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×#13232 |
| 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 | |
448 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,6,12,7,13,25,11)(3,14,18,32,19,30,28,20)(4,17,21,9,22,29,8,26)(5,10,15,27,24,16,23,31)(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);(1,3,15,9)(2,8,27,32)(4,16,14,11)(5,17,6,18)(7,19,23,26)(10,30,12,22)(13,21,31,20)(24,29,25,28);(1,4,6,21,7,22,25,8)(2,9,11,17,13,26,12,29)(3,16,18,31,19,10,28,27)(5,20,23,30,24,32,15,14)1802 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6220);