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) = C24×C4 |
| direct product | C3×#20513 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C22×D4 |
| commutator | C4 | C22×C3×C42 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C22×C3 | C2×C4×D4 |
| prime radical | C2×C3×#135 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C2×C3×#97 ◁ G | |
998 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)(8,9,13,18,14,16,12,26)(10,19,21,15,22,30,27,28)(11,17,25,32,23,31,24,36)(20,33,35,29,34,39,37,38);(4,5,7,6)(8,10)(9,15)(11,20)(12,27)(13,21)(14,22)(16,28)(17,29)(18,30)(19,26)(23,34)(24,37)(25,35)(31,38)(32,39)(33,36);(4,6,7,5)(9,16)(11,23)(15,28)(18,26)(19,30)(20,34)(24,25)(35,37);(8,11,13,25,14,23,12,24)(9,17,18,32,16,31,26,36)(10,20,21,35,22,34,27,37)(15,29,30,39,28,38,19,33)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 13501);