Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C22×#930 |
| Fitting | C24×C3×C4 ⋊ C2 |
| Sylow | C3 ⋊ C22×#121 |
| subgroup | G/H | |
|---|---|---|
| center | C24 | C22×D3 |
| commutator, cosocle | C22×C3 | C25 |
| Frattini | C22 | C24×D3 |
| Fitting | C24×C3×C4 | C2 |
| socle | C24×C3 | C23 |
| hypercenter | C24×C4 | D3 |
| hypocenter | C3 | C22×#121 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C24 ◁ C24×C4 | |
2528 conjugacy classes of subgroups; only the 725 normal subgroups are shown.
This group has 2528 conjugacy classes of subgroups — too many to draw. Only its 725 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) = 192 |
A minimal faithful permutation representation — 5 generators on 15 points:
(1,2)(3,4)(5,6,7)(8,9)(10,11);(1,2)(3,4)(12,13,14,15);(6,7)(9,11)(13,15);(3,4);(1,2)120 conjugacy classes — view the full character table.
| maximal subgroup of | #10686543, #10686627, #10686629, … |
| maximal normal subgroup of | #10686543, #10686627, #10686629, … |
| maximal quotient of | C22×#5212, C22×#5213, C22×#4940, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19797);