Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17386 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C27 |
| direct product | C3×#20991 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×C22⋊C4 |
| commutator | C23 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×#41 ◁ G | |
1674 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)(8,9,12,17,13,16,18,30)(10,14,21,26,20,27,24,31)(11,15,23,29,22,28,32,38)(19,25,34,37,33,36,35,39);(8,10,18,21,13,20,12,24)(9,14,17,31,16,27,30,26)(11,19,32,34,22,33,23,35)(15,25,29,39,28,36,38,37);(5,7)(8,11)(9,15)(10,19)(12,23)(13,22)(14,25)(16,28)(17,29)(18,32)(20,33)(21,34)(24,35)(26,37)(27,36)(30,38)(31,39);(11,22)(15,28)(19,33)(23,32)(25,36)(29,38)(34,35)(37,39)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 13979);