User:Tomruen/List of small groups

Notes

• "≈" should be $\cong$ or ≅ for isomorphic.
• × : Direct product
• ^ : wreath product, $\wr$ or ≀
  • $C_{2}\wr X_{n}$ = ²X_n
• Cn=Z_n ≈ [n]⁺ :Cyclic group, ${\mathfrak {C}}_{n}$, order n, also as $\mathbb {Z} _{n}$
  • Z_nm ≈ Z_n × Z_m, if gcd(n,m)=1.
• Dn=D_n ≈ [n] : Dihedral group, ${\mathfrak {D}}_{n}$, order n, also as Dih_n.
  • D_2n ≈ Z₂ × D_n, n odd.
  • D₂ ≈ Z₂ × D₁ ≈ Z₂ × Z₂.
• Sn=S_n ≈ [3^n-2] : Symmetric group, ${\mathfrak {S}}_{n}$, order n!
  • S2=S₂ ≈ D₁ ≈ [1], order 2
  • S3=S₃ ≈ D₃ ≈ [3], order 6
  • S4=S₄ ≈ [3,3] ≈ [4,3]⁺, order 24
  • S5=S₅ ≈ [3,3,3], order 120
• An=A_n ≈ [3^n-2]⁺: Alternating group, ${\mathfrak {A}}_{n}$, order n!/2
  • A2=A₂ ≈ Z₁ ≈ [ ]⁺, order 1
  • A3=A₃ ≈ Z₃ ≈ [3]⁺, order 3
  • A4=A₄ ≈ [3,3]⁺, order 12
  • A5=A₅ ≈ [3,3,3]⁺ ≈ [5,3]⁺, order 60
• Q8=Q=$Q_{2}$ ≈ <2,2,2> : Quaternion group, order 8, smallest dicyclic group
  • <l,m,n> : binary polyhedral group, order 4lmn/(6-n) (twice of polyhedral group)
    • 2D_2m=<2,2,m>, dicyclic group, order 4m
    • 2T=<2,3,3>, order 24, $2T=Q\rtimes \mathbb {Z} _{3}$ semidirect product
    • 2O=<2,3,4>, order 48
    • 2I=<2,3,5>, order 120
  • (l,m,n) : Triangular group
    • (p,q,2) : Polyhedral group
    • (n,2,2) : Dihedral group, D_n ≈ [n,2]⁺ ≈ [n], order 2n
    • (3,3,2), (4,3,2), (5,3,2), order 12, 24, 60.
    • ≈ [3,3]⁺, [4,3]⁺, [5,3]⁺
• [[p,q,p]⁺] = ( 2q, 4 | 2, p)

See also

• List of small groups

2D

Isomorphic type		Q-classes	Symmetry groups		Cyclic graph
Order.index (n)	Standard notation		[n]+ Zn	[n/2] Dn
1.1	Z1	1/1	[1]+
2.1	Z2	1/2,2/1	[2]+	[1]
3.1	Z3	4/1	[3]+
4.1	D4	2/2		[2]
4.2	Z4	3/1	[4]+
5	Z5		[5]+
6.1	Z6	4/3	[6]+
6.2	D6	4/2		[3]
7	Z7		[7]+
8.3	Z8		[8]+
8.4	D8	3/2		[4]
9	Z9		[9]+
10.1	Z10		[10]+
10.2	D10			[5]
11	Z11		[11]+
12.2	Z12		[12]+
12.3	D12	4/4		[6]
13	Z13		[13]+
14	Z14		[14]+
14	D14			[7]
15	Z15		[15]+
16	Z16		[16]+
16.12	D16			[8]
17	Z17		[17]+
18	Z18		[18]+
18	D18			[9]
19	Z19		[19]+
20	Z20		[20]+
20.3	D20			[10]
21	Z21		[21]+
22	Z22		[22]+
22	D22			[11]
23	Z23		[23]+
24	Z24		[24]+
24.12	D24			[12]

3D

Isomorphic type		Q-classes	Symmetry groups								Cyclic graph
Order.index (n)	Standard notation		[n]+	[n/2]	[n/4,2]	[n,2]+	Even: [n/2,2+]	[n/2+,2]	Even: [n+,2+]	[3,3], [4,3], [5,3]
			Zn	Z(n/2)v	D(n/4)h	Dnv	Z(n/2)h	D(n/2)d	Sn	T, Td, O, Th, Oh, I, Ih
1.1	Z1	1/1	[1]+
2.1	D2 ≈Z2	1/2,2/1,2/2	[2]+	[1]					[2+,2+]
3.1	Z3	5/1	[3]+
4.1	D4 ≈Z2×Z2	2/3,3/1,3/2		[2]	[2]	[2,2]+	[2+,2]	[2,2+]
4.2	Z4	4/1,4/2	[1,4]+						[4+,2+]
5	Z5		[5]+
6.1	Z6 ≈Z3×Z2	5/2,6/1,6/2	[6]+					[3+,3]	[2+,6+]
6.2	D6	5/3,5/4		[3]		[3,2]+
7	Z7		[7]+
8.1	D4×Z2 ≈Z2×Z2×Z2	3/3			[2,2]
8.2	Z4×Z2	4/3						[4+,2]
8.3	Z8		[8]+
8.4	D8	4/4,4/5,4/6		[4]		[4,2]+	[2+,4]
9	Z9		[9]+
10.1	Z10 ≈Z5×Z2		[10]+					[5+,2]	[10+,2+]
10.2	D10			[5]		[5,2]+
11	Z11		[11]+
12.1	Z6×Z2	6/3						[6+,2]
12.2	Z12		[12]+						[12+,22+]
12.3	D12	5/5,6/4,6/5,6/6		[6]		[6,2]+	[6,2+]
12.5	A4	7/1								[3,3]+
13	Z13		[13]+
14	Z14 ≈Z7×Z2		[14]+					[7+,2]	[14+,2+]
14	D14			[7]		[7,2]+
15	Z15		[15]+
16	Z8×Z2							[8+,2]
16	Z16		[16]+						[16+,2+]
16.12	D16			[8]		[8,2]+	[8,2+]
16.6	D8×Z2	4/7			[4,2]
17	Z17		[17]+
18	Z18 ≈Z9×Z2		[18]+					[9+,2]	[18+,2+]
18	D18			[9]		[9,2]+
19	Z19		[19]+
20	Z20		[20]+						[20+,2+]
20.3	D20 ≈D10×Z2			[10]	[5,2]	[10,2]+	[10,2+]
20	Z10×Z2							[10+,2]
21	Z21		[21]+
22	Z22 ≈Z11×Z2		[22]+					[11+,2]	[22+,2+]
22	D22			[11]		[11,2]+
23	Z23		[23]+
24	Z12×Z2							[12+,2]
24	Z24		[24]+						[24+,2+]
24.6	D12×Z2	6/7			[6,2]
24.10	A4×Z2	7/2								[4,3+]
24.12	D24			[12]		[12,2]+	[12,2+]
24.15	S4	7/3,7/4								[3,3], [4,3]+
48.36	S4×Z2	7/5								[4,3]
60	A5									[5,3]+
120	A5×Z2									[5,3]

4D

Isomorphic type		Q-classes	Symmetry groups	Cyclic graph	Cayley graph
Order.index (n)	Standard notation
2.1	Z2	01/02,02/01,02/02,03/01	[2]+, [2,2+,2+] [],[2+],[2+,2+], [2+,2+,2+]
3.1	Z3		[3]+
4.1	D4 ≈Z2×Z2		[2], [2,2+], [2,2]+ [2+,2,2+]
4.2	Z4		[4]+, [4+,2+], [4+,2+,2+]
5.1	Z5		[5]+, [5+,2+], [5+,2+,2+]
6.1	Z6 ≈Z3×Z2		[6]+, [6+,2+], [6+,2+,2+], [3+,2,2+]
6.2	D6 ≈S3		[3], [3,2+], [3,2+,2+]
7	Z7		[7]+
8.1	D4×Z2 ≈Z2×Z2×Z2		[2,2], [2,2,2+], [2,(2,2)+] [[2+,2,2+]]
8.2	Z4×Z2		[4+,2], [4+,2,2+], [4+,(2,2)+]
8.3	Z8	26/01	[8]+, [8+,2+], [8+,2+,2+]
8.4	D8 ≈Z2^S2		[4], [4,2+], [4,2+,2+], [(4,2)+,2+] [4,2]+, [(4,2)+,2+]
8.5	Q8≈<2,2,2>	32/01	?
9.1	Z3×Z3	32/01	[3+,2,3+]
9	Z9		[9]+
10.1	Z10 ≈Z5×Z2	27/02	[10]+, [10+,2+], [10+,2+,2+], [5+,2,2+]
10.2	D10	27/03	[5], [5,2+], [5,2+,2+]
11	Z11		[11]+
12.1	Z6×Z2		[6+,2], [6+,2,2+], [6+,(2,2)+]
12.2	Z12 ≈Z4×Z3		[12]+, [12+,2+], [12+,2+,2+], [4+,2,3+]
12.3	D6×Z2		[3,2], [3,2,2+], [3,(2,2)+]
12.4	Q12≈<2,2,3>		?
12.5	A4		[3,3]+
13	Z13		[13]+
14	Z14 ≈Z7×Z2		[14]+, [14+,2+], [14+,2+,2+] [7+,2], [7+,2,2+]
14	D14		[7], [7,2+], [7,2+,2+], [7,2]+
15	Z15 ≈Z5×Z3		[15]+, [5+,2,3+]
16.1	Z2×Z2×Z2×Z2 ≈D4×Z2×Z2 ≈D4×D4		[2,2,2]
16.2	Z4×Z2×Z2 ≈Z4×D4		[4+,2], [4+,2,2+] [4+,2,2]
16.3	Z4×Z4		[4+,2,4+]
16.6	D8×Z2		[4,2], [4,2,2+], [4,(2,2)+]
16.8	<2,2,2>2		?
16	<4,2,2>		?
16.9	(4,4|2,2)		[[2,2,2]+]
16.10	<2,2|4;2>		?
16.11	<2,2|2>		?
16.12	D16	26/02	[8], [8,2+], [8,2+,2+]
16.13	<-2,4|2>	32/03	?
16	Z8×Z2		[8+,2], [8+,2,2+]
16	Z16		[16]+, [2+,16+]
16	Q16=<2,2,4>
17	Z17		[17]+
18.1	Z6×Z3	23/01	[6+,2,3+]
18.3	D6×Z3	22/03,22/04,29/01	[3,2,3+]
18.4	D(Z3×Z3) ≈((3,3,3;2)) ?
18	Z18 ≈Z9×Z2		[18]+, [2+,18+] [9+,2], [9+,2,2+]
18	D18		[9], [9,2+], [9,2+,2+], [9,2]+
19	Z19		[19]+
20.3	D10×Z2 ≈D20	27/04	[5,2], [5,2,2+], [5,(2,2)+] [10], [10,2]+, [5,2+], [5,2,2+]
20.5		31/01	?
20	Z20 ≈Z5×Z4		[20]+, [20+,2+], [20+,2+,2+] [5+,2,4+]
20	Z10×Z2		[10+,2], [10+,2,2+]
20	Q20=<2,2,5>
21	Z21 ≈Z7×Z3		[21]+ [7+,2,3+]
22	Z22 ≈Z11×Z2		[22]+, [2+,22+] [11+,2,2+]
22	D22		[11], [11,2+], [11,2+,2+], [11,2]+
23	Z23		[23]+
24.1	Z6×Z2×Z2 ≈Z6×D4	15/08	[6+,2,2+] [6+,2,4]
24.2	Z12×Z2	20/05	[12+,2], [12+,2,2+]
24.4	D8×Z3	20/06,20/08,20/10,30/03	[4,2,3+]
24.5	Q8×Z3	33/01	?
24.6	D12×Z2 ≈D6xD4		[6,2], [6,2,2+], [6,(2,2)+] [3,2,2]
24.7	D6×Z4		[3,2,4+]
24.8	Q12×Z2	20/14	?
24.9	<-2,2,3>	33/02	?
24.10	A4×Z2	24/02,25/02,25/01	[3+,4], [3+,4,2+] [(3,3)+,2], [3,3,2]+
24.11	(4,6|2,2) ≈[3,2,2]	TE1: 20/12 TE2: 20/13 TE3: 30/04	[6,(2,2)+]
24.12	D24	20/07,20/11,28/02	[12], [12,2+], [12,2+,2+]
24.14	<2,3,3>	TE1: 32/05 TE2: 33/03	?
24.15	S4	TE1: 24/03 TE2: 24/04, 25/03, 25/04	[3,3] [3,4]+, [(3,3)+,2+]
24	Q24=<2,2,6>
24	Z12×Z2		[12+,2], [12+,2,2+]
24	Z24 ≈Z8×Z3		[24]+, [24+,2+], [24+,2+,2+] [8+,2,3+]
26	D26		[13], [13,2+], [13,2+,2+]
27	(3,3|3,3)
28	D28 ≈D14×Z2
28	<2,2,7>
30	D30
30	D10×C3		[5,2,3+]
30	D6×C5		[3,2,5+]
32.8	D8×Z2×Z2 ≈D8×D4	13/10	[4,2,2]
32.14	D8×Z4	19/03	[4,2,4+]
32.31	Z4^S2 = 4S2	32/08	?
32.33	(Z2×Z2)^S2 ≈D4^S2	18/05	?
32.34	D(Z4×Z4)	19/05
32.36		19/04	?
32.42		32/10	?
32.44		32/44	?
32.46		32/09	?
32.47		32/07	?
36.1	Z6×Z6	32/02	[6+,2,6+]
36.5	D6×Z6	22/06, 23/03,23/04, 23/06, 29/02	[3,2,6+]
36.6	Q12×Z3	30/05	?
36.9	D(C3×Z3)×Z2		?
36.13	D6×D6		[3,2,3]
36.14	(3,4,4;3)	29/04	[[3,2,3]+]
40.10		31/02	?
48.6	D8×Z6	20/18	[4,2,6+]
48.15	D12×Z2×Z2 ≈D12×D4	15/12	[6,2,2]
48.16	D12×Z4	20/15	[6,2,4+]
48.22	A4×Z2×Z2 ≈A4×D4	25/05	[3+,4,2], [(3,3)+,2]
48.24	(4,6|2,2)×Z2	20/21	[[2,3,2]]
48.25	D24×Z2	20/19	[12,2], [12,2,2+], [12,(2,2)+]
48.35		33/05	?
48.36	S4×Z2 ≈Z2^S3 = 2S3		[4,3], [4,3,2+]
48.38	D8×D6		[4,2,3]
48.47		33/04	?
48.49			?
48	<2,3,4>
60.13	A5		[5,3]+, [(5,3)+,2+] [3,3,3]+
64.154	D8×D8		[4,2,4]
64.250		32/13	?
64.252		32/15	?
64.259		32/14	?
64.261		32/12	?
72.11	D12×Z6	23/07	[6,2,6+]
72.17	D(Z6×Z6)	23/08	?
72.25	(4,6|2,2)×Z3	30/07	?
72.28	<2,3,3>×Z3	33/07	?
72.31	D12×D6		[6,2,3]
72.33		30/09	?
72.34	(3,4,4;3)×Z2	29/06	?
72.43		30/08	?
72.47	D6^S2	29/07,29/08	?
96.1		32/16	[31,1,1]+
96.2		33/08	?
96.3		33/09	?
96.4		33/10	?
96.5	S4×Z2×Z2 ≈S4×D4	25/11	[4,3,2]
96.6	D12×D8	20/22	[6,2,4]
120.1	S5 ≈(4,6|2,3)	TE1: 31/04 TE2: 31/05	[3,3,3], [[3,3,3]+]
120.2	A5×Z2		[5,3], [5,3,2]+, [(5,3)+,2] [[3,3,3]]+
120	<2,3,5>
128.1		32/17	?
144.1		33/11	?
144.2		30/10	?
144.3		30/11	?
144.4	(D6^S2)×Z2	29/09	?
144.5		30/12	?
144.6	D12×D12	23/11	[6,2,6]
192.1		32/18	?
192.2		32/19	?
192.3		32/20	?
192.4		33/12	?
240.1	S5×Z2	31/07	[[3,3,3]]
240	A5×Z2×Z2 ≈A5×D4		[5,3,2]
288.1		33/13	[3+,4,3+]
288.2	D12^S2	30/13	?
384.1	Z2^S4 ≈2S4	32/21	[4,3,3]
576.1		33/14	[3+,4,3]
576.2	3.2A4	33/15	[3,4,3]+
576		33/15	[[3+,4,3+]]
1152.1	3.2S4	33/16	[3,4,3]
1152			[[3,4,3]+]
1152			[[3,4,3]]+
2304			[[3,4,3]]
7200	2.(A5×A5)		[5,3,3]+
14400	2.(A5×A5).2		[5,3,3]