Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32941 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C23×C4 |
| direct product | C3×#33062 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C2×C3×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
764 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)(6,9)(7,8)(10,11)(12,13)(14,24)(15,19)(16,30)(17,34)(18,23)(20,27)(21,32)(22,25)(26,35)(28,37)(29,36)(31,39)(33,38)(40,41)(42,43);(5,8)(9,11)(12,14,25,29,18,27,17,15)(13,19,34,20,23,36,22,24)(16,26,32,41,33,42,31,28)(21,35,30,37,39,43,38,40);(4,6,7,10)(5,9,8,11)(12,15,17,27,18,29,25,14)(13,20,22,19,23,24,34,36)(16,28,31,42,33,41,32,26)(21,37,38,35,39,40,30,43);(12,16)(13,21)(14,26)(15,28)(17,31)(18,33)(19,35)(20,37)(22,38)(23,39)(24,40)(25,32)(27,42)(29,41)(30,34)(36,43)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 26050);