Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24468 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24×C4 |
| direct product | C3×#25596 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C3×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
1064 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 43 points:
(1,2,3)(4,5)(6,10)(7,8)(9,11)(12,13,16,20)(14,22,24,18)(15,19,27,35)(17,30,29,21)(23,32,39,38)(25,33,31,37)(26,40,41,34)(28,43,42,36);(5,8)(9,10)(12,14)(13,18)(15,26)(16,24)(17,31)(19,34)(20,22)(21,37)(23,28)(25,29)(27,41)(30,33)(32,36)(35,40)(38,43)(39,42);(12,15)(13,19)(14,23,24,39)(16,27)(17,28)(18,32,22,38)(20,35)(21,36)(25,26,31,41)(29,42)(30,43)(33,34,37,40);(4,6,7,11)(5,9,8,10)(13,20)(15,27)(18,22)(21,30)(23,39)(26,41)(28,42)(33,37)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 18584);