Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19755 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#19823 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | D42 |
| commutator, Frattini, cosocle | C42 | C24×C53 |
| prime radical | C53×#3325 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C42×C53 ◁ G | |
632 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2)(3,7)(4,11)(5,13)(6,9)(8,10)(12,15)(14,16)(18,22)(20,28)(23,29)(24,25)(26,32)(30,31)(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);(1,3,6,5)(2,7,9,13)(4,10,12,14)(8,15,16,11)(17,18)(19,23)(20,24)(21,25)(22,28)(26,30)(27,31)(29,32);(1,4,6,12)(2,8,9,16)(3,10,5,14)(7,15,13,11)(17,19,21,27)(18,23,25,31)(20,26,28,32)(22,29,24,30);(18,24)(20,28)(22,25)(23,30)(26,32)(29,31)2120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 12811);