Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#33738 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| socle | C22×C3 | C22×D8 |
| prime radical | C3×#466 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C2×C3×C42 ◁ C2×C3×C4×C8 ◁ G | |
510 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)(6,9)(7,8)(10,11)(12,13,17,21)(14,22,23,36)(15,27,29,42)(16,30,28,20)(18,31,32,25)(19,34,35,24)(26,41,39,38)(33,40,37,43);(5,8)(9,11)(14,23)(18,32)(22,36)(25,31)(26,39)(33,37)(38,41)(40,43);(12,14,24,40,16,26,29,31,17,23,34,43,28,39,15,25)(13,18,27,38,20,33,35,36,21,32,42,41,30,37,19,22);(4,6,7,10)(5,9,8,11)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26726);