LMFDB - Galois groups: statistics

The database currently contains $512{,}614$ transitive subgroups of $S_n$, including all subgroups (up to conjugacy) for $n \le 47$ and $n \ne 32$. Among the $2{,}801{,}324$ groups in degree $32$, all those with order less than $512$ or greater than $40{,}000{,}000{,}000$ are included.

Distribution of degrees and orders

		order
		1-15	16-31	32-63	64-127	128-255	256-511	512-1023	1024-2047	2048-65535	65536-40000000000	40000000000-	Total
degree	1-3	4											4
	1-3	100.00%											0.00%
	4-7	12	7	6	3	1	1	1		2			33
	4-7	36.36%	21.21%	18.18%	9.09%	3.03%	3.03%	3.03%		6.06%			0.01%
	8	5	9	11	9	7	3	2	2	2			50
	8	10.00%	18.00%	22.00%	18.00%	14.00%	6.00%	4.00%	4.00%	4.00%			0.01%
	9-11	5	8	10	12	14	10	10	6	6	6		87
	9-11	5.75%	9.20%	11.49%	13.79%	16.09%	11.49%	11.49%	6.90%	6.90%	6.90%		0.02%
	12	5	10	18	43	48	33	37	40	59	8		301
	12	1.66%	3.32%	5.98%	14.29%	15.95%	10.96%	12.29%	13.29%	19.60%	2.66%		0.06%
	13-15	4	5	9	7	10	18	17	17	55	30	4	176
	13-15	2.27%	2.84%	5.11%	3.98%	5.68%	10.23%	9.66%	9.66%	31.25%	17.05%	2.27%	0.03%
	16		14	52	130	251	330	304	248	532	91	2	1954
	16		0.72%	2.66%	6.65%	12.85%	16.89%	15.56%	12.69%	27.23%	4.66%	0.10%	0.38%
	17-23		17	39	64	92	136	149	158	921	732	40	2348
	17-23		0.72%	1.66%	2.73%	3.92%	5.79%	6.35%	6.73%	39.22%	31.18%	1.70%	0.46%
	24		15	44	144	375	778	1315	2315	11075	8868	71	25000
	24		0.06%	0.18%	0.58%	1.50%	3.11%	5.26%	9.26%	44.30%	35.47%	0.28%	4.88%
	25-31		19	34	64	145	201	249	365	2277	6383	548	10285
	25-31		0.18%	0.33%	0.62%	1.41%	1.95%	2.42%	3.55%	22.14%	62.06%	5.33%	2.01%
	32			51	369	1809	7322	8	8	1		2001	11569
	32			0.44%	3.19%	15.64%	63.29%	0.07%	0.07%	0.01%		17.30%	2.26%
	33-35			4	9	10	11	11	8	107	384	140	684
	33-35			0.58%	1.32%	1.46%	1.61%	1.61%	1.17%	15.64%	56.14%	20.47%	0.13%
	36			14	85	210	403	886	1425	14354	96392	7510	121279
	36			0.01%	0.07%	0.17%	0.33%	0.73%	1.17%	11.84%	79.48%	6.19%	23.66%
	37-39			5	10	10	11	9	11	87	187	63	393
	37-39			1.27%	2.54%	2.54%	2.80%	2.29%	2.80%	22.14%	47.58%	16.03%	0.08%
	40			14	51	130	246	544	779	17508	281759	14811	315842
	40			0.00%	0.02%	0.04%	0.08%	0.17%	0.25%	5.54%	89.21%	4.69%	61.61%
	41-47			17	35	57	114	156	185	1748	13777	6520	22609
	41-47			0.08%	0.15%	0.25%	0.50%	0.69%	0.82%	7.73%	60.94%	28.84%	4.41%
	Total	35	104	328	1035	3169	9617	3698	5567	48734	408617	31710	512614
	Total	0.01%	0.02%	0.06%	0.20%	0.62%	1.88%	0.72%	1.09%	9.51%	79.71%	6.19%

Distribution of solvability and degrees

		degree
		1-3	4-7	8	9-11	12	13-15	16	17-23	24	25-31	32	33-35	36	37-39	40	41-47	Total
solvable	no		9	5	29	36	70	49	542	807	2688	162	331	3252	72	24244	6517	38813
	no		27.27%	10.00%	33.33%	11.96%	39.77%	2.51%	23.08%	3.23%	26.14%	1.40%	48.39%	2.68%	18.32%	7.68%	28.82%	7.57%
	yes	4	24	45	58	265	106	1905	1806	24193	7597	11407	353	118027	321	291598	16092	473801
	yes	100.00%	72.73%	90.00%	66.67%	88.04%	60.23%	97.49%	76.92%	96.77%	73.86%	98.60%	51.61%	97.32%	81.68%	92.32%	71.18%	92.43%
	Total	4	33	50	87	301	176	1954	2348	25000	10285	11569	684	121279	393	315842	22609	512614
	Total	0.00%	0.01%	0.01%	0.02%	0.06%	0.03%	0.38%	0.46%	4.88%	2.01%	2.26%	0.13%	23.66%	0.08%	61.61%	4.41%

Distribution of primitivity and degrees

		degree
		1-3	4-7	8	9-11	12	13-15	16	17-23	24	25-31	32	33-35	36	37-39	40	41-47	Total
primitive	no		15	43	59	295	157	1932	2302	24995	10197	11567	672	121257	376	315834	22564	512265
	no		45.45%	86.00%	67.82%	98.01%	89.20%	98.87%	98.04%	99.98%	99.14%	99.98%	98.25%	99.98%	95.67%	100.00%	99.80%	99.93%
	yes	4	18	7	28	6	19	22	46	5	88	2	12	22	17	8	45	349
	yes	100.00%	54.55%	14.00%	32.18%	1.99%	10.80%	1.13%	1.96%	0.02%	0.86%	0.02%	1.75%	0.02%	4.33%	0.00%	0.20%	0.07%
	Total	4	33	50	87	301	176	1954	2348	25000	10285	11569	684	121279	393	315842	22609	512614
	Total	0.00%	0.01%	0.01%	0.02%	0.06%	0.03%	0.38%	0.46%	4.88%	2.01%	2.26%	0.13%	23.66%	0.08%	61.61%	4.41%

Distribution of number of arithmetic equivalent siblings and degrees

		degree
		1-3	4-7	8	9-11	12	13-15	16	17-23	24	25-31	32	33-35	36	37-39	40	41-47	Total
arithmetic equivalent count	not computed													80		235199	6032	241311
	not computed													0.07%		74.47%	26.68%	47.07%
	0	4	32	48	86	295	167	1784	2317	23882	10192	9255	670	120017	388	78879	16366	264382
	0	100.00%	96.97%	96.00%	98.85%	98.01%	94.89%	91.30%	98.68%	95.53%	99.10%	80.00%	97.95%	98.96%	98.73%	24.97%	72.39%	51.58%
	1		1	2	1	6	9	166	31	1106	81	2020	14	1154	5	1719	211	6526
	1		3.03%	4.00%	1.15%	1.99%	5.11%	8.50%	1.32%	4.42%	0.79%	17.46%	2.05%	0.95%	1.27%	0.54%	0.93%	1.27%
	2									11	11	6		22		39		89
	2									0.04%	0.11%	0.05%		0.02%		0.01%		0.02%
	3							4				270		6		6		286
	3							0.20%				2.33%		0.00%		0.00%		0.06%
	5									1	1							2
	5									0.00%	0.01%							0.00%
	7											18						18
	7											0.16%						0.00%
	Total	4	33	50	87	301	176	1954	2348	25000	10285	11569	684	121279	393	315842	22609	512614
	Total	0.00%	0.01%	0.01%	0.02%	0.06%	0.03%	0.38%	0.46%	4.88%	2.01%	2.26%	0.13%	23.66%	0.08%	61.61%	4.41%

Distribution of degrees and nilpotency classes

		nilpotency class
		-1	0	1	2	3	4	5	6	7	8	9	Total
degree	1-3	1	1	2									4
	1-3	25.00%	25.00%	50.00%									0.00%
	4-7	27		5	1								33
	4-7	81.82%		15.15%	3.03%								0.01%
	8	24		3	8	11	4						50
	8	48.00%		6.00%	16.00%	22.00%	8.00%						0.01%
	9-11	80		4	2	1							87
	9-11	91.95%		4.60%	2.30%	1.15%							0.02%
	12	298		2	1								301
	12	99.00%		0.66%	0.33%								0.06%
	13-15	173		3									176
	13-15	98.30%		1.70%									0.03%
	16	527		5	55	301	420	268	181	114	83		1954
	16	26.97%		0.26%	2.81%	15.40%	21.49%	13.72%	9.26%	5.83%	4.25%		0.38%
	17-23	2335		9	3	1							2348
	17-23	99.45%		0.38%	0.13%	0.04%							0.46%
	24	24974		3	8	11	4						25000
	24	99.90%		0.01%	0.03%	0.04%	0.02%						4.88%
	25-31	10040		11	14	48	54	52	16	24	16	10	10285
	25-31	97.62%		0.11%	0.14%	0.47%	0.53%	0.51%	0.16%	0.23%	0.16%	0.10%	2.01%
	32	2390		7	499	5048	2968	563	94				11569
	32	20.66%		0.06%	4.31%	43.63%	25.65%	4.87%	0.81%				2.26%
	33-35	681		3									684
	33-35	99.56%		0.44%									0.13%
	36	121264		4	8	3							121279
	36	99.99%		0.00%	0.01%	0.00%							23.66%
	37-39	390		3									393
	37-39	99.24%		0.76%									0.08%
	40	315816		3	8	11	4						315842
	40	99.99%		0.00%	0.00%	0.00%	0.00%						61.61%
	41-47	22596		9	3	1							22609
	41-47	99.94%		0.04%	0.01%	0.00%							4.41%
	Total	501616	1	76	610	5436	3454	883	291	138	99	10	512614
	Total	97.85%	0.00%	0.01%	0.12%	1.06%	0.67%	0.17%	0.06%	0.03%	0.02%	0.00%

Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Introduction

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Database

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Distribution of degrees and orders

Distribution of solvability and degrees

Distribution of primitivity and degrees

Distribution of number of arithmetic equivalent siblings and degrees

Distribution of degrees and nilpotency classes

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.