Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32696 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#32698 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #2749 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×#2002 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3×C4 ◁ G | |
594 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)(4,5,9,12,8,13,17,19)(6,10,11,18,14,7,15,16);(4,6)(5,10)(7,13)(8,14)(9,15)(11,17)(12,16)(18,19);(4,7)(5,11,13,15)(6,12,14,19)(8,10)(9,16)(17,18);(6,14)(9,17)(12,19)(16,18)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 25686);