Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13229 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C22×C4 |
| direct product | C53×#13229 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C53 | #1964 |
| commutator | C22×C4 | C22×C4×C53 |
| Frattini, cosocle | C2×#41 | C23×C53 |
| prime radical | C53×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C53 ◁ C2×C4×C53 ◁ C2×#131×C53 ◁ G | |
452 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)(3,13,18,28)(4,19)(5,23,14,10)(6,25)(7,12)(8,16)(9,21)(11,26)(15,24,27,30)(17,29,32,20)(22,31)(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);(1,3,14,16,7,18,5,9)(2,8,23,28,12,21,10,13)(4,15,29,25,22,27,20,11)(6,17,30,31,26,32,24,19);(1,4,6,21,7,22,26,8)(2,9,11,19,12,16,25,31)(3,15,17,10,18,27,32,23)(5,20,24,28,14,29,30,13)1802 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6217);