Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of #6258 |
| automorphism | Aut(G) = C2×((C3⋊S4)⋊C3)⋊C2 (order 864) · Inn(G) = C3⋊S4 · Out(G) = C2×D3 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
| direct product | C2×#6258 |
| Fitting | C23×C32 ⋊ D3 |
| derived | C22⋊H(3,𝔽3) ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | C3⋊S4 |
| commutator, cosocle, hypocenter | C22⋊H(3,𝔽3) | C22 |
| Frattini | C3 | C2×C3⋊S4 |
| Fitting | C23×C32 | D3 |
| socle | C23×C3 | C3⋊D3 |
| LCS | G ▷ C22⋊H(3,𝔽3) | |
| UCS | C1 ◁ C2×C3 | |
166 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 166 conjugacy classes of subgroups — too many to draw. Only its 23 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) = 18 |
A minimal faithful permutation representation — 3 generators on 15 points:
(1,2)(3,4,6,5)(8,10)(11,14)(13,15);(1,2)(7,8,10)(9,11,14)(12,15,13);(4,5,6)(8,11,15)(10,13,14)38 conjugacy classes — view the full character table.
| maximal subgroup of | #6258⋊C4, #11722271, #11722273, … |
| maximal normal subgroup of | #6258⋊C4, #11722271, #11722273, … |
| maximal quotient of | #11722380, C4×#6258, C4⋊#6258, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 538);