Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25×C4 |
| direct product | C3×#25735 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G | |
1466 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 43 points:
(1,2,3)(5,6)(12,13,17,21)(14,18,23,31)(15,25,28,40)(16,29,26,20)(19,33,34,27)(22,35,37,43)(24,39,36,32)(30,41,42,38);(4,5,7,6)(8,9)(10,11)(12,14)(13,18)(15,22)(16,24)(17,23)(19,30)(20,32)(21,31)(25,35)(26,36)(27,38)(28,37)(29,39)(33,41)(34,42)(40,43);(9,11)(12,15,26,33,17,28,16,27)(13,19,29,40,21,34,20,25)(14,22,36,41,23,37,24,38)(18,30,39,43,31,42,32,35);(4,6,7,5)(14,23)(18,31)(22,37)(24,36)(30,42)(32,39)(35,43)(38,41)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 18723);