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×#33553 |
| 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 ◁ C2×C3×C4 ◁ C2×C3×#41 ◁ G | |
1252 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)(6,10)(7,9)(8,11)(12,18)(13,15)(14,16)(17,19);(5,9)(6,12)(10,17)(11,16)(13,19)(15,18);(7,14)(13,19);(4,6,8,12)(5,10,11,18)(7,13,14,19)(9,15,16,17)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26541);