Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60457 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27×C4 |
| direct product | C3×#60624 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| prime radical | C3×#3500 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C4×#41 ◁ G | |
2038 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)(14,24)(15,19)(16,21)(17,32)(18,23)(20,27)(22,25)(26,38)(28,33)(29,34)(30,36)(31,42)(35,40)(37,39)(41,43);(5,8)(9,11)(12,14,25,29,18,27,17,15)(13,19,32,20,23,34,22,24)(16,26,39,41,30,40,31,28)(21,33,42,35,36,43,37,38);(4,6,7,10)(5,9,8,11)(12,15,17,27,18,29,25,14)(13,20,22,19,23,24,32,34)(16,28,31,40,30,41,39,26)(21,35,37,33,36,38,42,43);(12,16)(13,21)(14,26)(15,28)(17,31)(18,30)(19,33)(20,35)(22,37)(23,36)(24,38)(25,39)(27,40)(29,41)(32,42)(34,43);(16,30)(21,36)(26,40)(28,41)(31,39)(33,43)(35,38)(37,42)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53612);