Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C22×C4 |
| direct product | C53×#13242 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | C2×D8 |
| commutator | C2×C8 | C22×C4×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| socle | C22×C53 | C4×D8 |
| prime radical | C2×C53×#508 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C22×C8×C53 ◁ C2×C4×C8×C53 ◁ G | |
418 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)(5,6,11,18,13,19,12,31)(7,15,22,35,24,32,23,36)(8,25,21,33,27,34,20,14)(9,28,26,17,10,30,29,16)(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)(5,7,9,20,11,22,26,8,13,24,10,21,12,23,29,27)(6,14,17,32,31,34,28,35,19,33,16,15,18,25,30,36);(5,8,26,23,12,20,9,24,13,27,29,22,11,21,10,7)(6,15,30,33,18,35,16,34,19,32,28,14,31,36,17,25)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6230);