Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C53 · Schur multiplier (H2) = C25 |
| direct product | C2×C53×#1451 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42×C53 | C23 |
| commutator | C22 | C22×C42×C53 |
| Frattini, cosocle | C42 | C24×C53 |
| socle | C23×C53 | C2×C42 |
| prime radical | C24×C53 | C42 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C42×C53 ◁ G | |
966 conjugacy classes of subgroups.
| minimal degree | μ(G) = 71 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 4 generators on 71 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53)(54,55);(56,57,58,61,59,60,62,63)(65,68)(69,71);(64,65,66,69,67,68,70,71);(57,60)(61,63)5936 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 10305);