Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C22×C4 |
| direct product | C3×#27716 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C24×C3 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C3×C4×C22⋊C4 ◁ G | |
690 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,5)(6,9)(7,8)(10,11)(12,13)(14,16)(15,17)(18,19)(20,21,24,27,25,28,23,33)(22,26,31,34,32,35,30,29);(5,8)(9,11)(12,14,15,18)(13,16,17,19)(20,22,23,30,25,32,24,31)(21,26,33,29,28,35,27,34);(4,6,7,10)(5,9,8,11)(13,17)(16,19)(20,21)(22,29)(23,27)(24,33)(25,28)(26,30)(31,35)(32,34);(5,8)(9,11)(20,23,25,24)(21,27,28,33)(22,30,32,31)(26,34,35,29)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 20704);