Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C4 · Schur multiplier (H2) = C27×C4 |
| direct product | C3×#60129 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C25 |
| commutator | C22 | C24×C3×C4 |
| Frattini, cosocle | C2×C4 | C25×C3 |
| socle | C22×C3 | C24×C4 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ G | |
2038 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 43 points:
(1,2,3)(4,5)(6,9)(7,8)(10,11)(12,13,16,20,17,21,27,33)(14,18,24,30,22,28,35,39)(15,19,26,32,25,31,37,41)(23,29,36,40,34,38,42,43);(5,8)(9,11)(12,14)(13,18)(15,23)(16,24)(17,22)(19,29)(20,30)(21,28)(25,34)(26,36)(27,35)(31,38)(32,40)(33,39)(37,42)(41,43);(4,6,7,10)(5,9,8,11)(14,22)(18,28)(23,34)(24,35)(29,38)(30,39)(36,42)(40,43);(12,15)(13,19)(14,23)(16,26)(17,25)(18,29)(20,32)(21,31)(22,34)(24,36)(27,37)(28,38)(30,40)(33,41)(35,42)(39,43);(15,25)(19,31)(23,34)(26,37)(29,38)(32,41)(36,42)(40,43)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53117);