Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18414 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24×C4 |
| direct product | C3×#18434 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
982 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(5,10)(6,14)(7,18)(11,23)(12,21)(13,19)(15,27)(17,31)(20,30)(22,26)(24,34)(25,33)(28,29)(32,35);(4,5)(6,11)(7,19)(8,13)(9,12)(10,18)(14,26)(15,24)(16,25)(17,32)(20,33)(21,30)(22,29)(23,31)(27,35)(28,34);(6,15)(7,20)(10,21)(11,24)(14,27)(17,31)(18,30)(19,33)(22,34)(23,32)(26,35)(28,29);(4,6,16,27)(5,11,25,35)(7,17,30,31)(8,15,9,14)(10,22,33,34)(12,26,13,24)(18,29,20,28)(19,32,21,23)111 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 11422);