Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13313 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C53 · Schur multiplier (H2) = C22 |
| direct product | C53×#13470 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | #466 |
| commutator, Frattini, cosocle | C22×C8 | C23×C53 |
| prime radical | C22×D8×C53 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C22×C4×C53 ◁ C53×#405 ◁ G | |
326 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 89 points:
(5,6,12,17,11,18,10,28)(7,19,29,34,25,30,24,31)(8,26,9,16,27,14,20,15)(13,23,32,21,33,36,35,22)(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);(1,2)(3,4)(5,7,20,36,10,24,27,21,11,25,9,23,12,29,8,22)(6,13,26,34,17,32,16,30,18,33,14,31,28,35,15,19);(2,4)(6,14)(7,21)(8,20)(9,27)(10,12)(15,28)(16,17)(18,26)(19,34)(22,25)(23,29)(24,36)(30,31)(32,35)2120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6458);