Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C26 |
| direct product | C3×#22468 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | D42 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C2×C3×#578 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ G | |
1316 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)(8,9,13,19,15,11,14,18)(10,20,24,21,25,32,23,16)(12,26,29,35,30,17,28,34)(22,33,27,31,39,36,38,37);(8,10)(9,16)(11,21)(12,27)(13,24)(14,23)(15,25)(17,33)(18,32)(19,20)(22,28)(26,36)(29,39)(30,38)(31,34)(35,37);(5,7)(8,11,15,9)(10,21,25,16)(12,26,30,17)(13,19,14,18)(20,23,32,24)(22,37,39,31)(27,36,38,33)(28,35,29,34);(8,12)(9,17)(10,22)(11,26)(13,29)(14,28)(15,30)(16,31)(18,35)(19,34)(20,36)(21,37)(23,38)(24,27)(25,39)(32,33)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 15456);