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) = C2 |
| direct product | C3×#2183 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C8 | C23 |
| commutator | C22 | C22×C3×C8 |
| Frattini, cosocle | C2×C8 | C23×C3 |
| socle, prime radical | C22×C3 | C22×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C8 ◁ G | |
126 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,4,6)(3,7,8,5)(9,10,12,16,13,17,15,23,14,18,21,24,22,11,19,20)(25,26,27);(1,3,4,8)(2,5,6,7);(9,11,12,20,13,10,15,16,14,17,21,23,22,18,19,24)168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5474);