Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C42 · Schur multiplier (H2) = C28 |
| direct product | C22×C3×#340 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25×C3 | D4 |
| commutator | C4 | C22×C3×C42 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C25×C3 ◁ C24×C3×C4 ◁ G | |
3006 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2)(21,22,23);(3,4);(5,6)(7,8)(9,10,11,12)(14,18)(15,16)(19,20);(6,8)(13,14,16,19,17,20,15,18)336 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 13332);