Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13398 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C53 · Schur multiplier (H2) = C23 |
| direct product | C53×#13399 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | #481 |
| commutator, Frattini, cosocle | C22×C8 | C23×C53 |
| prime radical | C24×C53 | SD16 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C22×C4×C53 ◁ C22×M16×C53 ◁ G | |
406 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 89 points:
(5,6,10,14)(7,15,16,33)(8,19,20,36)(9,24,22,13)(11,26,27,17)(12,29,30,23)(18,31,25,34)(21,28,35,32)(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)(8,20)(12,30)(15,33)(17,26)(19,36)(21,35)(23,29)(31,34);(2,4)(5,7,9,25,10,16,22,18)(6,11,13,32,14,27,24,28)(8,21,23,26,20,35,29,17)(12,31,19,33,30,34,36,15)1802 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6387);