Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| automorphism | Aut(G) = C23×D33 (order 1728) · Inn(G) = D33 · Out(G) = C23 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
| direct product | D3×#626 |
| Sylow | C33 ⋊ C2×D4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D33 |
| commutator, Fitting, socle, cosocle | C2×C33 | C23 |
| hypocenter | C33 | C2×D4 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 | |
290 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 290 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 3 generators on 13 points:
(1,2,3,4)(5,6,7)(9,10)(11,12,13);(2,4)(8,9,10)(12,13);(6,7)45 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#3796, D3×#3798, #11724381, … |
| maximal normal subgroup of | D3×#3796, D3×#3798, #11724381, … |
| maximal quotient of | D3×#3714, D3×#3716, D3×#3717, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 598);