Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30428 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24×C4 |
| direct product | C3×#30481 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×#3013 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ G | |
984 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)(8,9,12,18,14,19,13,30)(10,20,22,16,23,32,29,15)(11,24,26,36,28,17,27,35)(21,37,25,31,38,34,39,33);(4,5)(6,7)(8,10,14,23)(9,15,19,16)(11,25,28,39)(12,29,13,22)(17,34,24,37)(18,32,30,20)(21,27,38,26)(31,36,33,35);(5,7)(8,10)(9,16)(11,21)(12,22)(13,29)(14,23)(15,19)(17,33)(18,32)(20,30)(24,31)(25,26)(27,39)(28,38)(34,36)(35,37);(8,11,14,28)(9,17,19,24)(10,21,23,38)(12,27,13,26)(15,31,16,33)(18,36,30,35)(20,37,32,34)(22,39,29,25)111 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 23469);