Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C53 · Schur multiplier (H2) = C22×C4 |
| direct product | C53×#8149 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42×C53 | C23 |
| commutator | C23 | C2×C42×C53 |
| Frattini, cosocle | C2×C42 | C23×C53 |
| socle | C23×C53 | C2×C42 |
| prime radical | C24×C53 | C42 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C2×C42×C53 ◁ G | |
554 conjugacy classes of subgroups.
| minimal degree | μ(G) = 73 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 3 generators on 73 points:
(1,2)(3,4)(5,6,9,11,8,7,10,12)(13,14,16,15,17,18,20,19)(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);(2,4)(6,7)(11,12)(13,14,16,15,17,18,20,19);(5,7,9,12,8,6,10,11)(13,15,16,18,17,19,20,14)4664 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 1137);