Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#12579 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | C2×#99 |
| commutator | C23 | C2×C42×C53 |
| Frattini, cosocle | C2×#41 | C23×C53 |
| socle | C2×C53 | C2×#342 |
| prime radical | C22×#41×C53 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C22×C4×C53 ◁ C22×#41×C53 ◁ G | |
736 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,4,9,6,8,15,21)(3,11,13,23,12,7,18,19)(5,10,16,24,17,22,28,30)(14,25,27,32,26,20,29,31)(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);(1,3,5,14,6,12,17,26)(2,7,10,20,8,11,22,25)(4,13,16,27,15,18,28,29)(9,23,24,32,21,19,30,31);(3,12)(7,11)(13,18)(14,26)(19,23)(20,25)(27,29)(31,32)2438 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 5567);