Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24×C4 |
| direct product | C3×#25609 |
| 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 | |
1052 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)(11,12)(13,15)(16,17,21,25)(18,22,27,35)(19,29,32,44)(20,33,30,24)(23,37,38,31)(26,39,41,47)(28,43,40,36)(34,45,46,42);(4,5)(6,7)(9,12)(13,14)(16,18)(17,22)(19,26)(20,28)(21,27)(23,34)(24,36)(25,35)(29,39)(30,40)(31,42)(32,41)(33,43)(37,45)(38,46)(44,47);(5,7)(16,19,30,37,21,32,20,31)(17,23,33,44,25,38,24,29)(18,26,40,45,27,41,28,42)(22,34,43,47,35,46,36,39);(8,10,11,15)(9,13,12,14)(18,27)(22,35)(26,41)(28,40)(34,46)(36,43)(39,47)(42,45)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 18597);