Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C2×C4 |
| direct product | C53×#9897 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C53 | C2×C22⋊C4 |
| commutator | C22×C4 | C22×C4×C53 |
| Frattini, cosocle | C2×C42 | C23×C53 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ C43×C53 ◁ G | |
430 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 13568 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,4,6)(3,7,8,5)(9,10,12,14,13,16,11,15)(17,18,21,24,20,23,25,31)(19,22,27,30,26,29,28,32)(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);(10,14)(11,12)(15,16)(17,19,25,27,20,26,21,28)(18,22,24,32,23,29,31,30);(1,3,4,8)(2,5,6,7)(18,23)(22,29)(24,31)(30,32)3074 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 2885);