Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#5432 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C2×S4 |
| commutator | C2×A4 | C22×C4 |
| Frattini | C22 | C22×S4 |
| Fitting | C26 | D3 |
| socle | C25 | C2×D3 |
| prime radical | C24×A4 | C2 |
| cosocle | C22×A4 | C23 |
| hypercenter | C24 | S4 |
| hypocenter | A4 | C2×C22⋊C4 |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C23 ◁ C24 | |
967 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 967 conjugacy classes of subgroups — too many to draw. Only its 92 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 — 3 generators on 14 points:
(3,4,6)(7,8)(9,10)(11,12)(13,14);(4,5)(8,10)(11,12,13,14);(1,2)56 conjugacy classes — view the full character table.
| maximal subgroup of | #11711294, #11711301, C4×#5432, … |
| maximal normal subgroup of | #11711294, #11711301, C4×#5432, … |
| maximal quotient of | C2×A4⋊#95, C4×#5432, #11711695, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18121);