Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10046 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#10061 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C3 | #120 |
| commutator, Frattini, cosocle | C2×C42 | C23×C3 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×C42 ◁ G | |
534 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3)(8,9,13,16)(10,17,19,22)(11,15,12,20)(14,21,23,18)(24,25,29,33)(26,34,37,38)(27,30,28,32)(31,35,39,36);(4,5)(6,7)(8,10,13,19)(9,14,16,23)(11,18,12,21)(15,22,20,17)(25,30)(27,28)(31,38)(32,33)(34,39)(35,36);(5,7)(8,11,13,12)(9,15,16,20)(10,18,19,21)(14,22,23,17)(24,26,29,37)(25,31,33,39)(27,36,28,35)(30,38,32,34)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 3049);