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moa(6) = 42
moa(6) = 42
Smallest order m with exactly six nonisomorphic groups. gnu(42) = 6. OEIS A046057 a(6) = 42.
Count the different kinds of group that have a given size. Size 42 has exactly six. No smaller size has exactly six. One of the six is a circle of forty-two. The other five are not.
What it supports
OEIS A046057: smallest order m > 0 for which there are n nonisomorphic finite groups. a(6) = 42. Conway, Dietrich, and O’Brien (2008) write this as moa(n), the minimal order attaining n. The SmallGroups census lists six isomorphism types of order 42 (GAP IDs 42,1 through 42,6). Groupprops names them: GA(1,7); C2 × (C7 ⋊ C3); S3 × C7; D14 × C3; D42; C42. Only the last is abelian. 41 and 43 are prime, so each has one group.
What it does not
Not six groups because Adams. Not a filler adjective (sphenic, Harshad, abundant). Not six separate objects. Mathematics happening at order 42 is not enough; the KEEP is the first time the group-number function hits six.
Calculation
A046057 a(6)=42. gnu(42)=6. SmallGroup(42,1) … SmallGroup(42,6).
Enumeration of finite groups by order. moa(n) is the least m with gnu(m)=n. Not a search for 42.
ExactnessExact
Form42
Unitgroup order
Strengthhard
When2008
Hung22 September 2026