Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | #41×#67 |
| Fitting | C22×C3×#41 ⋊ C2 |
| Sylow | C3 ⋊ D4×#41 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Frattini | C22 | C24×D3 |
| Fitting | C22×C3×#41 | C2 |
| hypercenter | C22×#41 | D3 |
| hypocenter | C3 | D4×#41 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#41 | |
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)(5,7);(2,3)(4,5)(6,7);(8,10,11,14)(9,13,12,15);(9,12)(13,15);(8,9)(10,13)(11,12)(14,15)90 conjugacy classes — view the full character table.
| maximal subgroup of | #10810854, #67×#104, #10810865, … |
| maximal normal subgroup of | #10810854, #67×#104, #10810865, … |
| maximal quotient of | #41×(C2×D3)⋊C4, #41×#217, #41×Dic3⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19848);