Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C3×#24462 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C2×C3×#578 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ G | |
1302 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)(8,9,14,18)(10,20,23,30)(11,24,28,16)(12,22,15,31)(13,19,32,17)(21,37,39,34)(25,33,38,26)(27,35,29,36);(8,10,14,23)(9,15,18,12)(11,25,28,38)(13,31,32,22)(16,34,24,37)(17,30,19,20)(21,27,39,29)(26,35,33,36);(5,7)(8,11,14,28)(9,16,18,24)(10,21,23,39)(12,26,15,33)(13,27,32,29)(17,35,19,36)(20,34,30,37)(22,38,31,25);(8,12)(9,10)(11,26)(13,20)(14,15)(16,21)(17,31)(18,23)(19,22)(24,39)(25,35)(27,34)(28,33)(29,37)(30,32)(36,38)129 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 17450);