Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C53 · Schur multiplier (H2) = C22×C4 |
| direct product | C53×#7965 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | #95 |
| commutator | C2×C4 | C2×C42×C53 |
| Frattini, cosocle | C2×C42 | C23×C53 |
| socle | C22×C53 | C2×#95 |
| prime radical | C24×C53 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C22×C42×C53 ◁ G | |
542 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,4,8,6,9,5,10)(3,11,13,15,14,7,12,16)(17,18,20,25,22,26,21,24)(19,27,28,23,30,31,29,32)(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);(1,3,5,13,6,14,4,12)(2,7,10,16,9,11,8,15)(17,19,21,29,22,30,20,28)(18,23,25,31,26,32,24,27);(1,4,6,5)(2,8,9,10)(3,12,14,13)(7,15,11,16)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 953);