Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1913 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C24×C4 |
| direct product | C3×#10941 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #386 |
| commutator | C23 | C2×C3×C42 |
| Frattini, cosocle | C23×C4 | C23×C3 |
| socle | C2×C3 | #1437 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C23×C3×C4 ◁ G | |
904 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3)(4,5,8,13,9,12,20,26)(6,10,16,23,15,22,19,27)(7,11,18,24,17,25,31,35)(14,28,30,34,29,21,32,33);(4,6)(5,10,12,22)(7,14)(8,19)(9,15)(11,21,25,28)(13,27,26,23)(16,20)(17,29)(18,32)(24,34,35,33)(30,31);(4,7,8,18,9,17,20,31)(5,11,13,24,12,25,26,35)(6,14,19,32,15,29,16,30)(10,21,27,34,22,28,23,33)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 3929);