Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×#23425 |
| 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 | |
1938 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)(4,5)(6,7)(12,13,18,23)(14,24,26,35)(15,29,32,36)(16,22,17,21)(19,28,34,27)(20,30,37,31)(25,39,40,33)(38,41,43,42);(5,7)(8,9)(10,11)(12,14)(13,19)(15,25)(16,28)(17,27)(18,26)(20,33)(21,35)(22,24)(23,34)(29,43)(30,42)(31,41)(32,40)(36,38)(37,39);(9,11)(12,15,18,32)(13,20,23,37)(14,25,26,40)(16,31,17,30)(19,33,34,39)(21,29,22,36)(24,38,35,43)(27,42,28,41);(14,26)(19,34)(24,35)(25,40)(27,28)(33,39)(38,43)(41,42)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 16413);