Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #392⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #1444⋊C3 |
| Fitting, Sylow | #1444 ⋊ C3 |
| derived | #95 ⋊ C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #127⋊C3 |
| commutator, hypocenter | #95 | C3×C4 |
| Frattini, socle | C24 | C2×A4 |
| Fitting | #1444 | C3 |
| prime radical | C2×#95⋊C3 | C2 |
| cosocle | C2×#95 | C2×C3 |
| LCS | G ▷ #95 | |
| UCS | C1 ◁ C22 | |
133 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2,3,4)(5,9,16,6,12,15,8,14,11,10,7,13);(5,10,8,6)(7,12)(11,15)(13,16)Polycyclic presentation:
2,3,2^6|a^2=ch,d^2=f,e^2=g,b^a=bfh,d^a=dfh,e^a=egh,d^b=efgh,e^b=defh,f^b=g,g^b=fgh,e^d=eh36 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 586);