Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C25 |
| direct product | C53×#18518 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | #120 |
| commutator, Frattini, cosocle | C42 | C24×C53 |
| socle | C22×C53 | C2×#120 |
| prime radical | C53×#3035 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C2×C42×C53 ◁ G | |
968 conjugacy classes of subgroups.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 4 generators on 77 points:
(2,5)(3,7)(6,8)(9,10)(11,21)(12,18)(13,19)(14,17)(15,20)(16,22)(23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77);(1,2,4,6)(3,8,7,5)(9,11,15,12)(10,16,20,24)(13,23,14,22)(17,18,19,21);(10,17)(13,14)(16,18)(19,20)(21,24)(22,23);(9,12,15,11)(10,18,20,21)(13,23,14,22)(16,19,24,17)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 11506);