Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C53 · Schur multiplier (H2) = C22×C4 |
| direct product | C53×#12642 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | C2×D8 |
| commutator | C8 | C2×C42×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| socle | C22×C53 | C4×D8 |
| prime radical | C2×#135×C53 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C22×C8×C53 ◁ C2×C4×C8×C53 ◁ G | |
430 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 89 points:
(5,6,10,17,13,19,12,28)(7,20,22,30,25,36,24,33)(8,15,11,16,9,18,27,34)(14,29,31,26,32,23,35,21)(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);(1,2,3,4)(5,7,11,23,12,24,8,26,13,25,27,29,10,22,9,21)(6,14,18,30,17,31,34,36,19,32,15,33,28,35,16,20);(5,8,10,11,13,9,12,27)(6,15,17,16,19,18,28,34)(7,21,22,29,25,26,24,23)(14,30,31,36,32,33,35,20)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 5630);