Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C2 |
| direct product | C3×#1328 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #99 |
| commutator | C23 | C2×C3×C8 |
| Frattini, cosocle | C2×M16 | C22×C3 |
| socle | C2×C3 | #324 |
| prime radical | C23×C3 | C2×C8 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C2×C3×C4 ◁ C2×C3×M16 ◁ G | |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 35 points:
(1,2,4,9,5,10,15,23,6,8,14,20,16,21,27,30)(3,7,12,17,13,19,26,28,11,18,24,22,25,29,32,31)(33,34,35);(1,3)(2,7,8,18)(4,12)(5,13)(6,11)(9,22,20,17)(10,19,21,29)(14,24)(15,26)(16,25)(23,31,30,28)(27,32)78 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 464);