Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C23×C4 |
| direct product | C3×#10871 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | #121 |
| commutator | C2×C4 | C2×C3×C42 |
| Frattini, cosocle | C23×C4 | C23×C3 |
| socle | C22×C3 | #390 |
| prime radical | C24×C3 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C23×C3×C4 ◁ G | |
746 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)(10,19)(11,24)(12,28)(13,17)(14,18)(15,21)(16,25)(20,29)(22,31)(23,30)(26,32)(27,38)(33,35)(34,37)(36,39);(8,10,13,21,14,23,12,22)(9,15,17,30,18,31,28,19)(11,20,26,34,27,36,25,35)(16,29,24,37,32,39,38,33);(8,11,12,25,14,27,13,26)(9,16,17,24,18,32,28,38)(10,20,22,35,23,36,21,34)(15,29,30,37,31,39,19,33)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 3859);