Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C53×#2133 ⋊ C2 |
| Sylow | C53 ⋊ #13606 |
320 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 320 conjugacy classes of subgroups — too many to draw. Only its 67 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)(54,56,59,65)(55,61,64,76)(57,66,69,79)(58,67,71,80)(60,72,62,74)(63,75,78,85)(68,81,70,82)(73,83,77,84);(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)(55,60)(57,68)(58,71)(61,74)(62,64)(66,81)(67,80)(69,70)(72,76)(73,77)(75,85)(79,82);(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)(54,55,57,63)(56,61,66,75)(58,62,68,77)(59,64,69,78)(60,70,73,71)(65,76,79,85)(67,74,81,84)(72,82,83,80)Polycyclic presentation:
2^8,53|a^2=e,c^2=h,d^2=g,f^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,i^a=i^52,c^b=cg,d^b=dg,e^b=efgh,f^b=fh,i^b=i^52,d^c=dh,e^d=egh,f^e=fh623 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 89960);