Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C53 · Schur multiplier (H2) = C28 |
| direct product | C53×#34348 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C53 | C24 |
| commutator | C22 | C24×C4×C53 |
| Frattini, cosocle | C23 | C25×C53 |
| socle | C23×C53 | C25 |
| prime radical | C24×C53 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C53 ◁ G | |
1898 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 5 generators on 89 points:
(1,2,3,4)(5,6)(7,10)(8,15)(9,12)(11,17)(13,20)(14,19)(16,18)(21,22)(23,29)(24,28)(25,27)(26,30)(31,35)(32,34)(33,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89);(5,7,9,14)(6,10,12,19)(8,16,17,13)(11,20,15,18)(21,23)(22,26)(24,31)(25,30)(27,29)(28,33)(32,36)(34,35);(5,8,9,17)(6,11,12,15)(7,13,14,16)(10,18,19,20)(23,30)(26,29)(31,36)(33,35);(6,12)(10,19)(11,15)(18,20)(22,27)(23,30)(28,34)(31,36);(21,24,25,32)(22,28,27,34)(23,31,30,36)(26,33,29,35)4028 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 27336);