Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C1 |
| direct product | C3×#1338 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C22⋊C4 |
| commutator | C2×C4 | C2×C3×C8 |
| Frattini, cosocle | C4×C8 | C22×C3 |
| socle, prime radical | C22×C3 | C22⋊C8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C3×C4×C8 ◁ G | |
86 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 2 generators on 27 points:
(2,5)(3,7)(6,8)(9,10,12,16,11,17,18,24,13,15,21,23,19,14,20,22)(25,26,27);(1,2,4,6)(3,8,7,5)(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)114 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 474);