Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×#423 ⋊ C2 |
| Sylow | C3 ⋊ D4×M16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C23×D3 |
| commutator, socle | C22×C3 | C23×C4 |
| Fitting | C3×#423 | C2 |
| prime radical | C22×D12 | C4 |
| cosocle | C2×C3×C4 | C24 |
| hypercenter | #423 | D3 |
| hypocenter | C3 | D4×M16 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C4 ◁ #423 | |
544 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 15 points:
(1,2,3)(9,10)(14,15);(2,3)(8,10,11,15,12,9,13,14);(2,3)(5,7);(4,5)(6,7)(8,9,11,14,12,10,13,15)Polycyclic presentation:
2^7,3|a^2=f,c^2=f,f^2=g,b^a=be,d^a=dg,h^b=h^2,d^c=dg,h^c=h^278 conjugacy classes — view the full character table.
| maximal subgroup of | #10689512, #10689516, #10689526, … |
| maximal normal subgroup of | #10689512, #10689516, #10689526, … |
| maximal quotient of | #10766608, #10766616, #10766625, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 11619);