Finite Group DatabasebetaFinite Group Databasebeta| order | 1568 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C49 · Schur multiplier (H2) = C25 |
| direct product | C22×Q8×C49 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C49 | C22 |
| commutator | C2 | C24×C49 |
| Frattini, cosocle | C2×C7 | C24×C7 |
| socle, prime radical | C23×C7 | C22×C7 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C49 ◁ G | |
234 conjugacy classes of subgroups.
| minimal degree | μ(G) = 61 |
| transitive degree | μt(G) = 1568 |
A minimal faithful permutation representation — 4 generators on 61 points:
(1,2)(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61);(3,4);(5,6,7,8)(9,12,11,10);(5,9,7,11)(6,10,8,12)980 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1568, 194);