Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23 |
| direct product | C2×B(4,𝔽2)⋊C3 |
| unambiguous semidirect product | (C2×B(4,𝔽2))⋊C3 |
| Fitting, Sylow | C2×B(4,𝔽2) ⋊ C3 |
| derived | H(5,𝔽2) ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C22 | #120⋊C3 |
| commutator, cosocle, hypocenter | H(5,𝔽2) | C22×C3 |
| Frattini | C23 | C22×A4 |
| Fitting | C2×B(4,𝔽2) | C3 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C22 | |
243 conjugacy classes of subgroups; only the 20 normal subgroups are shown.
This group has 243 conjugacy classes of subgroups — too many to draw. Only its 20 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 10 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 2 generators on 10 points:
(3,4,6)(5,9)(7,8,10);(1,2)(3,4)(5,6)(7,8)(9,10)32 conjugacy classes — view the full character table.
| maximal subgroup of | C4×B(4,𝔽2)⋊C3, #21482⋊C3, #32898⋊C3, … |
| maximal normal subgroup of | C4×B(4,𝔽2)⋊C3, #21482⋊C3, #32898⋊C3, … |
| maximal quotient of | C2×#1790⋊C3, C4×B(4,𝔽2)⋊C3, C2×#2029⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5833);