Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42 · Schur multiplier (H2) = C22 |
| direct product | C3×#1386 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #102 |
| commutator | C8 | C3×C42 |
| Frattini, cosocle | C22×C8 | C22×C3 |
| prime radical | C23×C3 | D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C22×C3×C8 ◁ G | |
140 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 140 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:
(1,2,3,4)(5,6)(7,15)(8,16)(9,12)(10,17)(11,14)(13,18)(19,20)(21,22,23);(5,7,9,12,10,14,18,16,11,17,19,20,15,6,8,13)96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 522);