Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#36150 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
2666 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 23 points:
(1,2,3)(8,9,13,11)(10,14,15,12)(16,17)(18,19)(20,22)(21,23);(5,7)(8,10)(9,12)(11,14)(13,15)(16,18,21,20)(17,19,23,22);(4,5)(6,7)(9,11)(10,15)(16,19,21,22)(17,20,23,18);(5,7)(8,11,13,9)(10,14,15,12)(16,20,21,18)(17,22,23,19);(4,5)(6,7)(8,12,13,14)(9,10,11,15)(16,17)(18,22)(19,20)(21,23)210 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 29138);