Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C53 · Schur multiplier (H2) = C24×C4 |
| direct product | C53×#17647 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | C2×C22⋊C4 |
| commutator | C2×C4 | C23×C4×C53 |
| Frattini, cosocle | C42 | C24×C53 |
| socle | C22×C53 | C22×C22⋊C4 |
| prime radical | C2×C53×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C22×C42×C53 ◁ G | |
1316 conjugacy classes of subgroups.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 4 generators on 69 points:
(1,2,3,6,4,5,7,8)(10,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69);(2,5)(3,7)(9,10)(11,14)(12,16)(13,15);(2,5)(6,8);(9,11,13,12)(10,14,15,16)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 10635);