Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C53 · Schur multiplier (H2) = C27 |
| direct product | C53×#34584 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C53 | C24 |
| commutator, Frattini, cosocle | C23 | C25×C53 |
| socle | C23×C53 | C25 |
| prime radical | C2×#121×C53 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C53 ◁ G | |
1774 conjugacy classes of subgroups.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 5 generators on 77 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,24)(20,22)(21,23)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77);(1,3)(2,5)(4,7)(6,8)(10,13)(14,16)(17,19,21,20)(18,22,23,24);(3,7)(5,8)(17,19,21,20)(18,22,23,24);(2,6)(5,8);(9,11,12,15)(10,14,13,16)(17,20,21,19)(18,22,23,24)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 27572);