Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×C3×#427 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | D4 |
| commutator | C4 | C23×C3×C4 |
| Frattini, cosocle | C2×C4 | C24×C3 |
| socle | C23×C3 | C2×D4 |
| prime radical | C24×C3 | D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ C23×C3×C4 ◁ G | |
512 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2)(3,4,5);(6,7,8,9)(11,15)(13,20)(16,21)(17,22)(18,24)(23,25);(10,11,13,15,14,17,20,22)(12,16,18,21,19,23,24,25);(1,2)(10,12,14,19)(11,16,17,23)(13,18,20,24)(15,21,22,25)168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17349);