Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×C3×#2158 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | D8 |
| commutator | C8 | C23×C3×C4 |
| Frattini, cosocle | C2×C8 | C24×C3 |
| socle | C23×C3 | C2×D8 |
| prime radical | C24×C3 | D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ C23×C3×C4 ◁ C23×C3×C8 ◁ G | |
694 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2)(39,40,41);(3,4,5,6)(7,8)(9,18)(10,15)(11,25)(12,27)(13,17)(14,21)(16,19)(20,31)(22,37)(23,33)(24,30)(26,38)(28,35)(29,36)(32,34);(7,9,19,34,12,22,26,36,13,23,24,35,10,20,11,21)(8,14,25,31,15,28,30,33,17,29,38,37,27,32,16,18);(7,10,13,12)(8,15,17,27)(9,20,23,22)(11,24,26,19)(14,28,29,32)(16,25,30,38)(18,31,33,37)(21,35,36,34)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26395);