Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of #3714 |
| automorphism | Aut(G) = C22×D32×D4 (order 1152) · Inn(G) = #626 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C3×#3714 |
| Sylow | C33 ⋊ D8 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | #626 |
| commutator, cosocle | C32×C4 | C22×C3 |
| Frattini | C4 | C3×D32 |
| Fitting | C33×C4 | C22 |
| socle | C2×C33 | D4 |
| hypercenter | C3×C4 | D32 |
| hypocenter | C32 | C3×D8 |
| LCS | G ▷ C32×C4 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2×C3 ◁ C3×C4 | |
134 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 134 conjugacy classes of subgroups — too many to draw. Only its 36 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:
(1,8,3,2,4,5,6,7)(9,10,11)(13,14)(15,16,17);(2,5)(3,6)(7,8)(12,13,14)(16,17)72 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#3714, #11722831, #11722832, … |
| maximal normal subgroup of | D3×#3714, #11722831, #11722832, … |
| maximal quotient of | C3×#63546, C3×#63548, C3×#63549, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 419);