Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10260 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#10278 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | #121 |
| commutator, Frattini, cosocle | C2×C42 | C23×C3 |
| prime radical | C22×C3×D4 | D4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×C42 ◁ G | |
562 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2,3)(4,5)(6,7)(8,9,13,10)(11,14,12,15)(16,17,20,23)(18,24,26,31)(19,27,29,22)(21,28,30,25);(5,7)(9,14)(10,15)(11,12);(8,10,13,9)(11,15,12,14)(16,18,20,26)(17,21,23,30)(19,28,29,25)(22,31,27,24)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 3266);