Finite Group DatabasebetaFinite Group Databasebeta| order | 568 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C71 · Schur multiplier (H2) = C2 |
| direct product | D4×C71 |
| unambiguous semidirect product | C4⋊(C2×C71) |
| unambiguous semidirect product | C22⋊(C2×C71) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C71 | C22 |
| commutator, Frattini, cosocle | C2 | C22×C71 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C71 ◁ G | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 75 |
| transitive degree | μt(G) = 284 |
A minimal faithful permutation representation — 2 generators on 75 points:
(1,2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75);(2,4)355 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(568, 9);