Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33758 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C23×C4 |
| direct product | C3×#33781 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| prime radical | C22×C3×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ C2×C3×C4×C8 ◁ G | |
598 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 43 points:
(1,2,3)(4,5)(6,10)(7,9)(8,11)(12,13,17,22)(14,18,23,33)(15,19,28,36)(16,29,27,21)(20,37,35,26)(24,34,38,30)(25,42,39,32)(31,40,43,41);(4,6,7,11)(5,8,9,10)(12,14)(13,18)(15,24)(16,25)(17,23)(19,34)(20,31)(21,32)(22,33)(26,41)(27,39)(28,38)(29,42)(30,36)(35,43)(37,40);(6,11)(8,10)(13,19)(14,23)(15,26)(16,27)(18,30)(20,21)(22,36)(24,40)(28,37)(29,35)(31,42)(32,43)(33,34)(38,41);(5,9)(8,10)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26769);