Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #7855 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C53 · Schur multiplier (H2) = C22 |
| direct product | C53×#12657 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | C2×C42 |
| commutator | C2×C4 | C2×C42×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| socle | C22×C53 | C2×#97 |
| prime radical | C23×C53 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ G | |
334 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)(7,17,20,32)(8,13)(9,14)(10,25)(11,16)(12,19,27,34)(15,22)(18,29,35,33)(21,28,36,26)(23,30)(24,31)(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);(2,4)(5,7,10,18,9,19,8,21,11,20,22,35,24,34,23,36)(6,12,15,28,14,17,30,33,16,27,25,26,31,32,13,29);(5,8,9,22,11,23,24,10)(6,13,14,15,16,30,31,25)(7,18,34,36,20,35,19,21)(12,26,32,33,27,28,17,29)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 5645);