Groups of order 5.2^n

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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 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