Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C3×D42 ⋊ C2 |
| Sylow | C3 ⋊ #3498 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C3×D42 | C2 |
| hypercenter | D42 | D3 |
| hypocenter | C3 | #3498 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ D42 | |
1748 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 5 generators on 15 points:
(2,3)(8,10,11,14)(9,13,12,15);(1,2,3)(9,12)(13,15);(4,5,6,7);(8,9)(10,13)(11,12)(14,15);(5,7)(9,12)(10,14)Polycyclic presentation:
2^7,3|c^2=g,e^2=f,b^a=bf,c^a=cg,e^a=ef,d^b=df,h^e=h^269 conjugacy classes — view the full character table.
| maximal subgroup of | #10771672, #10771678, #10771760, … |
| maximal normal subgroup of | #10771672, #10771678, #10771760, … |
| maximal quotient of | #10955285, #10955555, #10955557, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19184);