Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| automorphism | Aut(G) = C22×D32×D4 (order 1152) · Inn(G) = C2×D32 · Out(G) = C24 |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| direct product | C3×#3796 |
| Sylow | C33 ⋊ #41 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | C2×D32 |
| commutator, cosocle | C2×C32 | C23×C3 |
| Frattini | C2 | C2×C3×D32 |
| Fitting | C33×C4 | C22 |
| socle | C2×C33 | C23 |
| hypercenter | C3×C4 | D32 |
| hypocenter | C32 | C3×#41 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2×C3 ◁ C3×C4 | |
210 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 210 conjugacy classes of subgroups — too many to draw. Only its 64 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 — 3 generators on 17 points:
(1,7,4,3)(2,8,6,5)(9,10,11)(12,13,14)(15,16,17);(1,6,4,2)(3,5,7,8)(13,14);(3,7)(5,8)(16,17)72 conjugacy classes — view the full character table.
| maximal subgroup of | #11722857, #11722858, #11722859, … |
| maximal normal subgroup of | #11722857, #11722858, #11722859, … |
| maximal quotient of | C3×#63843, C3×#63846, C3×#63848, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 644);