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 |
| direct product | C3×#2168 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C8 | C23 |
| commutator | C22 | C22×C3×C8 |
| Frattini, cosocle | C2×C8 | C23×C3 |
| socle | C22×C3 | C22×C8 |
| prime radical | C2×C3×D4 | C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C8 ◁ G | |
164 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 164 conjugacy classes of subgroups — too many to draw. Only its 100 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) = 192 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2)(3,4)(5,6,8,12,9,13,11,19,10,14,17,20,18,7,15,16)(21,22,23);(2,4);(5,7,8,16,9,6,11,12,10,13,17,19,18,14,15,20)168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5459);