Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33605 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#33732 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| prime radical | C23×C3 | C2×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ C2×C3×C4×C8 ◁ G | |
482 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 43 points:
(1,2,3)(4,5,7,9)(6,10,11,8)(12,13,15,18)(14,22,24,36)(16,29,27,42)(17,30,28,21)(19,33,31,25)(20,35,32,23)(26,41,40,38)(34,39,37,43);(4,6,7,11)(5,8,9,10)(13,18)(19,31)(20,32)(21,30)(22,36)(29,42)(34,37)(38,41);(12,14,23,39,17,26,27,33,15,24,35,43,28,40,16,25)(13,19,29,38,21,34,32,36,18,31,42,41,30,37,20,22);(14,24)(19,31)(22,36)(25,33)(26,40)(34,37)(38,41)(39,43)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26720);