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×#20514 |
| 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)(8,9,13,18,14,16,12,27)(10,19,21,15,22,31,28,29)(11,23,26,38,24,17,25,32)(20,34,36,37,35,33,39,30);(4,5,7,6)(8,10)(9,15)(11,20)(12,28)(13,21)(14,22)(16,29)(17,30)(18,31)(19,27)(23,37)(24,35)(25,39)(26,36)(32,34)(33,38);(4,6,7,5)(9,16)(11,24)(15,29)(18,27)(19,31)(20,35)(25,26)(36,39);(8,11,13,26,14,24,12,25)(9,17,18,32,16,23,27,38)(10,20,21,36,22,35,28,39)(15,30,31,34,29,37,19,33)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 13502);