Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C8 · Schur multiplier (H2) = C23 |
| direct product | C3×#1282 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C22⋊C4 |
| commutator | C22 | C3×C4×C8 |
| Frattini, cosocle | C23×C4 | C22×C3 |
| socle | C22×C3 | #95 |
| prime radical | C24×C3 | C2×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C23×C3×C4 ◁ G | |
228 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,2,4,7,6,8,5,3)(9,10)(11,15,16,13)(12,14)(17,18,19);(1,3,5,8,6,7,4,2)(9,11)(10,13)(12,16)(14,15)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 418);