Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C25 |
| direct product | C53×#12766 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | B(4,𝔽2) |
| commutator | C2×D4 | C22×C4×C53 |
| Frattini, cosocle | C2×#41 | C23×C53 |
| socle | C2×C53 | #1790 |
| prime radical | C53×#3529 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C2×C4×C53 ◁ C2×#41×C53 ◁ G | |
760 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,6,11,7,12,20,29)(3,13,17,26,14,31,24,25)(4,18,5,9,21,10,23,30)(8,22,27,19,28,16,32,15)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(1,3)(2,8)(4,19)(5,22)(6,24)(7,14)(9,26)(10,31)(11,27)(12,28)(13,18)(15,21)(16,23)(17,20)(25,30)(29,32);(2,9)(3,14)(8,25)(10,11)(12,30)(13,27)(15,19)(16,22)(17,24)(18,29)(26,28)(31,32)1802 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 5754);