# Properties

 Label 35T4 Degree $35$ Order $70$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $D_{35}$

## Group action invariants

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

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$10$:  $D_{5}$
$14$:  $D_{7}$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 5: $D_{5}$

Degree 7: $D_{7}$

## 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1$ $35$ $2$ $( 2, 5)( 3, 4)( 6,31)( 7,35)( 8,34)( 9,33)(10,32)(11,26)(12,30)(13,29)(14,28) (15,27)(16,21)(17,25)(18,24)(19,23)(20,22)$ $5, 5, 5, 5, 5, 5, 5$ $2$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)$ $5, 5, 5, 5, 5, 5, 5$ $2$ $5$ $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19) (21,23,25,22,24)(26,28,30,27,29)(31,33,35,32,34)$ $7, 7, 7, 7, 7$ $2$ $7$ $( 1, 6,11,16,21,26,31)( 2, 7,12,17,22,27,32)( 3, 8,13,18,23,28,33) ( 4, 9,14,19,24,29,34)( 5,10,15,20,25,30,35)$ $35$ $2$ $35$ $( 1, 7,13,19,25,26,32, 3, 9,15,16,22,28,34, 5, 6,12,18,24,30,31, 2, 8,14,20, 21,27,33, 4,10,11,17,23,29,35)$ $35$ $2$ $35$ $( 1, 8,15,17,24,26,33, 5, 7,14,16,23,30,32, 4, 6,13,20,22,29,31, 3,10,12,19, 21,28,35, 2, 9,11,18,25,27,34)$ $35$ $2$ $35$ $( 1, 9,12,20,23,26,34, 2,10,13,16,24,27,35, 3, 6,14,17,25,28,31, 4, 7,15,18, 21,29,32, 5, 8,11,19,22,30,33)$ $35$ $2$ $35$ $( 1,10,14,18,22,26,35, 4, 8,12,16,25,29,33, 2, 6,15,19,23,27,31, 5, 9,13,17, 21,30,34, 3, 7,11,20,24,28,32)$ $7, 7, 7, 7, 7$ $2$ $7$ $( 1,11,21,31, 6,16,26)( 2,12,22,32, 7,17,27)( 3,13,23,33, 8,18,28) ( 4,14,24,34, 9,19,29)( 5,15,25,35,10,20,30)$ $35$ $2$ $35$ $( 1,12,23,34,10,16,27, 3,14,25,31, 7,18,29, 5,11,22,33, 9,20,26, 2,13,24,35, 6,17,28, 4,15,21,32, 8,19,30)$ $35$ $2$ $35$ $( 1,13,25,32, 9,16,28, 5,12,24,31, 8,20,27, 4,11,23,35, 7,19,26, 3,15,22,34, 6,18,30, 2,14,21,33,10,17,29)$ $35$ $2$ $35$ $( 1,14,22,35, 8,16,29, 2,15,23,31, 9,17,30, 3,11,24,32,10,18,26, 4,12,25,33, 6,19,27, 5,13,21,34, 7,20,28)$ $35$ $2$ $35$ $( 1,15,24,33, 7,16,30, 4,13,22,31,10,19,28, 2,11,25,34, 8,17,26, 5,14,23,32, 6,20,29, 3,12,21,35, 9,18,27)$ $7, 7, 7, 7, 7$ $2$ $7$ $( 1,16,31,11,26, 6,21)( 2,17,32,12,27, 7,22)( 3,18,33,13,28, 8,23) ( 4,19,34,14,29, 9,24)( 5,20,35,15,30,10,25)$ $35$ $2$ $35$ $( 1,17,33,14,30, 6,22, 3,19,35,11,27, 8,24, 5,16,32,13,29,10,21, 2,18,34,15, 26, 7,23, 4,20,31,12,28, 9,25)$ $35$ $2$ $35$ $( 1,18,35,12,29, 6,23, 5,17,34,11,28,10,22, 4,16,33,15,27, 9,21, 3,20,32,14, 26, 8,25, 2,19,31,13,30, 7,24)$ $35$ $2$ $35$ $( 1,19,32,15,28, 6,24, 2,20,33,11,29, 7,25, 3,16,34,12,30, 8,21, 4,17,35,13, 26, 9,22, 5,18,31,14,27,10,23)$ $35$ $2$ $35$ $( 1,20,34,13,27, 6,25, 4,18,32,11,30, 9,23, 2,16,35,14,28, 7,21, 5,19,33,12, 26,10,24, 3,17,31,15,29, 8,22)$

## Group invariants

 Order: $70=2 \cdot 5 \cdot 7$ Cyclic: no Abelian: no Solvable: yes GAP id: [70, 3]
 Character table:  2 1 1 . . . . . . . . . . . . . . . . . 5 1 . 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 7 1 . 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1a 2a 5a 5b 7a 35a 35b 35c 35d 7b 35e 35f 35g 35h 7c 35i 35j 35k 35l 2P 1a 1a 5b 5a 7b 35f 35h 35e 35g 7c 35k 35i 35l 35j 7a 35c 35a 35d 35b 3P 1a 2a 5b 5a 7c 35k 35i 35l 35j 7a 35b 35d 35a 35c 7b 35g 35e 35h 35f 5P 1a 2a 1a 1a 7b 7b 7b 7b 7b 7c 7c 7c 7c 7c 7a 7a 7a 7a 7a 7P 1a 2a 5b 5a 1a 5b 5a 5a 5b 1a 5b 5a 5a 5b 1a 5b 5a 5a 5b 11P 1a 2a 5a 5b 7c 35l 35k 35j 35i 7a 35a 35b 35c 35d 7b 35h 35g 35f 35e 13P 1a 2a 5b 5a 7a 35b 35d 35a 35c 7b 35f 35h 35e 35g 7c 35j 35l 35i 35k 17P 1a 2a 5b 5a 7c 35j 35l 35i 35k 7a 35c 35a 35d 35b 7b 35f 35h 35e 35g 19P 1a 2a 5a 5b 7b 35e 35f 35g 35h 7c 35l 35k 35j 35i 7a 35d 35c 35b 35a 23P 1a 2a 5b 5a 7b 35g 35e 35h 35f 7c 35j 35l 35i 35k 7a 35b 35d 35a 35c 29P 1a 2a 5a 5b 7a 35d 35c 35b 35a 7b 35h 35g 35f 35e 7c 35l 35k 35j 35i 31P 1a 2a 5a 5b 7c 35i 35j 35k 35l 7a 35d 35c 35b 35a 7b 35e 35f 35g 35h X.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 X.3 2 . 2 2 B B B B B C C C C C D D D D D X.4 2 . 2 2 C C C C C D D D D D B B B B B X.5 2 . 2 2 D D D D D B B B B B C C C C C X.6 2 . A *A 2 A *A *A A 2 A *A *A A 2 A *A *A A X.7 2 . *A A 2 *A A A *A 2 *A A A *A 2 *A A A *A X.8 2 . A *A D E L K F B J O P I C G N M H X.9 2 . A *A D F K L E B I P O J C H M N G X.10 2 . A *A C G N M H D F K L E B J O P I X.11 2 . A *A C H M N G D E L K F B I P O J X.12 2 . A *A B I P O J C G N M H D E L K F X.13 2 . A *A B J O P I C H M N G D F K L E X.14 2 . *A A D K E F L B P J I O C M G H N X.15 2 . *A A D L F E K B O I J P C N H G M X.16 2 . *A A C M G H N D L F E K B P J I O X.17 2 . *A A C N H G M D K E F L B O I J P X.18 2 . *A A B O I J P C M G H N D K E F L X.19 2 . *A A B P J I O C N H G M D L F E K A = E(5)^2+E(5)^3 = (-1-Sqrt(5))/2 = -1-b5 B = E(7)+E(7)^6 C = E(7)^2+E(7)^5 D = E(7)^3+E(7)^4 E = E(35)^6+E(35)^29 F = E(35)+E(35)^34 G = E(35)^11+E(35)^24 H = E(35)^4+E(35)^31 I = E(35)^16+E(35)^19 J = E(35)^9+E(35)^26 K = E(35)^13+E(35)^22 L = E(35)^8+E(35)^27 M = E(35)^17+E(35)^18 N = E(35)^3+E(35)^32 O = E(35)^12+E(35)^23 P = E(35)^2+E(35)^33