Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C53 ⋊ #12724 |
670 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 670 conjugacy classes of subgroups — too many to draw. Only its 127 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:
(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)(55,62,66,64)(57,72)(59,76)(61,77,79,82)(63,65,81,80)(67,78,84,83)(68,85)(70,75);(54,56,60,71)(55,61,66,79)(57,68,72,85)(58,74,73,69)(59,75,76,70)(62,77,64,82)(63,67,81,84)(65,78,80,83);(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)(54,55,59,65)(56,67,70,77)(57,63,73,62)(58,64,72,81)(60,66,76,80)(61,74,78,68)(69,83,85,79)(71,84,75,82)Polycyclic presentation:
2^8,53|a^2=f,b^2=h,c^2=gh,e^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,i^a=i^52,e^b=eh,f^b=fg,d^c=dh,f^c=fh,f^d=fh1178 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 85180);