Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C27×C3 · Schur multiplier (H2) = C221 |
| direct product | C25×C3×D4 |
| unambiguous semidirect product | C4⋊(C26×C3) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C26×C3 | C22 |
| commutator, Frattini, cosocle | C2 | C27×C3 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C26×C3 ◁ G | |
111198 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 7 generators on 17 points:
(1,2)(15,16,17);(3,4);(5,6);(7,8);(9,10);(11,12,13,14);(12,14)480 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 56083);