Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C25 |
| direct product | C53×#19079 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C53 |
| prime radical | C53×#3545 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G | |
906 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2,5,8)(3,9,10,16)(4,12,13,7)(6,14,15,11)(18,23)(19,26)(20,21)(24,32)(25,30)(27,31)(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);(4,13)(7,12)(9,16)(11,14)(17,18)(19,24)(20,30)(21,23)(22,25)(26,27)(28,32)(29,31);(1,3)(2,6)(4,14)(5,10)(7,16)(8,15)(9,12)(11,13)(17,18)(19,27)(20,23)(21,30)(22,25)(24,26)(28,32)(29,31);(17,19)(18,24)(20,28)(21,29)(22,26)(23,31)(25,27)(30,32)2120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 12067);