Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1902 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C23×C4 |
| direct product | C3×#10906 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #386 |
| commutator | C2×C4 | C2×C3×C42 |
| Frattini, cosocle | C23×C4 | C23×C3 |
| socle | C2×C3 | #1436 |
| prime radical | C24×C3 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C23×C3×C4 ◁ G | |
656 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)(4,5,8,14,10,15,9,13)(6,16,18,11,20,26,19,27)(7,21,23,28,24,29,22,12)(17,31,33,30,34,25,32,35);(4,6,9,19,10,20,8,18)(5,11,14,26,15,27,13,16)(7,17,22,32,24,34,23,33)(12,25,21,35,28,31,29,30);(4,7,8,23,10,24,9,22)(5,12,14,21,15,28,13,29)(6,17,18,33,20,34,19,32)(11,25,26,35,27,31,16,30)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 3894);