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×#61076 |
| 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 | |
2672 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)(8,9,13,16)(10,18,20,34)(11,15,24,32)(12,25,28,17)(14,26,30,21)(19,36,37,31)(22,39,38,29)(23,35,27,33);(8,10)(9,14)(11,22)(12,26)(13,20)(15,31)(16,30)(17,34)(18,25)(19,23)(21,28)(24,38)(27,37)(29,33)(32,36)(35,39);(5,7);(8,11,13,24)(9,15,16,32)(10,19,20,37)(12,27,28,23)(14,29,30,39)(17,35,25,33)(18,36,34,31)(21,38,26,22);(9,16)(10,20)(15,32)(17,25)(19,37)(21,26)(22,38)(33,35)165 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 54064);