Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C25 |
| direct product | C53×#12749 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | #447 |
| commutator | C2×C8 | C22×C4×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| prime radical | C2×C53×#578 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C22×C4×C53 ◁ C2×#131×C53 ◁ G | |
882 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 3 generators on 89 points:
(1,2,3,4)(5,6,10,15,11,16,9,14)(7,17,20,34,21,35,19,30)(8,22,25,32,26,31,24,13)(12,23,28,33,29,36,27,18)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89);(5,7)(6,12)(8,23)(9,19)(10,20)(11,21)(13,30)(14,27)(15,28)(16,29)(17,22)(18,24)(25,33)(26,36)(31,35)(32,34);(5,6,11,16)(7,12,21,29)(8,13,26,32)(9,15,10,14)(17,33,35,18)(19,28,20,27)(22,25,31,24)(23,30,36,34)2438 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 5737);