Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | D4×#239 |
| Fitting | C3×D4×Q8 ⋊ C2 |
| Sylow | C3 ⋊ D4×#41 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C3×D4×Q8 | C2 |
| hypercenter | D4×Q8 | D3 |
| hypocenter | C3 | D4×#41 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ D4×Q8 | |
1648 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)(8,10,11,14)(9,12,13,15);(8,9,11,13)(10,15,14,12);(2,3)(10,14)(12,15);(5,7);(4,5)(6,7)75 conjugacy classes — view the full character table.
| maximal subgroup of | #10771136, #10771144, #10771151, … |
| maximal normal subgroup of | #10771136, #10771144, #10771151, … |
| maximal quotient of | C22⋊C4×#239, C4⋊C4×#239, D4×#892, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19318);