Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C8 · Schur multiplier (H2) = C2×C4 |
| direct product | C3×#13266 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C8 | C2×D4 |
| commutator | C2×C4 | C22×C3×C8 |
| Frattini, cosocle | C22×C8 | C23×C3 |
| socle | C22×C3 | C2×C22⋊C8 |
| prime radical | C23×C3 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C8 ◁ C2×C3×C4×C8 ◁ G | |
342 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3)(4,5)(6,10)(7,8)(9,11)(12,13,16,21,17,20,30,33,18,22,31,36,15,29,24,39)(14,23,26,37,27,35,40,42,28,38,41,43,25,19,32,34);(5,8)(9,10)(12,14,24,32,15,25,31,41,18,28,30,40,17,27,16,26)(13,19,33,37,29,38,21,34,22,35,39,43,20,23,36,42);(4,6,7,11)(5,9,8,10)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6254);