Finite Group DatabasebetaFinite Group Databasebeta| order | 1568 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C72×C8 · Schur multiplier (H2) = C2×C7 |
| direct product | C72×C4⋊C8 |
| unambiguous semidirect product | C4⋊(C72×C8) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C72 | C22 |
| commutator | C2 | C2×C72×C8 |
| Frattini, cosocle | C2×C4 | C22×C72 |
| socle, prime radical | C22×C72 | C2×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C72 ◁ G | |
190 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 1568 |
A minimal faithful permutation representation — 2 generators on 26 points:
(1,2,3,4,5,6,7)(16,18)(19,20,21,23,22,24,25,26);(8,9,10,11,12,13,14)(15,16,17,18)980 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1568, 332);