Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C3×#23542 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C24×C3 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G | |
1362 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)(14,24,28,35)(15,29,31,39)(16,22,17,21)(19,27,36,26)(20,33,38,32)(25,41,42,34)(30,37,43,40);(8,9)(10,11)(12,14,18,28)(13,19,23,36)(15,30,31,43)(16,27,17,26)(20,37,38,40)(21,35,22,24)(25,32,42,33)(29,41,39,34);(5,7)(9,11)(12,15,16,32,18,31,17,33)(13,20,21,29,23,38,22,39)(14,25,26,43,28,42,27,30)(19,34,24,40,36,41,35,37);(15,31)(20,38)(25,42)(29,39)(30,43)(32,33)(34,41)(37,40)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 16530);