Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C2×C42 |
| unambiguous semidirect product | #2844⋊C3 |
| Fitting, derived, Sylow | #2844 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #5464 |
| commutator, Fitting, cosocle, hypocenter | #2844 | C3 |
| Frattini, socle | C23 | C22⋊A4 |
| LCS | G ▷ #2844 | |
| UCS | C1 ◁ C2 | |
91 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,3,7)(5,6,8)(10,14,21)(11,15,23)(12,13,20)(16,24,19)(17,22,18);(1,5,4,7)(2,8,6,3)(10,17)(11,21)(14,24)(15,16)(18,19)(22,23);(9,22,20,14)(10,19,16,18)(11,17,21,15)(12,23,13,24)Polycyclic presentation:
3,2^7|b^2=g,c^2=h,d^2=h,e^2=f,b^a=be,c^a=d,d^a=cdh,e^a=bf,f^a=g,g^a=fg,c^b=cf,d^b=dfg,d^c=dh,e^c=eg,e^d=ef14 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5870);