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×#12623 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C53 | #99 |
| commutator | C23 | C2×C42×C53 |
| Frattini, cosocle | C2×#41 | C23×C53 |
| socle | C2×C53 | C4×#99 |
| prime radical | C53×#525 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C53 ◁ C2×C8×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:
(1,2,5,10,7,4,9,16)(3,12,14,25,13,8,20,21)(6,11,18,24,19,17,23,29)(15,22,28,31,27,26,30,32)(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);(1,3,6,15)(2,8,11,22)(4,12,17,26)(5,14,18,28)(7,13,19,27)(9,20,23,30)(10,21,24,31)(16,25,29,32);(1,4,5,16,7,2,9,10)(3,12,14,25,13,8,20,21)(6,17,18,29,19,11,23,24)(15,26,28,32,27,22,30,31)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 5611);