Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | D3×D4×Q8 |
| Fitting | C3×D4×Q8 ⋊ C2 |
| Sylow | C3 ⋊ C2×D4×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C3×D4×Q8 | C2 |
| prime radical | C2×D3×D4 | C22 |
| hypercenter | D4×Q8 | D3 |
| hypocenter | C3 | C2×D4×Q8 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ D4×Q8 | |
1568 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 5 generators on 15 points:
(1,2,3)(8,10,11,15)(9,12,13,14);(8,9,11,13)(10,14,15,12);(5,7);(4,5)(6,7);(2,3)75 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1627, D3×#1637, D3×#1651, … |
| maximal normal subgroup of | D3×#1627, D3×#1637, D3×#1651, … |
| maximal quotient of | D3×Q8×C22⋊C4, D3×D4×C4⋊C4, D3×Q8×C4⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19334);