Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33607 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#33745 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| prime radical | C3×#466 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×#117 ◁ C3×#405 ◁ G | |
496 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)(12,13,16,21)(14,23,27,38)(15,20,18,30)(17,33,32,40)(19,34,24,28)(22,31,36,25)(26,37,29,43)(35,41,39,42);(5,8)(9,11)(12,13)(14,24)(15,30)(16,21)(17,22)(18,20)(19,27)(23,34)(25,33)(26,35)(28,38)(29,39)(31,40)(32,36)(37,42)(41,43);(12,14,25,41,18,29,32,34,16,27,31,42,15,26,17,28)(13,19,33,37,20,35,36,38,21,24,40,43,30,39,22,23);(4,6,7,10)(5,9,8,11)(12,15,16,18)(13,20,21,30)(14,26,27,29)(17,31,32,25)(19,35,24,39)(22,33,36,40)(23,37,38,43)(28,42,34,41)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26733);