Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32700 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#32700 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #2760 |
| commutator, Frattini, cosocle | C24 | C24×C3 |
| prime radical | C3×#2021 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C24×C3 ◁ G | |
778 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)(6,12,14,9)(7,15,17,10)(8,11)(13,19)(16,18);(4,6)(5,9)(7,16)(8,14)(10,18)(11,12)(13,17)(15,19);(4,7)(5,10)(6,13,14,16)(8,17)(9,18,12,19)(11,15);(9,12)(18,19)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 25688);