# Properties

 Label 20T3 Order $$20$$ n $$20$$ Cyclic No Abelian Yes Solvable Yes Primitive No $p$-group No Group: $C_2\times C_{10}$

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## Group action invariants

 Degree $n$ : $20$ Transitive number $t$ : $3$ Group : $C_2\times C_{10}$ Parity: $1$ Primitive: No Nilpotency class: $1$ Generators: (1,13,5,17,9,2,14,6,18,10)(3,15,8,20,11,4,16,7,19,12), (1,19,18,16,14,11,9,8,5,3)(2,20,17,15,13,12,10,7,6,4) $|\Aut(F/K)|$: $20$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
5:  $C_5$
10:  $C_{10}$ x 3

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 5: $C_5$

Degree 10: $C_{10}$ x 3

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

## Group invariants

 Order: $20=2^{2} \cdot 5$ Cyclic: No Abelian: Yes Solvable: Yes GAP id: [20, 5]
 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 10a 10b 5a 10c 10d 10e 5b 10f 2b 2c 10g 5c 10h 10i 10j 5d 2P 1a 1a 5a 5a 5b 5b 5c 5c 5d 5d 1a 1a 5a 5a 5b 5b 5c 5c 3P 1a 2a 10e 10d 5c 10g 10l 10k 5a 10c 2b 2c 10j 5d 10b 10a 10f 5b 5P 1a 2a 2b 2c 1a 2a 2c 2b 1a 2a 2b 2c 2a 1a 2c 2b 2a 1a 7P 1a 2a 10i 10h 5b 10f 10b 10a 5d 10j 2b 2c 10c 5a 10l 10k 10g 5c 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 1 -1 1 -1 -1 1 1 -1 1 -1 -1 1 -1 1 -1 1 X.4 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 X.5 1 -1 A -A -B B -/B /B -/A /A -1 1 A -A -B B /B -/B X.6 1 -1 B -B -/A /A -A A -/B /B -1 1 B -B -/A /A A -A X.7 1 -1 /B -/B -A A -/A /A -B B -1 1 /B -/B -A A /A -/A X.8 1 -1 /A -/A -/B /B -B B -A A -1 1 /A -/A -/B /B B -B X.9 1 -1 -/A /A -/B /B B -B -A A 1 -1 /A -/A /B -/B B -B X.10 1 -1 -/B /B -A A /A -/A -B B 1 -1 /B -/B A -A /A -/A X.11 1 -1 -B B -/A /A A -A -/B /B 1 -1 B -B /A -/A A -A X.12 1 -1 -A A -B B /B -/B -/A /A 1 -1 A -A B -B /B -/B X.13 1 1 A A -B -B /B /B -/A -/A -1 -1 -A -A B B -/B -/B X.14 1 1 B B -/A -/A A A -/B -/B -1 -1 -B -B /A /A -A -A X.15 1 1 /B /B -A -A /A /A -B -B -1 -1 -/B -/B A A -/A -/A X.16 1 1 /A /A -/B -/B B B -A -A -1 -1 -/A -/A /B /B -B -B X.17 1 1 -/A -/A -/B -/B -B -B -A -A 1 1 -/A -/A -/B -/B -B -B X.18 1 1 -/B -/B -A -A -/A -/A -B -B 1 1 -/B -/B -A -A -/A -/A X.19 1 1 -B -B -/A -/A -A -A -/B -/B 1 1 -B -B -/A -/A -A -A X.20 1 1 -A -A -B -B -/B -/B -/A -/A 1 1 -A -A -B -B -/B -/B 2 2 2 5 1 1 10k 10l 2P 5d 5d 3P 10i 10h 5P 2b 2c 7P 10e 10d X.1 1 1 X.2 -1 1 X.3 1 -1 X.4 -1 -1 X.5 /A -/A X.6 /B -/B X.7 B -B X.8 A -A X.9 -A A X.10 -B B X.11 -/B /B X.12 -/A /A X.13 /A /A X.14 /B /B X.15 B B X.16 A A X.17 -A -A X.18 -B -B X.19 -/B -/B X.20 -/A -/A A = -E(5) B = -E(5)^2