Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23400 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#23594 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
1038 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)(6,9)(7,8)(10,11)(12,13)(14,18)(15,25)(16,28)(17,21)(19,27)(20,31)(22,39)(23,37)(24,34)(26,38)(29,33)(30,36)(32,41)(35,42)(40,43);(5,8)(9,11)(12,14,17,24)(13,18,21,34)(15,22,26,40)(16,29,31,23)(19,35,36,32)(20,37,28,33)(25,39,38,43)(27,42,30,41);(12,15,17,26)(13,19,21,36)(14,22,24,40)(16,30,31,27)(18,32,34,35)(20,38,28,25)(23,42,29,41)(33,43,37,39);(4,6,7,10)(5,9,8,11)(15,26)(19,36)(22,40)(25,38)(27,30)(32,35)(39,43)(41,42)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 16582);