Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#60595 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| prime radical | C3×#3010 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C4×#41 ◁ G | |
2014 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)(14,23)(15,26)(16,19)(17,21)(18,32)(20,29)(22,31)(24,40)(25,35)(27,43)(28,37)(30,39)(33,36)(34,42)(38,41);(5,8)(9,11)(12,14,17,16)(13,19,21,23)(15,24,28,27)(18,25,31,30)(20,34,36,38)(22,35,32,39)(26,43,37,40)(29,41,33,42);(12,15,17,28)(13,20,21,36)(14,24,16,27)(18,33,31,29)(19,34,23,38)(22,37,32,26)(25,42,30,41)(35,40,39,43);(4,6,7,10)(5,9,8,11)(12,16,17,14)(13,19,21,23)(15,27,28,24)(18,30,31,25)(20,34,36,38)(22,35,32,39)(26,43,37,40)(29,42,33,41);(14,16)(19,23)(24,27)(25,30)(34,38)(35,39)(40,43)(41,42)183 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53583);