Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2883 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C53 · Schur multiplier (H2) = C210 |
| direct product | C2×C53×#2884 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C22×C22⋊C4 |
| commutator | C22 | C24×C4×C53 |
| Frattini, cosocle | C23 | C25×C53 |
| prime radical | C22×2−1+4×C53 | C2 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C23×Q8×C53 ◁ G | |
4818 conjugacy classes of subgroups.
| minimal degree | μ(G) = 87 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 5 generators on 87 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53)(54,55);(56,57)(58,66,69,62)(59,63)(60,64)(61,65)(67,80,83,74)(68,81,85,75)(70,76)(71,77)(72,78)(73,79)(82,87,84,86);(56,58)(57,62)(59,67)(60,68)(61,69)(63,74)(64,75)(65,66)(70,82)(71,83)(72,84)(73,85)(76,86)(77,80)(78,87)(79,81);(56,59,61,71)(57,63,65,77)(58,67,69,83)(60,72,73,70)(62,74,66,80)(64,78,79,76)(68,84,85,82)(75,87,81,86);(56,60,61,73)(57,64,65,79)(58,68,69,85)(59,70,71,72)(62,75,66,81)(63,76,77,78)(67,82,83,84)(74,86,80,87)4346 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 53057);