Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C22×C4 |
| direct product | C3×#8179 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | C2×D4 |
| commutator | C2×C4 | C2×C3×C42 |
| Frattini, cosocle | C2×C42 | C23×C3 |
| socle | C23×C3 | C2×C22⋊C4 |
| prime radical | C22×C3×D4 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ C3×C43 ◁ G | |
562 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2,3)(4,5,7,6)(9,13)(10,11)(14,15)(16,17,20,23)(18,24,25,31)(19,27,28,22)(21,29,30,26);(8,9,11,14,12,15,10,13)(16,18)(17,21,23,30)(19,26)(20,25)(22,24,27,31)(28,29);(4,6,7,5)(8,10,12,11)(9,14,15,13)(18,25)(21,30)(24,31)(26,29)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1167);