Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C53 · Schur multiplier (H2) = C25×C4 |
| direct product | C53×#20398 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C22×C22⋊C4 |
| commutator | C22 | C22×C42×C53 |
| Frattini, cosocle | C22×C4 | C24×C53 |
| prime radical | C23×D4×C53 | C22 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×#117×C53 ◁ G | |
1658 conjugacy classes of subgroups.
| minimal degree | μ(G) = 73 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 4 generators on 73 points:
(1,2,4,3)(6,10)(8,15)(11,16)(14,19)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73);(5,6,8,12,9,10,15,17)(7,11,14,18,13,16,19,20);(1,3,4,2)(7,13)(11,16)(14,19)(18,20);(5,7,9,13)(6,11,10,16)(8,14,15,19)(12,18,17,20)4346 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 13386);