Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1913 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C24×C4 |
| direct product | C3×#10940 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #386 |
| commutator | C23 | C2×C3×C42 |
| Frattini, cosocle | C23×C4 | C23×C3 |
| prime radical | C24×C3 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
898 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3)(5,7)(8,9)(10,14,19,26)(11,15)(12,23)(13,16)(17,24)(18,30,31,25)(20,34,33,27)(21,29)(22,36)(28,35)(32,37,39,38);(8,10,12,20,13,19,24,33)(9,14,17,27,16,26,23,34)(11,18,22,32,21,31,35,39)(15,25,28,37,29,30,36,38);(4,5,6,7)(8,11,13,21)(9,15,16,29)(10,18,19,31)(12,22,24,35)(14,25,26,30)(17,28,23,36)(20,32,33,39)(27,37,34,38)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 3928);