Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#5576 |
| Fitting | C2×C3×C4×Q8 ⋊ C2 |
| Sylow | C3 ⋊ C2×#548 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C23×D3 |
| commutator, cosocle | C22×C3 | C25 |
| Frattini | C22 | C24×D3 |
| Fitting | C2×C3×C4×Q8 | C2 |
| socle | C23×C3 | C24 |
| hypercenter | C2×C4×Q8 | D3 |
| hypocenter | C3 | C2×#548 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 | |
1620 conjugacy classes of subgroups; only the 573 normal subgroups are shown.
This group has 1620 conjugacy classes of subgroups — too many to draw. Only its 573 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 17 points:
(3,4,5)(6,7,8,9)(10,13,14,11)(12,16,17,15);(1,2)(10,12,14,17)(11,15,13,16);(10,11,14,13)(12,16,17,15);(4,5)(7,9);(1,2)90 conjugacy classes — view the full character table.
| maximal subgroup of | #10686819, #10761483, #10810563, … |
| maximal normal subgroup of | #10686819, #10761483, #10810563, … |
| maximal quotient of | C2×Q8×(C2×D3)⋊C4, C2×#82457, C2×#82460, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18388);