Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23924 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C22×C4 |
| direct product | C3×#29551 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C3×#376 ◁ G | |
650 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,5)(6,10)(7,9)(8,11)(20,21,26,30)(22,27,23,32)(24,29,25,34)(28,31,35,33);(4,6,7,11)(5,8,9,10)(12,13,15,16)(14,17,18,19)(20,22,26,23)(21,27,30,32)(24,33,25,31)(28,34,35,29);(13,16)(17,19)(20,23,26,22)(21,28,30,35)(24,33,25,31)(27,34,32,29);(5,9)(8,10)(12,14,15,18)(13,17,16,19)(20,24,26,25)(21,29,30,34)(22,31,23,33)(27,35,32,28)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 22539);