Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C23×C4 |
| direct product | C3×#10914 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C2×C22⋊C4 |
| commutator | C23 | C2×C3×C42 |
| Frattini, cosocle | C23×C4 | C23×C3 |
| socle | C22×C3 | C4×C22⋊C4 |
| prime radical | C24×C3 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C23×C3×C8 ◁ G | |
698 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)(4,5)(6,7)(8,9,12,16,13,17,24,30)(10,18,21,32,20,14,25,27)(11,15,22,28,23,29,35,38)(19,31,34,39,33,26,36,37);(8,10)(9,14,17,18)(11,19)(12,21)(13,20)(15,26,29,31)(16,27,30,32)(22,34)(23,33)(24,25)(28,37,38,39)(35,36);(5,7)(8,11,12,22,13,23,24,35)(9,15,16,28,17,29,30,38)(10,19,21,34,20,33,25,36)(14,26,27,37,18,31,32,39)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 3902);