Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #1304⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #1304⋊C3 |
| Fitting, derived, Sylow | #1304 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | #1304 | C3 |
| Frattini | C25 | A4 |
| socle | C22 | #95⋊C3 |
| LCS | G ▷ #1304 | |
| UCS | C1 | |
91 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 2 generators on 16 points:
(2,7,12)(3,10,14)(4,6,5)(8,16,11)(9,15,13);(1,7,6,16)(2,13)(3,10)(4,14,5,15)(8,12)(9,11)Polycyclic presentation:
3,2^7|b^2=d,c^2=f,b^a=bc,c^a=bdefg,d^a=efh,e^a=dfgh,f^a=dgh,g^a=gh,h^a=g,c^b=cde,e^b=eg,f^b=fh,d^c=dg,e^c=egh14 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4);