Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C8 · Schur multiplier (H2) = C24 |
| direct product | C2×C3×#323 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22⋊C4 |
| commutator | C22 | C22×C3×C8 |
| Frattini, cosocle | C22×C4 | C23×C3 |
| prime radical | C25×C3 | C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C23×C3×C4 ◁ G | |
456 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 456 conjugacy classes of subgroups — too many to draw. Only its 136 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2)(3,4,5);(7,10,11,13)(8,12)(14,15,17,19,18,20,16,21);(6,7)(8,10)(9,11)(12,13)(14,16)(15,19)(17,18)(20,21)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4747);