Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C1 |
| direct product | C3×#1369 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #106 |
| commutator | C8 | C2×C3×C8 |
| Frattini, cosocle | C4×C8 | C22×C3 |
| socle | C2×C3 | #334 |
| prime radical | C22×C3 | #108 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C2×C3×C8 ◁ C3×C4×C8 ◁ G | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 35 points:
(1,2,5,11,7,12,22,28,8,13,21,10,6,24,4,20)(3,14,16,25,18,29,15,26,19,27,23,32,17,31,30,9)(33,34,35);(1,3,4,15,6,17,21,16,8,19,22,30,7,18,5,23)(2,9,10,14,12,25,20,29,13,26,11,27,24,32,28,31)96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 505);