Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1913 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C23×C4 |
| direct product | C3×#10915 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #386 |
| commutator | C23 | C2×C3×C42 |
| Frattini, cosocle | C25 | C23×C3 |
| prime radical | C25×C3 | C23 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C25×C3 ◁ G | |
974 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3)(4,5,7,6)(8,9)(10,16)(11,14)(12,15)(13,18)(17,20)(19,21)(22,23);(8,10)(9,13,15,16)(11,17)(12,18)(14,20,21,22)(19,23);(4,6,7,5)(8,11)(9,14)(10,17)(12,19)(13,20)(15,21)(16,22)(18,23)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 3903);