Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C16 · Schur multiplier (H2) = C2 |
| direct product | C3×#1400 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C8 | C22⋊C4 |
| commutator | C22 | C2×C3×C16 |
| Frattini, cosocle | C2×C16 | C22×C3 |
| socle | C2×C3 | C22⋊C16 |
| prime radical | C23×C3 | C16 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C8 ◁ C2×C3×C16 ◁ G | |
74 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 35 points:
(1,2,3,8,4,9,12,20,5,10,13,21,14,22,25,30,6,7,11,17,15,18,23,27,16,19,24,28,26,29,31,32)(33,34,35);(2,7)(3,11)(9,18)(10,19)(12,23)(13,24)(22,29)(25,31)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 536);