Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13313 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C53 · Schur multiplier (H2) = C23 |
| direct product | C53×#13440 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | #466 |
| commutator, Frattini, cosocle | C22×C8 | C23×C53 |
| prime radical | C22×D8×C53 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C22×C4×C53 ◁ C53×#405 ◁ G | |
398 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)(5,6,11,17)(7,18,22,29)(8,12,9,25)(10,27,24,16)(13,19,31,20)(14,26,15,23)(21,28,34,32)(30,36,33,35)(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);(6,12)(7,19)(8,23)(9,26)(10,24)(14,16)(15,27)(17,25)(18,32)(20,22)(21,36)(28,29)(30,33)(34,35);(2,4)(5,7,8,19,24,34,26,36,11,22,9,20,10,21,23,35)(6,13,14,29,27,33,25,32,17,31,15,18,16,30,12,28)2120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6428);