Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | Q8×C4⋊Dic3 |
| Sylow | C3 ⋊ Q8×C4⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator | C22×C3 | C23×C4 |
| Fitting | C2×C3×C4×Q8 | C2 |
| socle, prime radical, cosocle | C23×C3 | C24 |
| hypercenter | C2×C4×Q8 | D3 |
| hypocenter | C3 | Q8×C4⋊C4 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 | |
568 conjugacy classes of subgroups; only the 333 normal subgroups are shown.
This group has 568 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) = 384 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(8,9,10,11);(2,3)(4,5,6,7)(9,11);(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 | #10689359, #10689363, #10689387, … |
| maximal normal subgroup of | #10689359, #10689363, #10689387, … |
| maximal quotient of | C4⋊C4×C4⋊Dic3, Q8×C3⋊#95, #10716733, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15034);