Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2010 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C27 |
| direct product | C2×C3×#2010 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | #127 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C22×C3 | #392 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C23×C3×C4 ◁ G | |
2190 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2)(19,20,21);(3,4)(5,11)(6,14)(7,10)(8,12)(9,16)(13,18)(15,17);(3,5)(4,8)(6,15)(7,13)(9,18)(10,17)(11,14)(12,16);(8,17)(11,18)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 16401);