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A133777 Number of isomorphism types of groups of order n!. 0
1, 1, 1, 2, 15, 47, 840, 4539 (list; graph; refs; listen; history; text; internal format)



This sequence is interesting in view of Cayley's theorem which says that every finite group with n elements is isomorphic to a subgroup of the symmetric group S_n whose number of elements is n!. Therefore a(n) - 1 gives the number of groups "competing" with S_n in this respect. The eighth term, a(7), i.e. the number of isomorphism types of groups of order 7!=5040, seems to be unknown.


Table of n, a(n) for n=0..7.

H. U. Besche, B. Eick and E. A. O'Brien, Number of isomorphism types of finite groups of given order

GAP package GrpConst


a(n) = A000001(n!). - M. F. Hasler, Dec 12 2010


a(0)=a(1)=1 because 0!=1!=1 and there is exactly one group of order one up to isomorphism.

a(2)=1 because there is exactly one group of order 2!=2, G={e,a} with a*a=e.

a(3)=2 because there are 2 groups of order 3!=6, namely the cyclic group Z/6Z and the nonabelian dihedral group of isometries of the triangle, generated by a rotation of 120° and a reflection w.r.t. an angle bisector. - M. F. Hasler, Dec 12 2010


Cf. A000001.

Sequence in context: A336209 A256328 A041719 * A025213 A290631 A116693

Adjacent sequences:  A133774 A133775 A133776 * A133778 A133779 A133780




Peter C. Heinig (algorithms(AT)gmx.de), Jan 02 2008


a(7) from Eric M. Schmidt, Sep 15 2014



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Last modified September 21 10:46 EDT 2021. Contains 347597 sequences. (Running on oeis4.)