Finite Group DatabasebetaFinite Group Databasebeta| order | 396 |
| isoclinism | isoclinic class of A4 |
| automorphism | Aut(G) = C2×C5×D3×S4 (order 1440) · Inn(G) = A4 · Out(G) = C22×C5×D3 |
| homology | Abelianization (H1) = C32×C11 · Schur multiplier (H2) = C2×C3 |
| Fitting, socle | C22×C3×C11 ⋊ C3 |
| derived, Sylow | C22 ⋊ C32×C11 |
| Hall | C22×C11 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C11 | A4 |
| commutator, cosocle, hypocenter | C22 | C32×C11 |
| Frattini | C1 | G |
| Fitting, socle | C22×C3×C11 | C3 |
| LCS | G ▷ C22 | |
| UCS | C1 ◁ C3×C11 | |
28 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 132 |
A minimal faithful permutation representation — 2 generators on 18 points:
(2,3,4)(8,9,10,11,12,13,14,15,16,17,18);(1,4)(2,3)(5,6,7)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(396, 24);