Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33764 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#33766 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| prime radical | C3×#3058 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×#41 ◁ C2×C3×#131 ◁ G | |
584 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)(14,18,23,31)(15,19,28,35)(16,29,27,21)(20,36,34,26)(24,30,37,41)(25,40,38,33)(32,43,42,39);(4,6)(5,8)(7,11)(9,10)(12,14,17,23)(13,18,22,31)(15,24,28,37)(16,25,27,38)(19,30,35,41)(20,32,34,42)(21,33,29,40)(26,39,36,43);(6,11)(8,10)(13,19)(15,26)(16,27)(18,30)(20,21)(22,35)(24,39)(25,38)(28,36)(29,34)(31,41)(32,33)(37,43)(40,42);(5,9)(8,10)(14,23)(18,31)(24,37)(25,38)(30,41)(32,42)(33,40)(39,43)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26754);