# Properties

 Label 20T44 Order $$160$$ n $$20$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $C_2\times C_2^4:C_5$

# Related objects

## Group action invariants

 Degree $n$ : $20$ Transitive number $t$ : $44$ Group : $C_2\times C_2^4:C_5$ Parity: $1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,5,9,14,18)(2,6,10,13,17)(3,7,12,16,20)(4,8,11,15,19), (1,2)(3,4)(5,6)(7,8)(9,19)(10,20)(11,12)(13,14)(15,16)(17,18) $|\Aut(F/K)|$: $4$

## Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 4: None

Degree 5: $C_5$

Degree 10: $C_{10}$, $C_2^4 : C_5$, $C_2 \times (C_2^4 : C_5)$

## Low degree siblings

10T14 x 3, 20T40 x 12, 20T41 x 6, 20T44 x 2, 20T46 x 3, 32T2133, 40T121 x 6, 40T122 x 6, 40T123 x 12, 40T141, 40T142 x 3

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy Classes

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

## Group invariants

 Order: $160=2^{5} \cdot 5$ Cyclic: No Abelian: No Solvable: Yes GAP id: [160, 235]
 Character table:  2 5 5 5 5 5 5 5 1 1 1 1 1 1 1 1 5 5 1 . . . . . . 1 1 1 1 1 1 1 1 1 1a 2a 2b 2c 2d 2e 2f 5a 10a 5b 10b 5c 10c 5d 10d 2g 2P 1a 1a 1a 1a 1a 1a 1a 5b 5b 5d 5d 5a 5a 5c 5c 1a 3P 1a 2a 2b 2c 2d 2e 2f 5c 10c 5a 10a 5d 10d 5b 10b 2g 5P 1a 2a 2b 2c 2d 2e 2f 1a 2g 1a 2g 1a 2g 1a 2g 2g 7P 1a 2a 2b 2c 2d 2e 2f 5b 10b 5d 10d 5a 10a 5c 10c 2g X.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 X.3 1 1 1 1 -1 -1 -1 A -A B -B /B -/B /A -/A -1 X.4 1 1 1 1 -1 -1 -1 B -B /A -/A A -A /B -/B -1 X.5 1 1 1 1 -1 -1 -1 /B -/B A -A /A -/A B -B -1 X.6 1 1 1 1 -1 -1 -1 /A -/A /B -/B B -B A -A -1 X.7 1 1 1 1 1 1 1 A A B B /B /B /A /A 1 X.8 1 1 1 1 1 1 1 B B /A /A A A /B /B 1 X.9 1 1 1 1 1 1 1 /B /B A A /A /A B B 1 X.10 1 1 1 1 1 1 1 /A /A /B /B B B A A 1 X.11 5 -3 1 1 -1 3 -1 . . . . . . . . -5 X.12 5 -3 1 1 1 -3 1 . . . . . . . . 5 X.13 5 1 -3 1 -1 -1 3 . . . . . . . . -5 X.14 5 1 -3 1 1 1 -3 . . . . . . . . 5 X.15 5 1 1 -3 -3 1 1 . . . . . . . . 5 X.16 5 1 1 -3 3 -1 -1 . . . . . . . . -5 A = E(5)^4 B = E(5)^3