Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C53 ⋊ #13269 |
308 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 308 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 85 points:
(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(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)(54,57,59,72,61,74,58,75)(55,62,63,70,66,82,65,80)(56,67,68,83,64,81,69,71)(60,73,77,79,78,84,76,85);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(54,56,60,65,59,69,77,66,61,64,78,63,58,68,76,55)(57,71,84,62,72,81,85,80,74,83,73,82,75,67,79,70);(54,55,60,64,59,65,77,68,61,66,78,56,58,63,76,69)(57,70,73,71,72,62,79,81,74,80,84,83,75,82,85,67)Polycyclic presentation:
2^8,53|a^2=f,b^2=def,c^2=e,d^2=h,e^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,e^a=eh,c^b=cg,e^b=eh,f^b=fh,d^c=dh,i^c=i^521190 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 88109);