Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C53 · Schur multiplier (H2) = C23×C4 |
| direct product | C53×#12573 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | #99 |
| commutator | C23 | C2×C42×C53 |
| Frattini, cosocle | C22×D4 | C23×C53 |
| socle | C22×C53 | C2×#99 |
| prime radical | C22×#41×C53 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C22×C4×C53 ◁ C22×#41×C53 ◁ G | |
838 conjugacy classes of subgroups.
| minimal degree | μ(G) = 73 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 73 points:
(5,6)(7,11)(8,15,16,10)(9,12)(13,19,20,17)(14,18)(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);(1,2,3,4)(6,10)(11,17)(12,15)(18,19);(5,7,9,14)(6,11,12,18)(8,13,16,20)(10,17,15,19)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 5561);