Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×C3×#1924 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C3×C4 | C23 |
| commutator | C23 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C24×C3 | C22×C4 |
| prime radical | C25×C3 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3×C4 ◁ G | |
1398 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)(8,9)(11,12,15,17,14,13,16,18);(7,8,10,9)(11,13,15,18,14,12,16,17);(4,6)(7,9,10,8)(12,13)(17,18)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 14033);