# Properties

 Label 20T1 Order $$20$$ n $$20$$ Cyclic Yes Abelian Yes Solvable Yes Primitive No $p$-group No Group: $C_{20}$

# Related objects

## Group action invariants

 Degree $n$ : $20$ Transitive number $t$ : $1$ Group : $C_{20}$ Parity: $-1$ Primitive: No Nilpotency class: $1$ Generators: (1,12,2,11)(3,14,4,13)(5,15,6,16)(7,17,8,18)(9,20,10,19), (1,4,6,8,9,12,13,16,18,20,2,3,5,7,10,11,14,15,17,19) $|\Aut(F/K)|$: $20$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
5:  $C_5$
10:  $C_{10}$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 5: $C_5$

Degree 10: $C_{10}$

## Low degree siblings

There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy Classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$ $20$ $1$ $20$ $( 1, 3, 6, 7, 9,11,13,15,18,19, 2, 4, 5, 8,10,12,14,16,17,20)$ $20$ $1$ $20$ $( 1, 4, 6, 8, 9,12,13,16,18,20, 2, 3, 5, 7,10,11,14,15,17,19)$ $5, 5, 5, 5$ $1$ $5$ $( 1, 5, 9,14,18)( 2, 6,10,13,17)( 3, 8,11,16,19)( 4, 7,12,15,20)$ $10, 10$ $1$ $10$ $( 1, 6, 9,13,18, 2, 5,10,14,17)( 3, 7,11,15,19, 4, 8,12,16,20)$ $20$ $1$ $20$ $( 1, 7,13,19, 5,12,17, 3, 9,15, 2, 8,14,20, 6,11,18, 4,10,16)$ $20$ $1$ $20$ $( 1, 8,13,20, 5,11,17, 4, 9,16, 2, 7,14,19, 6,12,18, 3,10,15)$ $5, 5, 5, 5$ $1$ $5$ $( 1, 9,18, 5,14)( 2,10,17, 6,13)( 3,11,19, 8,16)( 4,12,20, 7,15)$ $10, 10$ $1$ $10$ $( 1,10,18, 6,14, 2, 9,17, 5,13)( 3,12,19, 7,16, 4,11,20, 8,15)$ $4, 4, 4, 4, 4$ $1$ $4$ $( 1,11, 2,12)( 3,13, 4,14)( 5,16, 6,15)( 7,18, 8,17)( 9,19,10,20)$ $4, 4, 4, 4, 4$ $1$ $4$ $( 1,12, 2,11)( 3,14, 4,13)( 5,15, 6,16)( 7,17, 8,18)( 9,20,10,19)$ $10, 10$ $1$ $10$ $( 1,13, 5,17, 9, 2,14, 6,18,10)( 3,15, 8,20,11, 4,16, 7,19,12)$ $5, 5, 5, 5$ $1$ $5$ $( 1,14, 5,18, 9)( 2,13, 6,17,10)( 3,16, 8,19,11)( 4,15, 7,20,12)$ $20$ $1$ $20$ $( 1,15,10, 3,18,12, 6,19,14, 7, 2,16, 9, 4,17,11, 5,20,13, 8)$ $20$ $1$ $20$ $( 1,16,10, 4,18,11, 6,20,14, 8, 2,15, 9, 3,17,12, 5,19,13, 7)$ $10, 10$ $1$ $10$ $( 1,17,14,10, 5, 2,18,13, 9, 6)( 3,20,16,12, 8, 4,19,15,11, 7)$ $5, 5, 5, 5$ $1$ $5$ $( 1,18,14, 9, 5)( 2,17,13,10, 6)( 3,19,16,11, 8)( 4,20,15,12, 7)$ $20$ $1$ $20$ $( 1,19,17,15,14,11,10, 7, 5, 3, 2,20,18,16,13,12, 9, 8, 6, 4)$ $20$ $1$ $20$ $( 1,20,17,16,14,12,10, 8, 5, 4, 2,19,18,15,13,11, 9, 7, 6, 3)$

## Group invariants

 Order: $20=2^{2} \cdot 5$ Cyclic: Yes Abelian: Yes Solvable: Yes GAP id: [20, 2]
 Character table:  2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1a 2a 20a 20b 5a 10a 20c 20d 5b 10b 4a 4b 10c 5c 20e 20f 10d 5d 2P 1a 1a 10a 10a 5b 5b 10c 10c 5d 5d 2a 2a 5a 5a 10b 10b 5c 5c 3P 1a 2a 20c 20d 5c 10c 20g 20h 5a 10a 4b 4a 10d 5d 20a 20b 10b 5b 5P 1a 2a 4a 4b 1a 2a 4b 4a 1a 2a 4a 4b 2a 1a 4b 4a 2a 1a 7P 1a 2a 20e 20f 5b 10b 20a 20b 5d 10d 4b 4a 10a 5a 20g 20h 10c 5c 11P 1a 2a 20b 20a 5a 10a 20d 20c 5b 10b 4b 4a 10c 5c 20f 20e 10d 5d 13P 1a 2a 20d 20c 5c 10c 20h 20g 5a 10a 4a 4b 10d 5d 20b 20a 10b 5b 17P 1a 2a 20f 20e 5b 10b 20b 20a 5d 10d 4a 4b 10a 5a 20h 20g 10c 5c 19P 1a 2a 20h 20g 5d 10d 20f 20e 5c 10c 4b 4a 10b 5b 20d 20c 10a 5a X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 X.3 1 -1 A -A 1 -1 -A A 1 -1 A -A -1 1 -A A -1 1 X.4 1 -1 -A A 1 -1 A -A 1 -1 -A A -1 1 A -A -1 1 X.5 1 -1 B -B -/E /E -C C -D D A -A /D -/D /C -/C E -E X.6 1 -1 -/B /B -E E /C -/C -/D /D A -A D -D -C C /E -/E X.7 1 -1 C -C -/D /D /B -/B -/E /E A -A E -E -B B D -D X.8 1 -1 -/C /C -D D -B B -E E A -A /E -/E /B -/B /D -/D X.9 1 -1 /C -/C -D D B -B -E E -A A /E -/E -/B /B /D -/D X.10 1 -1 -C C -/D /D -/B /B -/E /E -A A E -E B -B D -D X.11 1 -1 /B -/B -E E -/C /C -/D /D -A A D -D C -C /E -/E X.12 1 -1 -B B -/E /E C -C -D D -A A /D -/D -/C /C E -E X.13 1 1 D D -E -E /E /E -/D -/D -1 -1 -D -D E E -/E -/E X.14 1 1 E E -/D -/D D D -/E -/E -1 -1 -E -E /D /D -D -D X.15 1 1 /E /E -D -D /D /D -E -E -1 -1 -/E -/E D D -/D -/D X.16 1 1 /D /D -/E -/E E E -D -D -1 -1 -/D -/D /E /E -E -E X.17 1 1 -/D -/D -/E -/E -E -E -D -D 1 1 -/D -/D -/E -/E -E -E X.18 1 1 -/E -/E -D -D -/D -/D -E -E 1 1 -/E -/E -D -D -/D -/D X.19 1 1 -E -E -/D -/D -D -D -/E -/E 1 1 -E -E -/D -/D -D -D X.20 1 1 -D -D -E -E -/E -/E -/D -/D 1 1 -D -D -E -E -/E -/E 2 2 2 5 1 1 20g 20h 2P 10d 10d 3P 20e 20f 5P 4a 4b 7P 20c 20d 11P 20h 20g 13P 20f 20e 17P 20d 20c 19P 20b 20a X.1 1 1 X.2 -1 -1 X.3 A -A X.4 -A A X.5 -/B /B X.6 B -B X.7 -/C /C X.8 C -C X.9 -C C X.10 /C -/C X.11 -B B X.12 /B -/B X.13 /D /D X.14 /E /E X.15 E E X.16 D D X.17 -D -D X.18 -E -E X.19 -/E -/E X.20 -/D -/D A = -E(4) = -Sqrt(-1) = -i B = -E(20) C = -E(20)^13 D = -E(5) E = -E(5)^2