Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17367 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C3×#20730 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×C22⋊C4 |
| commutator | C23 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C24×C3 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×#117 ◁ G | |
1166 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)(12,13,16,21,17,20,29,34)(14,18,24,32,23,31,39,42)(15,25,26,35,27,19,28,33)(22,36,38,43,37,30,40,41);(5,8)(9,11)(12,14)(13,18)(15,22)(16,24)(17,23)(19,30)(20,31)(21,32)(25,36)(26,38)(27,37)(28,40)(29,39)(33,41)(34,42)(35,43);(12,15,17,27)(13,19,20,25)(14,22,23,37)(16,28,29,26)(18,30,31,36)(21,35,34,33)(24,40,39,38)(32,43,42,41);(4,6,7,10)(5,9,8,11)(14,23)(18,31)(22,37)(24,39)(30,36)(32,42)(38,40)(41,43)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 13718);