Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23439 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C26 |
| direct product | C53×#23522 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24×C53 |
| prime radical | C22×#41×C53 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C23×C4×C53 ◁ G | |
1326 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 4 generators on 89 points:
(1,2)(3,4)(5,6)(7,15)(8,19)(9,23)(10,13)(11,17)(12,22)(14,25)(16,32)(18,30)(20,31)(21,29)(24,27)(26,35)(28,33)(34,36)(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);(5,7,10,18)(6,11,13,27)(8,20,22,16)(9,24,25,17)(12,28,19,26)(14,30,23,15)(21,33,36,35)(29,31,34,32);(2,4)(5,8,9,21,10,22,25,36)(6,12,14,29,13,19,23,34)(7,16,17,33,18,20,24,35)(11,26,15,31,27,28,30,32);(6,13)(11,27)(12,19)(14,23)(15,30)(26,28)(29,34)(31,32)2438 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 16510);