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) = C25 |
| direct product | C2×C3×#443 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C8 | D4 |
| commutator | C4 | C23×C3×C4 |
| Frattini, cosocle | C2×C4 | C24×C3 |
| socle | C22×C3 | C4×D4 |
| prime radical | C2×C3×#135 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C8 ◁ C22×C3×C8 ◁ G | |
472 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2)(3,4,5);(6,7,9,13,10,14,17,12)(8,11,15,19,16,20,21,18);(6,7)(8,11)(9,13)(10,14)(12,17)(15,19)(16,20)(18,21);(1,2)(6,8,9,15,10,16,17,21)(7,11,13,19,14,20,12,18)168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17393);