Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C53 · Schur multiplier (H2) = C210 |
| direct product | C22×C53×#413 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C53 | C22⋊C4 |
| commutator | C22 | C24×C4×C53 |
| Frattini, cosocle | C2×C4 | C25×C53 |
| socle | C23×C53 | C2×C22⋊C4 |
| prime radical | C23×#41×C53 | C2 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C53 ◁ C24×C4×C53 ◁ G | |
5118 conjugacy classes of subgroups.
| minimal degree | μ(G) = 73 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 5 generators on 73 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);(58,59,62,67,63,66,61,65)(60,68,69,73,70,64,71,72);(58,60)(59,64)(61,69)(62,71)(63,70)(65,72)(66,68)(67,73);(58,61,63,62)(59,65,66,67)(60,69,70,71)(64,72,68,73)4664 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 53061);