Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C53 · Schur multiplier (H2) = C25×C4 |
| direct product | C53×#20383 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8×C53 | C23 |
| commutator | C22 | C22×C42×C53 |
| Frattini, cosocle | C22×C4 | C24×C53 |
| socle | C23×C53 | C23×C4 |
| prime radical | C22×#41×C53 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8×C53 ◁ G | |
1622 conjugacy classes of subgroups.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 4 generators on 77 points:
(5,6)(7,8)(9,10,12,16,13,15,19,22)(11,14,17,21,18,20,23,24)(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);(1,2,3,4)(6,8);(9,11,12,17,13,18,19,23)(10,14,16,21,15,20,22,24);(10,15)(14,20)(16,22)(21,24)5936 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 13371);