Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C42 · Schur multiplier (H2) = C27 |
| direct product | C3×#20351 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C22×D4 |
| commutator | C4 | C22×C3×C42 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C22×C3 | C22×C4⋊C4 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C22×C3×#41 ◁ G | |
1978 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)(8,9,12,18,14,19,13,17)(10,20,22,15,24,29,23,30)(11,16,26,33,25,31,27,32)(21,34,36,28,35,39,37,38);(5,7)(8,10,13,23,14,24,12,22)(9,15,18,29,19,30,17,20)(11,21,27,37,25,35,26,36)(16,28,33,39,31,38,32,34);(8,11)(9,16)(10,21)(12,26)(13,27)(14,25)(15,28)(17,32)(18,33)(19,31)(20,34)(22,36)(23,37)(24,35)(29,39)(30,38);(11,25)(16,31)(21,35)(26,27)(28,38)(32,33)(34,39)(36,37)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 13339);