Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2238 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#33656 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #481 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| socle | C2×C3 | C2×#481 |
| prime radical | C3×#3054 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C22×C3×M16 ◁ G | |
556 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)(6,15)(7,12)(8,20)(9,24)(10,14)(11,18)(13,23)(16,30)(17,31)(19,29)(21,35)(22,28)(25,32)(26,33)(27,34);(8,21)(13,29)(15,31)(18,34)(19,23)(20,35)(22,33)(30,32);(4,6,9,25,10,17,7,16)(5,11,12,26,14,27,24,28)(8,22,23,34,21,33,19,18)(13,30,20,31,29,32,35,15);(4,7,10,9)(5,12,14,24)(6,16,17,25)(8,19,21,23)(11,26,27,28)(13,20,29,35)(15,30,31,32)(18,33,34,22)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26644);