Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C8 · Schur multiplier (H2) = C24 |
| direct product | C3×#12983 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C8 | C2×D4 |
| commutator | C2×C4 | C22×C3×C8 |
| Frattini, cosocle | C22×C8 | C23×C3 |
| socle | C22×C3 | C2×C22⋊C8 |
| prime radical | C3×#525 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C8 ◁ C23×C3×C8 ◁ G | |
508 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3)(4,5)(6,7)(8,9,12,17,13,16,26,29,14,18,27,32,11,25,20,35)(10,19,22,33,23,31,36,38,24,34,37,39,21,15,28,30);(8,10,20,28,11,21,27,37,14,24,26,36,13,23,12,22)(9,15,29,33,25,34,17,30,18,31,35,39,16,19,32,38);(5,7)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 5971);