Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C8 · Schur multiplier (H2) = C24 |
| direct product | C3×#12984 |
| 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 | |
488 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)(10,19,23,34,24,30,37,38,21,33,36,39,22,15,28,31);(8,10,20,28,11,22,27,36,14,21,26,37,13,24,12,23)(9,15,29,34,25,33,17,31,18,30,35,39,16,19,32,38);(5,7)(10,21)(15,30)(19,33)(22,24)(23,36)(28,37)(31,38)(34,39)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 5972);