(9,10,11)(13,14)(15,16,17);(1,5,7,2,4,8,3,6)(12,13,14)(16,17)108 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D32⋊C8, C3×#63729, C3×#63730, … |
| maximal normal subgroup of | C3×D32⋊C8, C3×#63729, C3×#63730, … |
| maximal quotient of | C3×#63542, C3×#63544, C3×#63553, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 415);