Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8603 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C2 |
| direct product | C53×#13107 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | #117 |
| commutator | C22×C4 | C22×C4×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| socle | C22×C53 | #432 |
| prime radical | C23×C53 | C2×M16 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C2×#97×C53 ◁ G | |
270 conjugacy classes of subgroups.
| minimal degree | μ(G) = 73 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 73 points:
(1,2)(3,4)(5,6,11,14,9,15,8,17,12,16,19,7,10,13,20,18)(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);(5,7,11,13,9,18,8,6,12,14,19,15,10,17,20,16);(2,4)(5,8,9,19,12,20,10,11)(6,7,13,17,16,14,15,18)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6095);