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×#27693 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #535 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C2×C3 | #2487 |
| prime radical | C3×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C22×C3×C4 ◁ G | |
634 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)(4,5,10,15,11,17,9,16)(6,18,28,32,21,35,20,12)(7,22,29,33,25,31,24,13)(8,26,19,30,27,34,23,14);(4,6,9,20,11,21,10,28)(5,12,15,18,17,32,16,35)(7,23,24,27,25,19,29,8)(13,30,22,34,33,14,31,26);(4,7)(5,13)(6,19)(8,20)(9,24)(10,29)(11,25)(12,30)(14,32)(15,22)(16,31)(17,33)(18,34)(21,23)(26,35)(27,28);(4,8,9,23,11,27,10,19)(5,14,15,26,17,30,16,34)(6,7,20,24,21,25,28,29)(12,31,18,13,32,22,35,33)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 20681);