Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9755 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C2×C8 |
| direct product | C53×#9769 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | #390 |
| commutator | C42 | C22×C4×C53 |
| Frattini, cosocle | C2×C42 | C23×C53 |
| prime radical | C2×#127×C53 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G | |
450 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2)(3,9)(4,12)(5,7)(6,10)(8,13)(11,15)(14,16)(17,18,22,19,21,25,26,24)(20,28,30,23,29,27,32,31)(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);(1,3,4,10,5,11,13,16)(2,6,8,9,7,14,12,15)(17,19,26,25,21,24,22,18)(20,27,32,31,29,28,30,23);(2,7)(6,14)(8,12)(9,15)(17,20,26,32,21,29,22,30)(18,23,19,27,25,31,24,28)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 2757);