Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C32 · Schur multiplier (H2) = C24 |
| direct product | C2×C3×C22⋊C32 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C16 | C22 |
| commutator | C2 | C22×C3×C32 |
| Frattini, cosocle | C2×C16 | C23×C3 |
| socle | C23×C3 | C2×C16 |
| prime radical | C24×C3 | C16 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C16 ◁ G | |
342 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2)(39,40,41);(3,4)(5,6)(7,8,9,13,10,14,17,23,11,15,18,24,20,26,29,33,12,16,19,25,21,27,30,34,22,28,31,35,32,36,37,38);(4,6)480 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6540);