Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C1 |
| direct product | C3×#1373 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C8 | D8 |
| commutator | C8 | C2×C3×C8 |
| Frattini, cosocle | C4×C8 | C22×C3 |
| socle | C2×C3 | #335 |
| prime radical | C22×C3 | #108 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C8 ◁ C2×C3×C8 ◁ C3×C4×C8 ◁ G | |
62 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 35 points:
(1,2,5,10,6,12,4,20,7,13,15,9,21,11,22,27)(3,14,17,24,18,29,16,8,19,23,30,25,31,28,32,26)(33,34,35);(1,3,15,30,6,18,22,32,7,19,5,17,21,31,4,16)(2,8,20,28,11,24,10,23,13,26,27,29,12,25,9,14)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 509);