Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C53 · Schur multiplier (H2) = C22×C4 |
| direct product | C53×#22316 |
| 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 | |
630 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 13568 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2,4,6)(3,7,8,5)(10,12)(11,15)(17,18,21,19,22,25,30,24)(20,23,29,26,28,31,27,32)(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);(9,10,13,12)(11,16,15,14)(17,19,21,25,22,24,30,18)(20,26,29,31,28,32,27,23);(1,3,4,8)(2,5,6,7)(9,11)(10,14)(12,16)(13,15)(17,20,22,28)(18,23,25,31)(19,26,24,32)(21,29,30,27);(9,12,13,10)(11,16,15,14)(17,21,22,30)(18,24,25,19)(20,27,28,29)(23,26,31,32)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 15304);