Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C8×C16 · Schur multiplier (H2) = C4 |
| direct product | C3×#7329 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4×C16 | C22 |
| commutator | C2 | C3×C8×C16 |
| Frattini, cosocle | C4×C16 | C22×C3 |
| socle, prime radical | C22×C3 | C4×C16 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4×C16 ◁ G | |
126 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 2 generators on 43 points:
(1,2,3)(12,13,14,19,15,20,22,31,16,21,23,32,25,33,34,41,17,18,24,28,26,29,35,38,27,30,36,39,37,40,42,43);(4,5,6,8,7,9,10,11)(13,18)(19,28)(20,29)(21,30)(31,38)(32,39)(33,40)(41,43)480 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 317);