Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24985 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#29232 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #555 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C2×C3 | #2572 |
| prime radical | C3×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C22×C3×C4 ◁ G | |
552 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(4,5,8,12,9,13,7,10)(6,14,15,19,16,11,17,18);(5,10,13,12)(6,15,16,17)(11,14)(18,19);(4,6)(5,11)(7,15)(8,17)(9,16)(10,18)(12,19)(13,14);(5,10,13,12)(11,18,14,19)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 22220);