Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33551 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C3×#33551 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×D4 | C24×C3 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C23×C3 ◁ C22×C3×D4 ◁ G | |
1326 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)(7,10)(8,11)(9,14)(13,18)(15,17)(16,19);(4,6)(5,9)(7,14)(8,13)(10,12)(11,16)(15,19)(17,18);(7,15)(14,19);(6,13)(9,16)(12,18)(14,19)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26539);