Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C25 |
| direct product | C53×#12748 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | #447 |
| commutator | C2×C8 | C22×C4×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| prime radical | C2×C53×#576 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C22×C4×C53 ◁ C2×#131×C53 ◁ G | |
934 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 89 points:
(1,2,3,4)(5,6,10,16,11,17,9,15)(7,18,21,13,22,32,20,33)(8,23,26,14,27,34,25,35)(12,28,30,36,31,24,29,19)(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)(8,24)(9,20)(10,21)(11,22)(13,14)(15,29)(16,30)(17,31)(18,23)(19,26)(25,36)(27,28)(32,34)(33,35);(5,7)(6,13)(8,19)(9,21)(10,20)(11,22)(12,14)(15,32)(16,18)(17,33)(23,30)(24,26)(25,28)(27,36)(29,34)(31,35)2438 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 5736);