Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C53 · Schur multiplier (H2) = C28 |
| direct product | C4×C53×#563 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C53 | C24 |
| commutator | C22 | C24×C4×C53 |
| Frattini, cosocle | C23 | C25×C53 |
| socle | C23×C53 | C25 |
| prime radical | C22×D4×C53 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C53 ◁ G | |
2024 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 5 generators on 81 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);(58,59)(60,62)(61,63)(64,65)(66,67,70,73)(68,74,76,71)(69,72,78,81)(75,80,77,79);(58,60)(59,62)(61,64)(63,65)(66,68,70,76)(67,71,73,74)(69,77,78,75)(72,80,81,79);(59,63)(60,64)(66,69,70,78)(67,72,73,81)(68,75,76,77)(71,79,74,80);(59,63)(62,65)(67,73)(68,76)(72,81)(75,77)4028 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 27342);