Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C22×D3×D8 |
| Fitting | C22×C3×D8 ⋊ C2 |
| Sylow | C3 ⋊ C23×D8 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | D3×D4 |
| commutator, cosocle | C3×C4 | C25 |
| Frattini | C4 | C24×D3 |
| Fitting | C22×C3×D8 | C2 |
| socle | C23×C3 | C2×D4 |
| hypercenter | C22×D8 | D3 |
| hypocenter | C3 | C23×D8 |
| LCS | G ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C22×C4 ◁ C22×D8 | |
1976 conjugacy classes of subgroups; only the 565 normal subgroups are shown.
This group has 1976 conjugacy classes of subgroups — too many to draw. Only its 565 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 5 generators on 15 points:
(1,2)(3,4)(5,6,7)(8,9)(10,15)(11,13)(12,14);(1,2)(3,4)(9,12)(10,14)(13,15);(6,7);(3,4);(1,2)84 conjugacy classes — view the full character table.
| maximal subgroup of | #10710827, #10710848, D3×#1879, … |
| maximal normal subgroup of | #10710827, #10710848, D3×#1879, … |
| maximal quotient of | C22×D3×#102, C22×D3×#107, C2×C4×D3×D8, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19745);