Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13313 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#13394 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #466 |
| commutator, Frattini, cosocle | C22×C8 | C23×C3 |
| prime radical | C2×C3×#41 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×#131 ◁ G | |
392 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3)(8,9,15,21)(10,22,25,34)(11,26,23,16)(12,20,14,18)(13,29,32,38)(17,35,37,27)(19,28,39,31)(24,36,30,33);(4,5)(6,7)(8,10)(9,16)(11,12)(13,30)(14,23)(15,25)(17,29)(18,34)(19,36)(20,22)(21,26)(24,32)(27,31)(28,35)(33,39)(37,38);(5,7)(8,11,12,10,15,23,14,25)(9,17,18,33,21,37,20,36)(13,30,28,27,32,24,31,35)(16,19,22,38,26,39,34,29)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6382);