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×#25801 |
| 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 | |
1444 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)(10,21)(11,16)(12,27)(13,19)(14,20)(15,23)(17,28)(18,25)(22,36)(24,31)(26,30)(29,33)(32,38)(34,39)(35,37);(8,10,13,23,14,11,25,24)(9,15,18,21,20,31,19,16)(12,22,28,37,29,26,39,38)(17,30,27,35,34,36,33,32);(4,5,6,7)(8,11)(9,16)(10,14)(12,26)(13,24)(15,18)(17,32)(19,31)(20,21)(22,29)(23,25)(27,30)(28,38)(33,36)(34,35)(37,39);(8,12,13,28,14,29,25,39)(9,17,18,27,20,34,19,33)(10,22,23,37,11,26,24,38)(15,30,21,35,31,36,16,32)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 18789);