Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33607 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C25 |
| direct product | C53×#33613 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C53 |
| prime radical | C53×#3061 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×#41×C53 ◁ C2×#131×C53 ◁ G | |
662 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 4 generators on 89 points:
(1,2)(3,4)(5,6,10,13)(7,17,20,33)(8,14,23,31)(9,26,28,36)(11,29,27,16)(12,30,25,19)(15,32,22,18)(21,35,34,24)(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);(5,7,18,14,11,21,28,30,10,20,32,31,27,34,9,19)(6,12,26,24,16,8,22,33,13,25,36,35,29,23,15,17);(2,4)(6,13)(8,23)(12,25)(15,22)(16,29)(17,33)(24,35)(26,36);(5,8)(6,14)(7,15)(9,24)(10,23)(11,25)(12,27)(13,31)(16,30)(17,32)(18,33)(19,29)(20,22)(21,26)(28,35)(34,36)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 26601);