Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1828 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C26 |
| direct product | C3×#10770 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #381 |
| commutator | C23 | C2×C3×C42 |
| Frattini, cosocle | C23×C4 | C23×C3 |
| prime radical | C24×C3 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
1162 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)(8,9,12,17,13,16,26,31)(10,14,20,29,19,28,24,32)(11,21,22,33,23,15,25,30)(18,34,36,39,35,27,37,38);(8,10)(9,14,16,28)(11,18)(12,24)(13,19)(15,27,21,34)(17,32,31,29)(20,26)(22,37)(23,35)(25,36)(30,39,33,38);(5,7)(8,11,13,23)(9,15,16,21)(10,18,19,35)(12,25,26,22)(14,27,28,34)(17,33,31,30)(20,37,24,36)(29,39,32,38)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 3758);