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) = C23×C4 |
| direct product | C3×#10917 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #386 |
| commutator | C23 | C2×C3×C42 |
| Frattini, cosocle | C23×C4 | C23×C3 |
| socle | C2×C3 | #1436 |
| prime radical | C24×C3 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C23×C3×C4 ◁ G | |
676 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)(4,5,8,12,9,13,20,26)(6,14,17,28,16,10,21,23)(7,11,19,25,18,24,31,34)(15,22,30,33,29,27,32,35);(4,6)(5,10,13,14)(7,15)(8,17)(9,16)(11,22,24,27)(12,23,26,28)(18,29)(19,30)(20,21)(25,33,34,35)(31,32);(4,7,8,19,9,18,20,31)(5,11,12,25,13,24,26,34)(6,15,17,30,16,29,21,32)(10,22,23,33,14,27,28,35)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 3905);