Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C4 · Schur multiplier (H2) = C29 |
| direct product | C3×#60140 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C8 | C24 |
| commutator | C22 | C24×C3×C4 |
| Frattini, cosocle | C2×C4 | C25×C3 |
| socle | C22×C3 | C24×C4 |
| prime radical | C3×#3449 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C8 ◁ G | |
2722 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 39 points:
(1,2,3)(4,5)(6,7)(8,9,12,17,13,16,23,29)(10,14,20,26,19,25,32,36)(11,15,21,28,22,27,33,37)(18,24,31,35,30,34,38,39);(5,7)(8,10)(9,14)(11,18)(12,20)(13,19)(15,24)(16,25)(17,26)(21,31)(22,30)(23,32)(27,34)(28,35)(29,36)(33,38)(37,39);(8,11,12,21,13,22,23,33)(9,15,17,28,16,27,29,37)(10,18,20,31,19,30,32,38)(14,24,26,35,25,34,36,39);(10,19)(14,25)(18,30)(20,32)(24,34)(26,36)(31,38)(35,39);(9,16)(14,25)(15,27)(17,29)(24,34)(26,36)(28,37)(35,39)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53128);