Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C26 |
| direct product | C3×#20852 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C22×D4 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C22×C3 | C22×C4⋊C4 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C22×C3×#41 ◁ G | |
1314 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3)(4,5)(6,7)(8,9,13,17,14,18,12,26)(10,19,21,15,22,29,27,30)(11,16,25,32,23,31,24,36)(20,33,35,28,34,39,37,38);(8,10,13,21,14,22,12,27)(9,15,17,29,18,30,26,19)(11,20,25,35,23,34,24,37)(16,28,32,39,31,38,36,33);(5,7)(8,11)(9,16)(10,20)(12,24)(13,25)(14,23)(15,28)(17,32)(18,31)(19,33)(21,35)(22,34)(26,36)(27,37)(29,39)(30,38);(11,23)(16,31)(20,34)(24,25)(28,38)(32,36)(33,39)(35,37)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 13840);