Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18562 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#18601 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| prime radical | C22×C3×D4 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
910 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,5,9,12)(6,13,16,19)(7,11,8,17)(10,14,18,15)(20,21,25,29)(22,30,31,34)(23,28,24,27)(26,33,35,32);(4,6,9,16)(5,10,12,18)(7,15,8,14)(11,19,17,13)(20,22,25,31)(21,26,29,35)(23,33,24,32)(27,34,28,30);(21,27)(23,24)(26,34)(28,29)(30,35)(32,33);(4,7,9,8)(5,11,12,17)(6,14,16,15)(10,13,18,19)(22,31)(26,35)(30,34)(32,33)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 11589);