Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
| direct product | D8×S4 |
| Fitting | C22×D8 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D4×S4 |
| commutator, cosocle | C4×A4 | C23 |
| Frattini | C4 | C22×S4 |
| Fitting | C22×D8 | D3 |
| socle | C23 | D3×D4 |
| hypercenter | D8 | S4 |
| hypocenter | A4 | C2×D8 |
| LCS | G ▷ C4×A4 ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ D8 | |
337 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 337 conjugacy classes of subgroups — too many to draw. Only its 36 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) = 24 |
A minimal faithful permutation representation — 3 generators on 12 points:
(1,2,4,3);(2,3,4)(6,9)(7,12)(10,11);(5,6)(7,9)(8,11)(10,12)35 conjugacy classes — view the full character table.
| maximal subgroup of | #11714200, #11714201, S4×D16, … |
| maximal normal subgroup of | #11714200, #11714201, S4×D16, … |
| maximal quotient of | #11712268, #11712269, #11712270, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18032);