Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C53 · Schur multiplier (H2) = C2×C4 |
| direct product | C53×#12596 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | #117 |
| commutator | C2×C4 | C2×C42×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| socle | C22×C53 | #432 |
| prime radical | C23×C53 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C2×#97×C53 ◁ G | |
310 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 89 points:
(5,6,10,13,9,16,8,22,11,15,24,32,26,14,25,31)(7,17,20,34,19,12,23,28,21,30,36,33,35,29,18,27)(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)(6,12,15,29)(8,23)(9,19)(10,20)(11,21)(13,28,32,27)(14,17,16,30)(18,25)(22,33,31,34)(24,36)(26,35);(5,8,9,24,11,25,26,10)(6,13,14,31,15,32,16,22)(7,18,19,20,21,23,35,36)(12,27,17,33,29,28,30,34)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 5584);