Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C2×#4741 |
| Fitting | C2×C3×#99 ⋊ C2 |
| Sylow | C3 ⋊ C2×B(4,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #882 |
| commutator, cosocle | C23×C3 | C24 |
| Frattini | C23 | C23×D3 |
| Fitting | C2×C3×#99 | C2 |
| socle | C22×C3 | #120 |
| hypercenter | C2×#99 | D3 |
| hypocenter | C3 | C2×B(4,𝔽2) |
| LCS | G ▷ C23×C3 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C2×#99 | |
1018 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 1018 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 4 generators on 13 points:
(1,2)(3,4,5)(7,10,11,13)(8,12);(1,2)(6,7)(8,10)(9,11)(12,13);(4,5)(8,12)(10,13);(1,2)54 conjugacy classes — view the full character table.
| maximal subgroup of | #10712446, #10712448, #10712460, … |
| maximal normal subgroup of | #10712446, #10712448, #10712460, … |
| maximal quotient of | C2×#68284, C2×#68106, C2×#68286, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 11260);