Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C53 · Schur multiplier (H2) = C25×C4 |
| direct product | C53×#10736 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C53 | C2×C22⋊C4 |
| commutator | C23 | C2×C42×C53 |
| Frattini, cosocle | C25 | C23×C53 |
| prime radical | C25×C53 | C23 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ C24×C4×C53 ◁ G | |
1510 conjugacy classes of subgroups.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 77 points:
(1,2,3,4)(5,6)(7,8)(9,10)(11,14,18,21)(12,15)(13,16)(17,20,23,24)(19,22)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77);(9,11)(10,14,16,21)(12,17)(13,18)(15,20,22,24)(19,23);(6,8)(9,12,13,19)(10,15,16,22)(11,17,18,23)(14,20,21,24)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 3724);