Finite Group DatabasebetaFinite Group Databasebeta| order | 432 |
| isoclinism | isoclinic class of #89101 |
| automorphism | Aut(G) = S4×GO+(4,𝔽2) (order 1728) · Inn(G) = G · Out(G) = C22 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting, socle | C22×C32 ⋊ C2×D3 |
| derived | C32×A4 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C32×A4 | C22 |
| Fitting, socle | C22×C32 | C2×D3 |
| LCS | G ▷ C32×A4 | |
| UCS | C1 | |
164 conjugacy classes of subgroups; only the 20 normal subgroups are shown.
This group has 164 conjugacy classes of subgroups — too many to draw. Only its 20 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 10 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 10 points:
(1,2,4)(5,6)(9,10);(2,3)(6,7)(8,9,10)Polycyclic presentation:
2^2,3^3,2^2|c^a=c^2,d^a=d^2,f^a=g,g^a=f,d^b=d^2,e^b=e^2,f^c=g,g^c=fg24 conjugacy classes — view the full character table.
| maximal subgroup of | #157⋊S4, D32⋊S4, D32×S4, … |
| maximal normal subgroup of | #157⋊S4, D32⋊S4, D32×S4, … |
| maximal quotient of | #11724206, #11724207, #11724208, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(432, 748);