Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1665 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22×C4 |
| direct product | C3×#9732 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | #121 |
| commutator | C42 | C22×C3×C4 |
| Frattini, cosocle | C2×C42 | C23×C3 |
| prime radical | C2×C3×#127 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×C42 ◁ G | |
586 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3)(5,7)(9,14)(10,15)(11,12)(16,17,18,21,19,22,23,20);(4,5,6,7)(8,9,11,14,13,15,12,10)(17,20)(18,23)(21,22);(8,10,12,15,13,14,11,9)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 2720);