Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C53 · Schur multiplier (H2) = C24×C4 |
| direct product | C53×#20512 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | C22×D4 |
| commutator | C4 | C22×C42×C53 |
| Frattini, cosocle | C22×C4 | C24×C53 |
| socle | C22×C53 | C2×C4×D4 |
| prime radical | C2×#135×C53 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C2×#97×C53 ◁ G | |
1014 conjugacy classes of subgroups.
| minimal degree | μ(G) = 73 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 4 generators on 73 points:
(5,6,9,13,10,7,14,12)(8,11,17,19,16,15,20,18)(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);(1,2,4,3)(5,7)(6,10)(8,15)(9,12)(11,16)(13,14)(17,18)(19,20);(1,3,4,2)(8,16)(11,15)(17,20)(18,19);(5,8,9,17,10,16,14,20)(6,11,13,19,7,15,12,18)4664 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 13500);