Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23994 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#27691 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #535 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C2×C3 | #2487 |
| prime radical | C3×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C22×C3×C4 ◁ G | |
640 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,5,10,13,9,14,8,22)(6,15,23,26,16,29,18,11)(7,19,24,27,21,33,20,12)(17,31,34,35,30,25,32,28);(4,6,8,18,9,16,10,23)(5,11,13,15,14,26,22,29)(7,17,20,32,21,30,24,34)(12,25,19,28,27,31,33,35);(6,16)(17,30)(18,23)(32,34);(4,7,8,20,9,21,10,24)(5,12,13,19,14,27,22,33)(6,17,18,32,16,30,23,34)(11,25,15,28,26,31,29,35)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 20679);