Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42 · Schur multiplier (H2) = C22 |
| direct product | C3×#1394 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #99 |
| commutator | C2×C4 | C3×C42 |
| Frattini, cosocle | C22×Q8 | C22×C3 |
| prime radical | C23×C3 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3 ◁ C22×C3×Q8 ◁ G | |
172 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 172 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 23 points:
(5,6,8,12,9,13,17,20)(7,11,14,19,15,10,16,18)(21,22,23);(1,2,3,4)(6,10,13,11)(12,19,20,18)78 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 530);