Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C2 |
| direct product | C53×#9633 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | #121 |
| commutator | C42 | C22×C4×C53 |
| Frattini, cosocle | C2×C42 | C23×C53 |
| socle | C22×C53 | #389 |
| prime radical | C23×C53 | #125 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C2×C42×C53 ◁ G | |
268 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,5,8,6,9,4,13)(3,10,11,7,12,15,14,16)(17,18,22,26,23,27,21,31)(19,24,28,32,29,25,20,30)(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);(1,3,5,11,6,12,4,14)(2,7,8,15,9,16,13,10)(17,19,22,28,23,29,21,20)(18,24,26,32,27,25,31,30);(17,20,21,29,23,28,22,19)(18,25,26,30,27,24,31,32)3074 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 2621);