Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Fitting, derived, Sylow | C42×Q8 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C22⋊(C42⋊C3) |
| commutator, Fitting, cosocle, hypocenter | C42×Q8 | C3 |
| Frattini, socle | C23 | C22⋊A4 |
| LCS | G ▷ C42×Q8 | |
| UCS | C1 ◁ C2 | |
139 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,5,3)(6,7,8)(9,10,14)(11,15,20)(12,16,19)(13,17,18);(9,11,13,12)(14,19,18,20);(1,7,4,5)(2,3,6,8)(9,13)(10,16,17,15)(11,12)(14,19,18,20)Polycyclic presentation:
3,2^7|b^2=f,c^2=h,d^2=g,e^2=fh,b^a=bc,c^a=bfgh,d^a=eg,e^a=de,f^a=gh,g^a=fh,h^a=f,c^b=cfg32 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5860);