Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42 · Schur multiplier (H2) = C22 |
| direct product | C3×#1383 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #106 |
| commutator | C8 | C3×C42 |
| Frattini, cosocle | C22×C8 | C22×C3 |
| prime radical | C23×C3 | SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C22×C3×C8 ◁ G | |
136 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 80 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:
(6,10,11,16)(7,13)(8,15)(12,19,18,17)(14,20)(21,22,23);(1,2,3,4)(5,6,7,10,8,12,13,17,9,11,14,16,15,18,20,19)96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 519);