Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C25 |
| direct product | C2×C3×#342 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22⋊C4 |
| commutator | C22 | C2×C3×C42 |
| Frattini, cosocle | C24 | C23×C3 |
| prime radical | C25×C3 | C22 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C25×C3 ◁ G | |
680 conjugacy classes of subgroups; only the 232 normal subgroups are shown.
This group has 680 conjugacy classes of subgroups — too many to draw. Only its 232 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 17 points:
(1,2)(3,4,5);(11,14,15,17)(12,16);(6,7,8,9)(10,11)(12,14)(13,15)(16,17)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5030);