Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13307 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#13320 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C23×C3 |
| prime radical | C2×C3×#472 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×#131 ◁ G | |
540 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)(9,15)(10,20)(11,23)(12,13)(16,32)(17,24)(18,30)(19,31)(21,25)(22,28)(26,27)(29,34)(33,36)(35,38);(4,5)(6,7)(8,9,14,19)(10,16,22,35)(11,24,27,30)(12,18,13,17)(15,26,31,23)(20,36,28,34)(21,38,25,32)(29,37,33,39);(5,7)(8,10)(9,16)(11,25)(12,28)(13,20)(14,22)(15,29)(17,36)(18,34)(19,35)(21,27)(23,39)(24,32)(26,37)(30,38)(31,33)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6308);