Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C4 · Schur multiplier (H2) = C210 |
| direct product | C23×C3×#107 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25×C3 | D4 |
| commutator | C4 | C24×C3×C4 |
| Frattini, cosocle | C2×C4 | C25×C3 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C25×C3 ◁ C24×C3×C4 ◁ G | |
3614 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 5 generators on 21 points:
(1,2)(19,20,21);(3,4);(5,6);(7,8,9,10)(12,15)(13,16)(17,18);(11,12,16,18,14,17,13,15)336 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53090);