Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10374 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C53 · Schur multiplier (H2) = C22 |
| direct product | C53×#10377 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | #393 |
| commutator, Frattini, cosocle | C23×C4 | C23×C53 |
| prime radical | C24×C53 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C23×C4×C53 ◁ G | |
422 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)(3,6)(4,11,13,7)(5,8)(9,16,12,14)(10,15)(17,18,20,22)(19,25,26,32)(21,29,28,24)(23,30,31,27)(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);(1,3,5,10)(2,6,8,15)(4,12)(7,14)(9,13)(11,16)(18,22)(23,31)(24,29)(25,32);(1,4,5,13)(2,7,8,11)(3,9,10,12)(6,14,15,16)(17,19,20,26)(18,23,22,31)(21,30,28,27)(24,32,29,25)1961 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 3365);