Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13068 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#13068 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C53 | #1893 |
| commutator | C2×C8 | C22×C4×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| prime radical | C2×C53×#472 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C53 ◁ C22×C4×C53 ◁ C2×#131×C53 ◁ G | |
470 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2)(3,14)(4,19,22,28)(5,17)(6,11)(7,13)(8,12)(9,21,26,23)(10,15)(16,20,27,24)(18,25)(29,31,30,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(1,3,15,13,6,18,25,11,7,10,14,2,8,17,5,12)(4,16,30,26,23,31,20,28,22,27,29,9,21,32,24,19);(1,4)(2,9)(3,16)(5,24)(6,23)(7,22)(8,21)(10,27)(11,28)(12,19)(13,26)(14,29)(15,30)(17,32)(18,31)(20,25)2438 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6056);