Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33522 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#33631 |
| 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 ◁ C3×#117 ◁ C3×#405 ◁ G | |
556 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)(4,5,7,9)(6,10,11,8)(12,13)(14,18)(15,19)(16,29)(17,22)(20,26)(21,27)(23,31)(24,30)(25,39)(28,35)(32,38)(33,37)(34,40)(36,41)(42,43);(4,6,7,11)(5,8,9,10)(12,14)(13,18)(15,24)(16,25)(17,23)(19,30)(20,32)(21,33)(22,31)(26,38)(27,37)(28,36)(29,39)(34,42)(35,41)(40,43);(13,19)(15,26)(16,27)(18,30)(20,21)(22,35)(24,38)(25,37)(28,40)(29,34)(31,41)(32,33)(36,43)(39,42);(14,23)(18,31)(24,36)(25,37)(30,41)(32,42)(33,39)(38,43)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26619);