Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C24 |
| direct product | C2×C3×#341 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | D4 |
| commutator | C4 | C2×C3×C42 |
| Frattini, cosocle | C22×C4 | C23×C3 |
| socle | C23×C3 | C4⋊C4 |
| prime radical | C24×C3 | Q8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ C23×C3×C4 ◁ G | |
392 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2)(3,4,5);(10,11,13,17,14,18,22,16)(12,19,20,15,21,23,25,24);(6,7,8,9)(10,12,13,20,14,21,22,25)(11,15,16,19,18,24,17,23)168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5028);