Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C24×C4 |
| direct product | C3×#10912 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C2×C22⋊C4 |
| commutator | C23 | C2×C3×C42 |
| Frattini, cosocle | C25 | C23×C3 |
| prime radical | C25×C3 | C23 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C24×C3×C4 ◁ G | |
1266 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(4,5)(6,7)(8,9,11,10)(12,13)(14,18)(15,17)(16,19);(12,14)(13,16,17,18)(15,19);(5,7)(8,10,11,9)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 3900);