Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C2×#5032 |
| Fitting | C2×C3×#120 ⋊ C2 |
| Sylow | C3 ⋊ C2×B(4,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #937 |
| commutator, cosocle | C23×C3 | C24 |
| Frattini | C23 | C23×D3 |
| Fitting | C2×C3×#120 | C2 |
| socle | C22×C3 | #120 |
| hypercenter | C2×#120 | D3 |
| hypocenter | C3 | C2×B(4,𝔽2) |
| LCS | G ▷ C23×C3 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C2×#120 | |
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);(1,2)(6,7)(8,10)(9,11)(12,13);(4,5)(8,12)(10,13);(1,2)60 conjugacy classes — view the full character table.
| maximal subgroup of | #10712445, #10712447, #10712459, … |
| maximal normal subgroup of | #10712445, #10712447, #10712459, … |
| maximal quotient of | C2×#70968, C2×#69696, C2×#70972, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15179);