Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C53 ⋊ #12953 |
416 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 416 conjugacy classes of subgroups — too many to draw. Only its 109 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)(55,63)(57,74)(58,75)(59,60)(62,68)(64,66)(65,78)(67,76)(69,77)(70,71)(73,85)(79,82)(80,84)(81,83);(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)(54,56)(55,62,67,80)(57,73)(58,77)(59,71)(60,70)(61,72)(63,68,76,84)(64,81,65,82)(66,83,78,79)(69,75)(74,85);(54,55,59,66,61,67,60,78)(56,68,70,81,72,84,71,82)(57,63,74,65,58,76,75,64)(62,73,79,69,80,77,83,85)Polycyclic presentation:
2^8,53|a^2=f,b^2=fg,d^2=g,e^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gh,d^b=dg,e^b=egh,f^b=fh,g^b=gh,i^b=i^52,d^c=dgh,e^c=egh,f^c=fh,g^c=gh872 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 86284);