Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C2×C42 |
| unambiguous semidirect product | #2736⋊C3 |
| Fitting, derived, Sylow | #2736 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #5464 |
| commutator, Fitting, cosocle, hypocenter | #2736 | C3 |
| Frattini, socle | C23 | C22⋊A4 |
| LCS | G ▷ #2736 | |
| UCS | C1 ◁ C2 | |
135 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,6)(3,10,20)(4,13,23)(5,9,17)(7,16,11)(8,19,12)(14,21,15)(18,24,22);(7,18)(13,21)(15,23)(16,24);(1,22,5,11)(2,13)(3,4,12,14)(6,20,17,19)(7,10)(8,18)(9,21)(15,24,23,16)Polycyclic presentation:
3,2^7|b^2=f,c^2=g,b^a=cf,c^a=bc,d^a=e,e^a=de,f^a=g,g^a=fg,h^a=fgh,d^b=dh,e^b=egh,d^c=dfg,e^c=efgh16 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5864);