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×#8098 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | C2×C22⋊C4 |
| commutator | C2×C4 | C2×C42×C53 |
| Frattini, cosocle | C2×C42 | C23×C53 |
| socle | C22×C53 | C2×#95 |
| prime radical | C24×C53 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C22×C42×C53 ◁ G | |
742 conjugacy classes of subgroups.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 69 points:
(1,2,4,6,5,3,8,7)(9,10)(11,13)(12,15)(14,16)(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);(1,3)(2,5)(4,7)(6,8)(9,11,13,10,14,15,12,16);(1,4,5,8)(2,6,3,7)(9,12,14,13)(10,15,16,11)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 1086);