Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C25 |
| direct product | C53×#11134 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | #397 |
| commutator | C22×C4 | C22×C4×C53 |
| Frattini, cosocle | C23×C4 | C23×C53 |
| prime radical | C24×C53 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C23×C4×C53 ◁ G | |
838 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 89 points:
(1,2,3,4)(5,6,10,14,11,15,9,16)(7,17,21,29,20,28,19,12)(8,22,25,32,26,13,24,31)(18,30,34,27,33,35,23,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89);(5,7,9,19,11,20,10,21)(6,12,14,17,15,29,16,28)(8,23,24,33,26,34,25,18)(13,30,31,27,22,35,32,36);(5,8,11,26)(6,13,15,22)(7,18,20,33)(9,25,10,24)(12,27,29,36)(14,32,16,31)(17,30,28,35)(19,34,21,23)2438 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 4122);