Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #1273⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C4 |
| unambiguous semidirect product | #1273⋊C3 |
| Fitting, derived, Sylow | #1273 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C82⋊C3 |
| commutator, Fitting, cosocle, hypocenter | #1273 | C3 |
| Frattini | C2×C42 | A4 |
| socle | C23 | C42⋊C3 |
| LCS | G ▷ #1273 | |
| UCS | C1 ◁ C2 | |
47 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2,7)(3,12,17)(4,11,19)(5,8,18)(6,10,20)(9,21,14)(13,22,16)(15,23,24);(1,14,4,13,6,15,5,3)(2,9,8,12,10,23,11,22)(7,19,20,18)(16,21,17,24)Polycyclic presentation:
3,2^7|b^2=d,c^2=f,d^2=g,e^2=g,f^2=h,b^a=bc,c^a=bdefh,d^a=ef,e^a=df,f^a=dh,g^a=gh,h^a=g,c^b=cdeg32 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 3);