Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| Fitting | C3×#386 ⋊ C2 |
| Sylow | C3 ⋊ #2297 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator | C22×C3 | C23×C4 |
| Fitting | C3×#386 | C2 |
| socle, cosocle | C23×C3 | C24 |
| prime radical | C22×#237 | C2 |
| hypercenter | #386 | D3 |
| hypocenter | C3 | #2297 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #386 | |
936 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 19 points:
(2,3)(12,14,15,18)(13,17,16,19);(1,2,3)(13,16)(17,19);(4,6,7,5)(9,11);(4,5,7,6)(8,9,10,11)(12,13)(14,17)(15,16)(18,19)Polycyclic presentation:
2^7,3|a^2=g,b^2=eg,d^2=f,b^a=be,c^a=cf,h^d=h^284 conjugacy classes — view the full character table.
| maximal subgroup of | #10710128, #10711428, #10794448, … |
| maximal normal subgroup of | #10710128, #10711428, #10794448, … |
| maximal quotient of | #10716299, #10716324, #10716326, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 7234);