Groups of order 2^m.3^n

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 $p^{a}q^{b}$-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	49487367289

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	49487367289