Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42 · Schur multiplier (H2) = C22 |
| direct product | C3×#1392 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #99 |
| commutator | C23 | C3×C42 |
| Frattini, cosocle | C2×#41 | C22×C3 |
| socle | C2×C3 | #342 |
| prime radical | C2×C3×#41 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C2×C3×C4 ◁ C2×C3×#41 ◁ G | |
158 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 158 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 35 points:
(1,2,4,9,6,8,15,21)(3,11,12,23,14,7,18,19)(5,10,16,24,17,22,28,30)(13,25,26,32,27,20,29,31)(33,34,35);(1,3,5,13,6,14,17,27)(2,7,10,20,8,11,22,25)(4,12,16,26,15,18,28,29)(9,23,24,32,21,19,30,31)78 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 528);