Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C53 · Schur multiplier (H2) = C29 |
| direct product | C53×#60144 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | C25 |
| commutator | C22 | C24×C4×C53 |
| Frattini, cosocle | C2×C4 | C25×C53 |
| socle | C22×C53 | C24×C4 |
| prime radical | C2×C53×#532 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ G | |
2778 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)(5,6,9,13,10,7,11,15)(8,14,17,20,16,12,18,19)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73);(2,4)(8,16)(12,14)(17,18)(19,20);(5,7,9,15,10,6,11,13)(8,14,17,20,16,12,18,19);(6,7)(12,14)(13,15)(19,20);(5,8)(6,12)(7,14)(9,17)(10,16)(11,18)(13,19)(15,20)4664 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 53132);