Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19697 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C23×C8 |
| direct product | C53×#19703 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C24×C53 |
| prime radical | C53×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G | |
704 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2,5,8)(3,9,10,16)(4,12,13,7)(6,14,15,11)(18,22)(19,25)(20,28)(23,26)(24,30)(27,31)(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);(3,10)(6,15)(9,16)(11,14)(17,18,20,22,21,24,28,30)(19,26,25,32,27,23,31,29);(1,3,4,11,5,10,13,14)(2,6,7,9,8,15,12,16)(17,19,20,25,21,27,28,31)(18,23,22,29,24,26,30,32);(19,27)(23,26)(25,31)(29,32)2120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 12691);