Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#60573 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C3 |
| socle | C22×C3 | C23×D4 |
| prime radical | C3×#3013 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C3×C4×#41 ◁ G | |
2004 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 43 points:
(1,2,3)(4,5)(6,9)(7,8)(10,11)(12,13,17,21)(14,18,23,32)(15,25,27,35)(16,28,30,20)(19,29,34,26)(22,36,37,43)(24,40,39,33)(31,41,42,38);(5,8)(9,11)(12,14)(13,18)(15,22)(16,24)(17,23)(19,31)(20,33)(21,32)(25,36)(26,38)(27,37)(28,40)(29,41)(30,39)(34,42)(35,43);(12,15,17,27)(13,19,21,34)(14,22,23,37)(16,29,30,26)(18,31,32,42)(20,35,28,25)(24,41,39,38)(33,43,40,36);(4,6,7,10)(5,9,8,11);(14,23)(18,32)(22,37)(24,39)(31,42)(33,40)(36,43)(38,41)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53561);