# Properties

 Label 40T36 Degree $40$ Order $80$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $D_{10}.C_2^2$

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## Group action invariants

 Degree $n$: $40$ Transitive number $t$: $36$ Group: $D_{10}.C_2^2$ Parity: $1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $4$ Generators: (1,14,2,13)(3,15,4,16)(5,12,6,11)(7,9,8,10)(17,40,18,39)(19,38,20,37)(21,35,22,36)(23,34,24,33)(25,30,26,29)(27,32,28,31), (1,9,19,27,36,3,11,18,26,33,2,10,20,28,35,4,12,17,25,34)(5,16,21,31,38,7,14,23,30,39,6,15,22,32,37,8,13,24,29,40), (1,6,11,13,20,21,25,30,36,37,2,5,12,14,19,22,26,29,35,38)(3,7,10,15,17,24,27,31,33,39,4,8,9,16,18,23,28,32,34,40)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$8$:  $C_2^3$
$10$:  $D_{5}$
$16$:  $Q_8:C_2$
$20$:  $D_{10}$ x 3

Resolvents shown for degrees $\leq 10$

## Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 5: $D_{5}$

Degree 8: $Q_8:C_2$

Degree 10: $D_{10}$ x 3

Degree 20: 20T8

## Low degree siblings

There are no siblings with degree $\leq 10$
A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1$ $10$ $2$ $( 3, 4)( 5,38)( 6,37)( 7,40)( 8,39)( 9,34)(10,33)(11,35)(12,36)(13,30)(14,29) (15,32)(16,31)(17,27)(18,28)(19,25)(20,26)(23,24)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $10$ $2$ $( 1, 3)( 2, 4)( 5,39)( 6,40)( 7,38)( 8,37)( 9,36)(10,35)(11,34)(12,33)(13,32) (14,31)(15,29)(16,30)(17,26)(18,25)(19,27)(20,28)(21,23)(22,24)$ $4, 4, 4, 4, 4, 4, 4, 4, 4, 4$ $2$ $4$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,11,10,12)(13,16,14,15)(17,19,18,20)(21,23,22,24) (25,27,26,28)(29,31,30,32)(33,35,34,36)(37,40,38,39)$ $4, 4, 4, 4, 4, 4, 4, 4, 4, 4$ $5$ $4$ $( 1, 5, 2, 6)( 3, 7, 4, 8)( 9,39,10,40)(11,37,12,38)(13,35,14,36)(15,33,16,34) (17,32,18,31)(19,29,20,30)(21,26,22,25)(23,28,24,27)$ $20, 20$ $4$ $20$ $( 1, 5,11,14,20,22,25,29,36,38, 2, 6,12,13,19,21,26,30,35,37)( 3, 8,10,16,17, 23,27,32,33,40, 4, 7, 9,15,18,24,28,31,34,39)$ $4, 4, 4, 4, 4, 4, 4, 4, 4, 4$ $5$ $4$ $( 1, 6, 2, 5)( 3, 8, 4, 7)( 9,40,10,39)(11,38,12,37)(13,36,14,35)(15,34,16,33) (17,31,18,32)(19,30,20,29)(21,25,22,26)(23,27,24,28)$ $20, 20$ $4$ $20$ $( 1, 7,11,15,20,24,25,31,36,39, 2, 8,12,16,19,23,26,32,35,40)( 3, 5,10,14,17, 22,27,29,33,38, 4, 6, 9,13,18,21,28,30,34,37)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $10$ $2$ $( 1, 7)( 2, 8)( 3, 6)( 4, 5)( 9,37)(10,38)(11,40)(12,39)(13,34)(14,33)(15,35) (16,36)(17,29)(18,30)(19,31)(20,32)(21,28)(22,27)(23,25)(24,26)$ $20, 20$ $4$ $20$ $( 1, 9,19,27,36, 3,11,18,26,33, 2,10,20,28,35, 4,12,17,25,34)( 5,16,21,31,38, 7,14,23,30,39, 6,15,22,32,37, 8,13,24,29,40)$ $10, 10, 10, 10$ $2$ $10$ $( 1,11,20,25,36, 2,12,19,26,35)( 3,10,17,27,33, 4, 9,18,28,34)( 5,14,22,29,38, 6,13,21,30,37)( 7,15,24,31,39, 8,16,23,32,40)$ $5, 5, 5, 5, 5, 5, 5, 5$ $2$ $5$ $( 1,12,20,26,36)( 2,11,19,25,35)( 3, 9,17,28,33)( 4,10,18,27,34) ( 5,13,22,30,38)( 6,14,21,29,37)( 7,16,24,32,39)( 8,15,23,31,40)$ $20, 20$ $4$ $20$ $( 1,13,25,37,12,22,35, 6,20,30, 2,14,26,38,11,21,36, 5,19,29)( 3,15,27,39, 9, 23,34, 7,17,31, 4,16,28,40,10,24,33, 8,18,32)$ $20, 20$ $4$ $20$ $( 1,15,25,39,12,23,35, 7,20,31, 2,16,26,40,11,24,36, 8,19,32)( 3,14,27,38, 9, 21,34, 5,17,29, 4,13,28,37,10,22,33, 6,18,30)$ $20, 20$ $4$ $20$ $( 1,17,35,10,26, 3,19,34,12,28, 2,18,36, 9,25, 4,20,33,11,27)( 5,24,37,15,30, 7,21,40,13,32, 6,23,38,16,29, 8,22,39,14,31)$ $10, 10, 10, 10$ $2$ $10$ $( 1,19,36,11,26, 2,20,35,12,25)( 3,18,33,10,28, 4,17,34, 9,27)( 5,21,38,14,30, 6,22,37,13,29)( 7,23,39,15,32, 8,24,40,16,31)$ $5, 5, 5, 5, 5, 5, 5, 5$ $2$ $5$ $( 1,20,36,12,26)( 2,19,35,11,25)( 3,17,33, 9,28)( 4,18,34,10,27) ( 5,22,38,13,30)( 6,21,37,14,29)( 7,24,39,16,32)( 8,23,40,15,31)$ $4, 4, 4, 4, 4, 4, 4, 4, 4, 4$ $2$ $4$ $( 1,21, 2,22)( 3,24, 4,23)( 5,26, 6,25)( 7,27, 8,28)( 9,32,10,31)(11,30,12,29) (13,36,14,35)(15,33,16,34)(17,39,18,40)(19,38,20,37)$ $4, 4, 4, 4, 4, 4, 4, 4, 4, 4$ $2$ $4$ $( 1,23, 2,24)( 3,21, 4,22)( 5,28, 6,27)( 7,26, 8,25)( 9,29,10,30)(11,32,12,31) (13,33,14,34)(15,35,16,36)(17,37,18,38)(19,39,20,40)$

## Group invariants

 Order: $80=2^{4} \cdot 5$ Cyclic: no Abelian: no Solvable: yes GAP id: [80, 42]
 Character table:  2 4 3 4 3 3 4 2 4 2 3 2 3 3 2 2 2 3 3 3 3 5 1 . 1 . 1 . 1 . 1 . 1 1 1 1 1 1 1 1 1 1 1a 2a 2b 2c 4a 4b 20a 4c 20b 2d 20c 10a 5a 20d 20e 20f 10b 5b 4d 4e 2P 1a 1a 1a 1a 2b 2b 10a 2b 10a 1a 10b 5b 5b 10b 10b 10a 5a 5a 2b 2b 3P 1a 2a 2b 2c 4a 4c 20d 4b 20e 2d 20f 10b 5b 20a 20b 20c 10a 5a 4d 4e 5P 1a 2a 2b 2c 4a 4b 4d 4c 4e 2d 4a 2b 1a 4d 4e 4a 2b 1a 4d 4e 7P 1a 2a 2b 2c 4a 4c 20d 4b 20e 2d 20f 10b 5b 20a 20b 20c 10a 5a 4d 4e 11P 1a 2a 2b 2c 4a 4c 20a 4b 20b 2d 20c 10a 5a 20d 20e 20f 10b 5b 4d 4e 13P 1a 2a 2b 2c 4a 4b 20d 4c 20e 2d 20f 10b 5b 20a 20b 20c 10a 5a 4d 4e 17P 1a 2a 2b 2c 4a 4b 20d 4c 20e 2d 20f 10b 5b 20a 20b 20c 10a 5a 4d 4e 19P 1a 2a 2b 2c 4a 4c 20a 4b 20b 2d 20c 10a 5a 20d 20e 20f 10b 5b 4d 4e X.1 1 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 1 1 X.3 1 -1 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 1 -1 X.5 1 -1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 1 -1 1 1 -1 1 X.6 1 1 1 -1 -1 -1 -1 -1 1 1 -1 1 1 -1 1 -1 1 1 -1 1 X.7 1 1 1 -1 -1 1 1 1 -1 -1 -1 1 1 1 -1 -1 1 1 1 -1 X.8 1 1 1 1 1 -1 -1 -1 -1 -1 1 1 1 -1 -1 1 1 1 -1 -1 X.9 2 . -2 . . A . -A . . . -2 2 . . . -2 2 . . X.10 2 . -2 . . -A . A . . . -2 2 . . . -2 2 . . X.11 2 . 2 . -2 . B . -B . *B -*B -*B *B -*B B -B -B -2 2 X.12 2 . 2 . -2 . *B . -*B . B -B -B B -B *B -*B -*B -2 2 X.13 2 . 2 . -2 . -*B . *B . B -B -B -B B *B -*B -*B 2 -2 X.14 2 . 2 . -2 . -B . B . *B -*B -*B -*B *B B -B -B 2 -2 X.15 2 . 2 . 2 . B . B . -*B -*B -*B *B *B -B -B -B -2 -2 X.16 2 . 2 . 2 . *B . *B . -B -B -B B B -*B -*B -*B -2 -2 X.17 2 . 2 . 2 . -*B . -*B . -B -B -B -B -B -*B -*B -*B 2 2 X.18 2 . 2 . 2 . -B . -B . -*B -*B -*B -*B -*B -B -B -B 2 2 X.19 4 . -4 . . . . . . . . C -C . . . *C -*C . . X.20 4 . -4 . . . . . . . . *C -*C . . . C -C . . A = -2*E(4) = -2*Sqrt(-1) = -2i B = -E(5)-E(5)^4 = (1-Sqrt(5))/2 = -b5 C = -2*E(5)^2-2*E(5)^3 = 1+Sqrt(5) = 1+r5