Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C43×C53 · Schur multiplier (H2) = C22×C42 |
| direct product | C4×C53×#344 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C53 | C22⋊C4 |
| commutator | C22 | C43×C53 |
| Frattini, cosocle | C23×C4 | C23×C53 |
| socle | C22×C53 | C2×#95 |
| prime radical | C24×C53 | C22×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42×C53 ◁ C22×C42×C53 ◁ G | |
906 conjugacy classes of subgroups.
| minimal degree | μ(G) = 73 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 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)(60,66,67,63)(61,65)(62,64)(68,72,73,70)(69,71);(58,60,61,68,62,67,69,73)(59,63,65,70,64,66,71,72)4664 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 3698);