Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of #6196 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
| direct product | C2×#6200 |
| Sylow | H(3,𝔽3) ⋊ C2×D4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | ((C3⋊D3)⋊C3)⋊C2 |
| commutator, cosocle | C2×H(3,𝔽3) | C23 |
| Frattini | C2×C3 | C2×D32 |
| Fitting | C22×H(3,𝔽3) | C22 |
| socle | C22×C3 | D32 |
| hypocenter | H(3,𝔽3) | C2×D4 |
| LCS | G ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C22 | |
221 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 221 conjugacy classes of subgroups — too many to draw. Only its 45 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 3 generators on 15 points:
(1,2)(3,4,5,6)(8,14,12,10,11,15)(9,13);(4,6)(7,8,10)(9,11,14,13,12,15);(1,2)38 conjugacy classes — view the full character table.
| maximal subgroup of | #11721100, #11721102, #11721103, … |
| maximal normal subgroup of | #11721100, #11721102, #11721103, … |
| maximal quotient of | C2×#88436, #11721068, #11721069, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 322);