Groups of order 2^m.3^n

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This article discusses groups whose order has prime factors in the set \{ 2, 3 \} and no others. Another way of putting this is that the article discusses finite \pi-groups where \pi = \{ 2, 3\}.

There are only two prime factors of this number. Order has only two prime factors implies solvable (by Burnside's p^aq^b-theorem) and hence all groups of this order are solvable groups (specifically, finite solvable groups). Another way of putting this is that the order is a solvability-forcing number. In particular, there is no simple non-abelian group of this order.

Orders of interest

The clickable links in the table are to pages devoted to groups of that order.

The rows represent values of m and corresponding values of 2^m. The columns represent values of n and corresponding values of 3^n.

m 2^m generic n n = 0 n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 n = 7.
-- -- 3^n = 1 3^n = 3 3^n = 9 3^n = 27 3^n = 81 3^n = 243 3^n = 729 3^n = 2187
generic generic 2^m 3.2^m 3^2.2^m 3^3.2^m 3^4.2^m 3^5.2^m 3^6.2^m 3^7.2^m
0 1 3^n 1 3 9 27 81 243 729 2187
1 2 2.3^n 2 6 18 54 162 486 1458 4374
2 4 2^2.3^n 4 12 36 108 324 972 2916 8748
3 8 2^3.3^n 8 24 72 216 648 1944 5832
4 16 2^4.3^n 16 48 144 432 1296 3888
5 32 2^5.3^n 32 96 288 864
6 64 2^6.3^n 64 192 576 1728
7 128 2^7.3^n 128 384 1152
8 256 2^8.3^n 256 768
9 512 2^9.3^n 512 1536
10 1024 2^10.3^n 1024

Counts of groups

Counts of all groups

The rows give values of m, 2^m and the columns give values of n, 2^n. Each cell value is the total number of isomorphism classes of groups of order 2^m \cdot 3^n.

Note that for small values of m,n, the order of magnitude of the values depends roughly on m + n (so it is roughly of the same order of magnitude as we move along a north-east-to-south-west diagonal). This fails to hold for larger values of m,n.

m 2^m n = 0 n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 n = 7.
-- -- 3^n = 1 3^n = 3 3^n = 9 3^n = 27 3^n = 81 3^n = 243 3^n = 729 3^n = 2187
0 1 1 1 2 5 15 67 504 9310
1 2 1 2 5 15 55 261 1798
2 4 2 5 14 45 176 900
3 8 5 15 50 177 757 3973
4 16 14 52 197 775 3609
5 32 51 231 1045 4725
6 64 267 1543 8681 47937
7 128 2328 20169 157877
8 256 56092 1090235
9 512 10494213 408641062
10 1024 49487365422

Counts of nilpotent groups

See number of nilpotent groups equals product of number of groups of order each maximal prime power divisor, which in turn follows from equivalence of definitions of finite nilpotent group.

Thus, the number of nilpotent groups of order 2^m \cdot 3^n = (number of groups of order 2^m) \times (number of groups of order 3^n)

m 2^m n = 0 n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 n = 7.
-- -- 3^n = 1 3^n = 3 3^n = 9 3^n = 27 3^n = 81 3^n = 243 3^n = 729 3^n = 2187
0 1 1 1 2 5 15 67 504 9310
1 2 1 1 2 5 15 67 504 9310
2 4 2 2 4 10 30 134 1008 18620
3 8 5 5 10 25 75 335 2520 46550
4 16 14 14 28 70 210 938 7056 130340
5 32 51 51 102 255 765 3417 25704 474810
6 64 267 267 534 1335 4005 17889 134568 2485770
7 128 2328
8 256 56092
9 512 10494213
10 1024 49487365422