Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C53 · Schur multiplier (H2) = C2×C4 |
| direct product | C53×#12668 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C53 | #106 |
| commutator | C8 | C2×C42×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| socle | C2×C53 | C4×#106 |
| prime radical | C23×C53 | #135 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C53 ◁ C22×C8×C53 ◁ C2×C4×C8×C53 ◁ G | |
308 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,6,12,7,14,4,10,8,13,5,11,20,28,21,29)(3,15,17,27,18,30,22,26,19,25,16,32,24,31,23,9)(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)(2,9,13,26)(4,16)(5,22)(6,23)(7,24)(8,19)(10,25,29,15)(11,30,12,31)(14,32,28,27)(17,21)(18,20);(1,4,20,6,8,21,7,5)(2,10,28,12,13,29,14,11)(3,16,18,23,19,17,24,22)(9,25,27,31,26,15,32,30)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 5656);