Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9345 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C3×#9352 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #386 |
| commutator | C22×C4 | C22×C3×C4 |
| Frattini, cosocle | C2×C42 | C23×C3 |
| prime radical | C2×C3×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
414 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3)(4,5,7,10,9,13,8,12)(6,14,15,11,16,18,17,19)(20,21,24,29,25,28,23,27)(22,26,30,33,31,34,32,35);(5,10,13,12)(11,18,19,14)(20,22,23,30,25,31,24,32)(21,26,29,35,28,34,27,33);(4,6,9,16)(5,11,13,19)(7,15,8,17)(10,18,12,14)(20,23,25,24)(21,27,28,29)(22,30,31,32)(26,33,34,35)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 2340);