Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C53 · Schur multiplier (H2) = C22×C4 |
| direct product | C53×#12651 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | #442 |
| commutator | C8 | C2×C42×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| socle | C2×C53 | #1762 |
| prime radical | C2×#135×C53 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C22×C4×C53 ◁ C2×#97×C53 ◁ G | |
412 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,6,13,9,15,8,26)(3,16,19,32,21,27,24,29)(4,11,23,12,5,14,7,25)(10,20,28,18,31,17,30,22)(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);(1,3,7,20,6,24,4,22,9,21,23,17,8,19,5,18)(2,10,14,29,26,28,12,16,15,31,11,32,13,30,25,27);(1,4,6,23,9,5,8,7)(2,11,13,12,15,14,26,25)(3,17,19,22,21,20,24,18)(10,27,28,29,31,16,30,32)2438 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 5639);