Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C24 |
| direct product | C2×C3×#1301 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | C22⋊C4 |
| commutator | C2×C4 | C2×C3×C42 |
| Frattini, cosocle | C2×C42 | C23×C3 |
| socle | C23×C3 | #95 |
| prime radical | C24×C3 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ C22×C3×C42 ◁ G | |
666 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2)(27,28,29);(4,8)(5,6)(9,10)(11,12,14,17,15,18,22,25)(13,19,21,26,20,16,23,24);(3,4,6,9,7,10,5,8)(11,13)(12,16,18,19)(14,21)(15,20)(17,24,25,26)(22,23)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1063);