Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
| direct product | C3×#61096 |
| 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×#532 ◁ G | |
2686 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)(8,9,13,17)(10,15,20,32)(11,16,24,34)(12,25,28,18)(14,26,30,21)(19,31,37,33)(22,39,38,29)(23,36,27,35);(8,10)(9,14)(11,22)(12,26)(13,20)(15,25)(16,33)(17,30)(18,32)(19,23)(21,28)(24,38)(27,37)(29,35)(31,34)(36,39);(5,7)(8,10)(9,15)(11,19)(12,21)(13,20)(14,25)(16,31)(17,32)(18,30)(22,23)(24,37)(26,28)(27,38)(29,35)(33,34)(36,39);(8,11,13,24)(9,16,17,34)(10,19,20,37)(12,27,28,23)(14,29,30,39)(15,31,32,33)(18,36,25,35)(21,38,26,22);(9,17)(10,20)(16,34)(18,25)(19,37)(21,26)(22,38)(35,36)165 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 54084);