Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C43 · Schur multiplier (H2) = C22×C42 |
| direct product | C3×#10711 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C8 | C22⋊C4 |
| commutator | C22 | C3×C43 |
| Frattini, cosocle | C23×C4 | C23×C3 |
| socle | C22×C3 | C4×C22⋊C4 |
| prime radical | C24×C3 | C22×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C8 ◁ C23×C3×C8 ◁ G | |
842 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)(8,9)(10,18,20,14)(11,15)(12,17)(13,16)(19,30,32,25)(21,31,34,26)(22,28)(23,27)(24,29)(33,38,39,36)(35,37);(4,5,6,7)(8,10)(9,14)(11,19)(12,21)(13,20)(15,25)(16,18)(17,26)(22,33)(23,32)(24,34)(27,30)(28,36)(29,31)(35,39)(37,38);(8,11,12,22,13,23,24,35)(9,15,17,28,16,27,29,37)(10,19,21,33,20,32,34,39)(14,25,26,36,18,30,31,38)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 3699);