Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×Q8×S4 |
| Fitting | C23×Q8 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×S4 |
| commutator, cosocle | C2×A4 | C24 |
| Frattini | C2 | C23×S4 |
| Fitting | C23×Q8 | D3 |
| socle | C24 | C22×D3 |
| prime radical | C22×S4 | C22 |
| hypercenter | C2×Q8 | S4 |
| hypocenter | A4 | C22×Q8 |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C22 ◁ C2×Q8 | |
811 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 811 conjugacy classes of subgroups — too many to draw. Only its 116 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 4 generators on 14 points:
(3,4,6,5);(1,2)(4,5,6)(7,9,10,14)(8,11,12,13);(7,8,10,12)(9,13,14,11);(1,2)50 conjugacy classes — view the full character table.
| maximal subgroup of | S4×#101, S4×#103, #11712911, … |
| maximal normal subgroup of | S4×#101, S4×#103, #11712911, … |
| maximal quotient of | C2×Q8×SL(2,ℤ/4), C2×C4⋊C4×S4, C2×#85302, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20057);