Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #1309⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C4 |
| unambiguous semidirect product | #1309⋊C3 |
| Fitting, derived, Sylow | #1309 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | #1309 | C3 |
| Frattini | C2×C42 | A4 |
| socle | C22 | #95⋊C3 |
| LCS | G ▷ #1309 | |
| UCS | C1 | |
36 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)(3,12,16)(4,11,22)(5,8,17)(6,10,18)(9,19,15)(13,21,23)(14,20,24);(1,15,4,14,6,13,5,3)(2,9,8,20,10,21,11,12)(7,17)(16,19)(18,22)(23,24)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=bdefgh,d^a=efh,e^a=dfgh,f^a=dg,g^a=gh,h^a=g,c^b=cdeg,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, 6);