Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | Q8×#217 |
| Sylow | C3 ⋊ Q8×C22⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator | C22×C3 | C23×C4 |
| Fitting | C23×C3×Q8 | C2 |
| socle, cosocle | C23×C3 | C24 |
| prime radical | C24×C3 | C23 |
| hypercenter | C23×Q8 | D3 |
| hypocenter | C3 | Q8×C22⋊C4 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C23×Q8 | |
820 conjugacy classes of subgroups; only the 333 normal subgroups are shown.
This group has 820 conjugacy classes of subgroups — too many to draw. Only its 333 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(9,11);(2,3)(4,5,6,7)(8,9)(10,11);(12,14,15,19)(13,16,17,18);(12,13,15,17)(14,18,19,16)90 conjugacy classes — view the full character table.
| maximal subgroup of | #10691625, #10691629, #10691653, … |
| maximal normal subgroup of | #10691625, #10691629, #10691653, … |
| maximal quotient of | C4⋊C4×#217, Q8×C3⋊#95, #10716725, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16639);