Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25×C4 |
| direct product | C3×#25806 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C2×C3×#578 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ G | |
1402 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3)(5,7)(8,9,14,19)(10,16,22,30)(11,23,26,37)(12,27,13,18)(15,28,31,21)(17,25,33,24)(20,34,36,29)(32,38,39,35);(8,10,13,28,14,22,12,21)(9,15,27,30,19,31,18,16)(11,20,25,38,26,36,24,35)(17,29,37,39,33,34,23,32);(4,5,6,7)(8,10)(9,16)(11,20)(12,21)(13,28)(14,22)(15,27)(17,32)(18,31)(19,30)(23,34)(24,35)(25,38)(26,36)(29,37)(33,39);(8,11,13,25,14,26,12,24)(9,17,27,37,19,33,18,23)(10,20,28,38,22,36,21,35)(15,29,30,39,31,34,16,32)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 18794);