Finite Group DatabasebetaFinite Group Databasebeta| order | 688 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C43 · Schur multiplier (H2) = C22 |
| direct product | #41×C43 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C43 | C22 |
| commutator, Frattini, cosocle | C2 | C23×C43 |
| socle | C2×C43 | C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C43 ◁ G | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 344 |
A minimal faithful permutation representation — 3 generators on 51 points:
(1,2)(3,6)(4,5)(7,8)(9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51);(2,5)(6,8);(1,3,4,7)(2,6,5,8)430 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(688, 40);