Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #15257 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×#15762 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C3 | C24 |
| commutator, Frattini, cosocle | C24 | C24×C3 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
2018 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(5,7)(9,11)(16,17,20,19)(18,22,23,21);(13,15)(16,18,20,23)(17,21,19,22);(8,9)(10,11)(12,13,14,15);(4,5)(6,7)(8,9)(10,11)(16,19,20,17)(18,21,23,22)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 8750);