Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60367 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
| direct product | C3×#60424 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| prime radical | C3×#3449 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×#117 ◁ G | |
2674 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 39 points:
(1,2,3)(5,7)(8,9)(10,14)(11,21)(12,24)(13,16)(15,22)(17,26)(18,32)(19,28)(20,36)(23,31)(25,30)(27,34)(29,35)(33,39)(37,38);(4,5,6,7)(8,10)(9,14)(11,18)(12,20)(13,19)(15,27)(16,28)(17,29)(21,32)(22,34)(23,33)(24,36)(25,37)(26,35)(30,38)(31,39);(8,11,13,23)(9,15,16,30)(10,18,19,33)(12,25,26,22)(14,27,28,38)(17,31,24,21)(20,37,35,34)(29,39,36,32);(10,19)(14,28)(18,33)(20,35)(27,38)(29,36)(32,39)(34,37);(9,16)(14,28)(15,30)(17,24)(21,31)(27,38)(29,36)(32,39)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53412);