Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C4 · Schur multiplier (H2) = C210 |
| direct product | C2×C3×#2366 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C3 | C24 |
| commutator | C22 | C24×C3×C4 |
| Frattini, cosocle | C23 | C25×C3 |
| prime radical | C22×C3×#127 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
5062 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 17 points:
(1,2)(15,16,17);(3,4,6,5)(8,10)(11,12,14,13);(4,5)(11,13,14,12);(7,8,9,10);(3,5,6,4)300 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 27127);