Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×Q8×Dic6 |
| Sylow | C3 ⋊ C2×Q82 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C23×D3 |
| commutator, cosocle | C22×C3 | C25 |
| Frattini | C22 | C24×D3 |
| Fitting | C2×C3×C4×Q8 | C2 |
| socle, prime radical | C23×C3 | C24 |
| hypercenter | C2×C4×Q8 | D3 |
| hypocenter | C3 | C2×Q82 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 | |
1112 conjugacy classes of subgroups; only the 621 normal subgroups are shown.
This group has 1112 conjugacy classes of subgroups — too many to draw. Only its 621 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 21 points:
(3,4,5)(6,8,9,13)(7,10,11,12);(1,2)(4,5)(6,7,9,11)(8,12,13,10);(14,16,17,21)(15,18,19,20);(14,15,17,19)(16,20,21,18);(1,2)90 conjugacy classes — view the full character table.
| maximal subgroup of | #10686953, #10686954, #10686956, … |
| maximal normal subgroup of | #10686953, #10686954, #10686956, … |
| maximal quotient of | C2×Q8×Dic3⋊C4, C2×C4⋊C4×Dic6, C2×Q8×C4⋊Dic3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18376);