Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C43×C53 · Schur multiplier (H2) = C24×C43 |
| direct product | C42×C22⋊C4×C53 |
| unambiguous semidirect product | C22⋊(C43×C53) |
| subgroup | G/H | |
|---|---|---|
| center | C22×C42×C53 | C22 |
| commutator | C2 | C2×C43×C53 |
| Frattini, cosocle | C24 | C24×C53 |
| socle | C24×C53 | C24 |
| prime radical | C25×C53 | C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C42×C53 ◁ G | |
3342 conjugacy classes of subgroups.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 4 generators on 69 points:
(1,2,3,4)(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);(5,6,7,8);(9,10,11,12)(13,14)(15,16);(14,16)8480 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6735);