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×#30483 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ G | |
1000 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)(8,9,13,18,15,19,14,31)(10,20,22,33,11,23,25,16)(12,26,28,36,30,17,29,35)(21,37,27,32,24,34,38,39);(4,5)(6,7)(8,10,15,11)(9,16,19,33)(12,27,30,38)(13,25,14,22)(17,34,26,37)(18,23,31,20)(21,29,24,28)(32,36,39,35);(5,7)(8,11,15,10)(9,16,19,33)(12,24,30,21)(13,25,14,22)(17,32,26,39)(18,20,31,23)(27,28,38,29)(34,36,37,35);(8,12,15,30)(9,17,19,26)(10,21,11,24)(13,29,14,28)(16,32,33,39)(18,36,31,35)(20,37,23,34)(22,38,25,27)111 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 23471);