Groups of order 5.2^n

This article discusses the groups of order $5\cdot 2^{n}$, where $n$ varies over nonnegative integers. Note that any such group has a 5-Sylow subgroup which is cyclic group:Z5, and a 2-Sylow subgroup, which is of order $2^{n}$. Further, because order has only two prime factors implies solvable, any such group is a solvable group.

Number of groups of small orders

Exponent $n$	Value $2^{n}$	Value $5\cdot 2^{n}$	Number of groups of order $5\cdot 2^{n}$	Reason/explanation/list
0	1	5	1	only cyclic group:Z5, see equivalence of definitions of group of prime order
1	2	10	2	cyclic group:Z10 and dihedral group:D10; see classification of groups of an order two times a prime or classification of groups of order a product of two distinct primes
2	4	20	5	See groups of order 20
3	8	40	14	See groups of order 40
4	16	80	52	See groups of order 80
5	32	160	238	See groups of order 160
6	64	320	1640	See groups of order 320
7	128	640	21541	See groups of order 640
8	256	1280	1116461	See groups of order 1280