Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#13009 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | #481 |
| commutator | C2×C8 | C22×C4×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| prime radical | C24×C53 | SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C22×C4×C53 ◁ C22×M16×C53 ◁ G | |
496 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 89 points:
(1,2,4,3)(5,6,9,15,26,28,23,14,11,13,24,32,10,16,8,22)(7,17,20,12,27,30,18,34,19,35,36,29,21,33,25,31)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89);(1,3,4,2)(5,7,10,27,11,19,26,21)(6,12,28,31,13,29,16,34)(8,20,24,18,23,36,9,25)(14,33,32,35,22,30,15,17);(6,13)(8,23)(9,24)(12,29)(16,28)(18,25)(20,36)(31,34)2120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 5997);