Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23924 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C3×#23943 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C24×C3 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G | |
1158 conjugacy classes of subgroups.
| minimal degree | μ(G) = 47 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 47 points:
(1,2,3)(8,9)(10,14)(11,13)(12,15)(16,17,21,25)(18,22,27,36)(19,29,31,39)(20,32,34,24)(23,33,38,30)(26,40,41,47)(28,44,43,37)(35,45,46,42);(4,5)(6,7)(8,10,11,15)(9,12,13,14)(16,18)(17,22)(19,26)(20,28)(21,27)(23,35)(24,37)(25,36)(29,40)(30,42)(31,41)(32,44)(33,45)(34,43)(38,46)(39,47);(5,7)(16,19,21,31)(17,23,25,38)(18,26,27,41)(20,33,34,30)(22,35,36,46)(24,39,32,29)(28,45,43,42)(37,47,44,40);(9,13)(12,14)(18,27)(22,36)(26,41)(28,43)(35,46)(37,44)(40,47)(42,45)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 16931);