Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C53 · Schur multiplier (H2) = C22 |
| direct product | C53×#13162 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C53 | C2×D4 |
| commutator | C2×C4 | C22×C8×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| socle | C22×C53 | C8×D4 |
| prime radical | C23×C53 | #131 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C53 ◁ C22×C16×C53 ◁ G | |
290 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)(5,6,10,14,11,15,8,22,12,16,24,31,9,26,25,36)(7,13,19,29,20,28,23,35,21,30,33,27,18,34,17,32)(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);(2,4)(5,7,10,19,11,20,8,23,12,21,24,33,9,18,25,17)(6,13,14,29,15,28,22,35,16,30,31,27,26,34,36,32);(5,8,9,10,12,25,11,24)(6,14,15,22,16,31,26,36)(7,17,18,33,21,23,20,19)(13,27,28,32,30,29,34,35)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6150);