Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61058 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
| direct product | C3×#61061 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×#525 ◁ G | |
2674 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 39 points:
(1,2,3)(4,5)(6,7)(8,9)(10,18)(11,15)(12,25)(13,16)(14,21)(17,28)(19,34)(20,36)(22,39)(23,35)(24,32)(26,29)(27,33)(30,38)(31,37);(8,10,13,19)(9,14,16,29)(11,22,24,38)(12,26,28,21)(15,31,32,36)(17,34,25,18)(20,27,37,23)(30,35,39,33);(5,7)(10,19)(14,29)(18,34)(20,37)(21,26)(22,38)(30,39)(31,36);(8,11,13,24)(9,15,16,32)(10,20,19,37)(12,27,28,23)(14,30,29,39)(17,35,25,33)(18,36,34,31)(21,38,26,22);(9,16)(14,29)(15,32)(17,25)(18,34)(30,39)(31,36)(33,35)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 54049);