Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23×C4 |
| direct product | A4×#127 |
| Fitting, Sylow | C22×#127 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×A4 |
| commutator, socle, cosocle | C24 | C23×C3 |
| Fitting | C22×#127 | C3 |
| hypercenter | #127 | A4 |
| hypocenter | C22 | C3×#127 |
| LCS | G ▷ C24 ▷ C22 | |
| UCS | C1 ◁ C22 ◁ #127 | |
680 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 680 conjugacy classes of subgroups — too many to draw. Only its 78 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 12 points:
(2,3,4)(9,12,11,10);(1,4)(2,3)(5,8,7,6);(6,8)(10,12)56 conjugacy classes — view the full character table.
| maximal subgroup of | A4×#331, A4×#353, A4×#453, … |
| maximal normal subgroup of | A4×#331, A4×#353, A4×#453, … |
| maximal quotient of | A4×#384, A4×#390, A4×#392, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18168);