Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C53 · Schur multiplier (H2) = C213 |
| direct product | C2×C53×#3531 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C53 | C24 |
| commutator, Frattini, cosocle | C22 | C26×C53 |
| socle | C23×C53 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C53 ◁ G | |
6554 conjugacy classes of subgroups.
| minimal degree | μ(G) = 87 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 6 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)(58,61)(59,66)(60,63)(62,68)(64,71)(65,70)(67,69)(72,73)(74,80)(75,79)(76,78)(77,81)(82,86)(83,85)(84,87);(56,58,60,65)(57,61,63,70)(59,67,68,64)(62,71,66,69)(72,74)(73,77)(75,82)(76,81)(78,80)(79,84)(83,87)(85,86);(56,59,60,68)(57,62,63,66)(58,64,65,67)(61,69,70,71)(74,81)(77,80)(82,87)(84,86);(57,63)(61,70)(62,66)(69,71)(73,78)(74,81)(79,85)(82,87);(72,75,76,83)(73,79,78,85)(74,82,81,87)(77,84,80,86)4028 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 55681);