Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C8 · Schur multiplier (H2) = C22×C4 |
| direct product | C3×#1831 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | C23 |
| commutator | C22 | C22×C3×C8 |
| Frattini, cosocle | C22×C4 | C23×C3 |
| socle, prime radical | C23×C3 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ G | |
260 conjugacy classes of subgroups; only the 160 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 160 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) = 384 |
A minimal faithful permutation representation — 3 generators on 23 points:
(2,3)(5,6,7,9,8,10,11,12)(13,14)(15,17)(16,18)(19,20)(21,22,23);(1,2,4,3)(13,15,16,19)(14,17,18,20);(1,3,4,2)(14,18)(17,20)168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5122);