Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of #625 |
| automorphism | Aut(G) = C22×#64241 (order 1152) · Inn(G) = D32 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| direct product | C3×#3798 |
| Sylow | C33 ⋊ #41 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C4 | D32 |
| commutator, cosocle | C2×C32 | C23×C3 |
| Frattini | C2 | C2×C3×D32 |
| Fitting | C33×C4 | C22 |
| socle | C2×C33 | C23 |
| hypocenter | C32 | C3×#41 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C3×C4 | |
214 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 214 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)(2,8,5,6)(9,10,11)(12,13,14)(15,16,17);(2,5)(6,8)(13,14);(1,2)(3,6)(4,5)(7,8)(16,17)90 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#63731, #11722762, #11722821, … |
| maximal normal subgroup of | C3×#63731, #11722762, #11722821, … |
| maximal quotient of | C3×#63850, C3×#63851, C3×#63852, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 646);