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 | #2276 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C22⋊(C42⋊C3) |
| commutator, Fitting, cosocle, hypocenter | #2276 | C3 |
| Frattini, socle | C23 | C22⋊A4 |
| LCS | G ▷ #2276 | |
| UCS | C1 ◁ C2 | |
133 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,5,8)(3,7,6)(9,10,14)(11,16,18)(12,17,19)(13,15,20);(1,7,4,8)(2,5,6,3)(10,15)(14,20)(16,17)(18,19);(2,6)(3,5)(10,16,15,17)(14,19,20,18)Polycyclic presentation:
3,2^7|b^2=f,c^2=fg,d^2=g,e^2=fh,b^a=cf,c^a=bc,d^a=beh,e^a=bcdefg,f^a=fg,g^a=f,h^a=gh,e^b=efh,d^c=dfh28 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5862);