Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4×C8 · Schur multiplier (H2) = C23×C4 |
| direct product | C3×#8144 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C42 | C23 |
| commutator | C22 | C2×C3×C4×C8 |
| Frattini, cosocle | C2×C42 | C23×C3 |
| socle | C23×C3 | C2×C42 |
| prime radical | C24×C3 | C42 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C42 ◁ G | |
678 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)(4,5)(6,7)(8,9,12,14,11,10,13,15);(5,7)(9,10)(14,15)(16,17,18,20,19,21,22,23);(8,10,12,15,11,9,13,14)336 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1132);