Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C42 · Schur multiplier (H2) = C27 |
| direct product | C2×C3×#1728 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C3×C4 | C23 |
| commutator | C22 | C22×C3×C42 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C24×C3 | C22×C4 |
| prime radical | C25×C3 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3×C4 ◁ G | |
2782 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2)(19,20,21);(3,4)(5,6)(11,12,13,16,14,15,17,18);(4,6)(7,8,9,10);(12,15)(16,18)336 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 13321);