Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23 |
| direct product | C2×#393⋊C3 |
| unambiguous semidirect product | (C2×#393)⋊C3 |
| Fitting, Sylow | C2×#393 ⋊ C3 |
| derived | C22×Q8 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #120⋊C3 |
| commutator, cosocle, hypocenter | C22×Q8 | C22×C3 |
| Frattini | C23 | C22×A4 |
| Fitting | C2×#393 | C3 |
| socle | C24 | C2×A4 |
| LCS | G ▷ C22×Q8 | |
| UCS | C1 ◁ C22 | |
241 conjugacy classes of subgroups; only the 31 normal subgroups are shown.
This group has 241 conjugacy classes of subgroups — too many to draw. Only its 31 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 18 points:
(3,4,6)(7,9,8)(11,12)(13,16,17,15,14,18);(1,2)(3,9,7,4)(5,8,10,6)(12,18)(15,16)36 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#393⋊C3, #14301⋊C3, #14609⋊C3, … |
| maximal normal subgroup of | C4×#393⋊C3, #14301⋊C3, #14609⋊C3, … |
| maximal quotient of | C2×#1446⋊C3, C4×#393⋊C3, #14873⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5758);