Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C82 · Schur multiplier (H2) = C2 |
| direct product | C3×#7330 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C8 | C42 |
| commutator | C4 | C3×C82 |
| Frattini, cosocle | C4×C16 | C22×C3 |
| socle, prime radical | C22×C3 | #346 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C8 ◁ G | |
110 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)(12,13,14,19,15,21,24,34,16,22,25,32,26,36,38,42,17,20,23,33,27,35,37,43,28,18,29,30,39,31,40,41);(4,5,6,8,7,9,10,11)(13,18,20,22)(14,23)(19,32,33,30)(21,31,35,36)(24,37)(25,29)(34,42,43,41)(38,40)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 318);