Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#23366 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C53 | C24 |
| commutator | C23 | C23×C4×C53 |
| Frattini, cosocle | C22×C4 | C24×C53 |
| socle, prime radical | C23×C53 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C53 ◁ G | |
616 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 13568 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,14,12,16,11,15)(17,18)(19,25)(20,24)(21,23)(22,27)(26,32)(28,30)(29,31)(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);(3,8)(5,7)(9,11,12,13)(10,14,16,15)(17,19,21,27)(18,22,23,25)(20,28,29,26)(24,32,31,30);(1,3,4,8)(2,5,6,7)(9,11,12,13)(10,15,16,14)(18,23)(22,25)(24,31)(30,32);(2,6)(5,7)(9,11,12,13)(10,15,16,14)(17,20,21,29)(18,24,23,31)(19,26,27,28)(22,30,25,32)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 16354);