Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C27 |
| direct product | C22×D4×A4 |
| unambiguous semidirect product | C4⋊(C23×A4) |
| Fitting, Sylow | C24×D4 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C22×A4 |
| commutator, cosocle | C23 | C24×C3 |
| Frattini | C2 | C24×A4 |
| Fitting | C24×D4 | C3 |
| socle | C25 | C22×C3 |
| hypercenter | C22×D4 | A4 |
| hypocenter | C22 | C22×C3×D4 |
| LCS | G ▷ C23 ▷ C22 | |
| UCS | C1 ◁ C23 ◁ C22×D4 | |
2152 conjugacy classes of subgroups; only the 234 normal subgroups are shown.
This group has 2152 conjugacy classes of subgroups — too many to draw. Only its 234 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 — 4 generators on 12 points:
(1,2)(3,4)(6,7,8)(9,10)(11,12);(1,2)(3,4)(5,8)(6,7)(10,12);(3,4);(1,2)80 conjugacy classes — view the full character table.
| maximal subgroup of | A4×#386, A4×#390, A4×#392, … |
| maximal normal subgroup of | A4×#386, A4×#390, A4×#392, … |
| maximal quotient of | C22×A4×C22⋊C4, C22×A4×C4⋊C4, C2×C4×D4×A4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20102);