Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C53 · Schur multiplier (H2) = C23 |
| direct product | C53×#12608 |
| 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 | C24×C53 | M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C2×#97×C53 ◁ G | |
462 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 89 points:
(1,2,4,3)(5,6)(7,17,20,28)(8,14)(9,15)(10,16)(11,13)(12,19,25,18)(21,31,34,36)(22,29)(23,27)(24,30)(26,33,35,32)(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);(5,7,9,21,10,19,24,33,11,20,23,34,8,18,22,32)(6,12,15,26,16,17,30,31,13,25,27,35,14,28,29,36);(1,3,4,2)(6,13)(7,18,20,19)(9,23)(12,17,25,28)(14,16)(21,33,34,32)(22,24)(26,36,35,31)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 5596);