Classification of groups of order sixteen times a prime

We split the classification of groups of order $16p$, $p$ prime into nine cases. Some of these cases themselves have their own sub-cases, which you can see on their pages.

• Classification of groups of order 32 ($p=2$, which is also classified as a group of prime-fifth order.)
• Classification of groups of order sixteen times a prime congruent to 1 modulo 16
• Classification of groups of order sixteen times a prime congruent to 3 modulo 16
• Classification of groups of order sixteen times a prime congruent to 5 modulo 16
• Classification of groups of order sixteen times a prime congruent to 7 modulo 16
• Classification of groups of order sixteen times a prime congruent to 9 modulo 16
• Classification of groups of order sixteen times a prime congruent to 11 modulo 16
• Classification of groups of order sixteen times a prime congruent to 13 modulo 16
• Classification of groups of order sixteen times a prime congruent to 15 modulo 16

Summary of results

Number of groups up to isomorphism

There are a few special cases to be dealt with manually:

• In the case $p=2$, that is, the groups of order 32, there are 51 groups.
• In the case $p=3$, that is, the groups of order 48, there are 52 groups.
• In the case $p=5$, that is, the groups of order 80, there are 52 groups.
• In the case $p=7$, that is, the groups of order 112, there are 43 groups.

All other cases depend only on the value of the prime modulo $16$, and the value is constant on the residue classes:

• When $p\equiv 1\mod 16$, there are 54 groups.
• When $p\equiv 3,7,11,15\mod 16$ (aside from the cases $p=3,7$ dealt with above), there are 42 groups.
• When $p\equiv 5,13\mod 16$ (aside from the case $p=5$ dealt with above), there are 51 groups.
• When $p\equiv 9\mod 16$, there are 53 groups.

List of groups

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See also

• Classification of groups of order a power of two times a prime