# Properties

 Label 28T35 Order $$196$$ n $$28$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $C_7:D_7.C_2$

## Group action invariants

 Degree $n$ : $28$ Transitive number $t$ : $35$ Group : $C_7:D_7.C_2$ Parity: $-1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,9)(2,10)(3,11)(4,12)(5,6)(7,8)(13,25)(14,26)(15,27)(16,28)(17,22)(18,21)(19,24)(20,23), (1,28,21,8)(2,27,22,7)(3,25,4,26)(5,24,18,11)(6,23,17,12)(9,20,14,15)(10,19,13,16) $|\Aut(F/K)|$: $14$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 7: None

Degree 14: 14T12

## Low degree siblings

14T12 x 4, 28T35 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $7, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $4$ $7$ $( 3, 8,12,16,20,24,27)( 4, 7,11,15,19,23,28)$ $7, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $4$ $7$ $( 3,12,20,27, 8,16,24)( 4,11,19,28, 7,15,23)$ $7, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $4$ $7$ $( 3,16,27,12,24, 8,20)( 4,15,28,11,23, 7,19)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $49$ $2$ $( 1, 2)( 3, 4)( 5,25)( 6,26)( 7,27)( 8,28)( 9,22)(10,21)(11,24)(12,23)(13,18) (14,17)(15,20)(16,19)$ $4, 4, 4, 4, 4, 4, 4$ $49$ $4$ $( 1, 3, 2, 4)( 5, 7,25,27)( 6, 8,26,28)( 9,11,22,24)(10,12,21,23)(13,15,18,20) (14,16,17,19)$ $4, 4, 4, 4, 4, 4, 4$ $49$ $4$ $( 1, 4, 2, 3)( 5,27,25, 7)( 6,28,26, 8)( 9,24,22,11)(10,23,21,12)(13,20,18,15) (14,19,17,16)$ $7, 7, 7, 7$ $4$ $7$ $( 1, 6,10,14,18,22,25)( 2, 5, 9,13,17,21,26)( 3, 8,12,16,20,24,27) ( 4, 7,11,15,19,23,28)$ $7, 7, 7, 7$ $4$ $7$ $( 1, 6,10,14,18,22,25)( 2, 5, 9,13,17,21,26)( 3,12,20,27, 8,16,24) ( 4,11,19,28, 7,15,23)$ $7, 7, 7, 7$ $4$ $7$ $( 1, 6,10,14,18,22,25)( 2, 5, 9,13,17,21,26)( 3,16,27,12,24, 8,20) ( 4,15,28,11,23, 7,19)$ $7, 7, 7, 7$ $4$ $7$ $( 1, 6,10,14,18,22,25)( 2, 5, 9,13,17,21,26)( 3,20, 8,24,12,27,16) ( 4,19, 7,23,11,28,15)$ $7, 7, 7, 7$ $4$ $7$ $( 1, 6,10,14,18,22,25)( 2, 5, 9,13,17,21,26)( 3,24,16, 8,27,20,12) ( 4,23,15, 7,28,19,11)$ $7, 7, 7, 7$ $4$ $7$ $( 1,10,18,25, 6,14,22)( 2, 9,17,26, 5,13,21)( 3,12,20,27, 8,16,24) ( 4,11,19,28, 7,15,23)$ $7, 7, 7, 7$ $4$ $7$ $( 1,10,18,25, 6,14,22)( 2, 9,17,26, 5,13,21)( 3,16,27,12,24, 8,20) ( 4,15,28,11,23, 7,19)$ $7, 7, 7, 7$ $4$ $7$ $( 1,10,18,25, 6,14,22)( 2, 9,17,26, 5,13,21)( 3,20, 8,24,12,27,16) ( 4,19, 7,23,11,28,15)$ $7, 7, 7, 7$ $4$ $7$ $( 1,14,25,10,22, 6,18)( 2,13,26, 9,21, 5,17)( 3,16,27,12,24, 8,20) ( 4,15,28,11,23, 7,19)$

## Group invariants

 Order: $196=2^{2} \cdot 7^{2}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [196, 8]
 Character table:  2 2 . . . 2 2 2 . . . . . . . . . 7 2 2 2 2 . . . 2 2 2 2 2 2 2 2 2 1a 7a 7b 7c 2a 4a 4b 7d 7e 7f 7g 7h 7i 7j 7k 7l 2P 1a 7b 7c 7a 1a 2a 2a 7i 7k 7e 7h 7j 7l 7g 7f 7d 3P 1a 7c 7a 7b 2a 4b 4a 7l 7f 7k 7j 7g 7d 7h 7e 7i 5P 1a 7b 7c 7a 2a 4a 4b 7i 7k 7e 7h 7j 7l 7g 7f 7d 7P 1a 1a 1a 1a 2a 4b 4a 1a 1a 1a 1a 1a 1a 1a 1a 1a 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 J -J 1 1 1 1 1 1 1 1 1 X.4 1 1 1 1 -1 -J J 1 1 1 1 1 1 1 1 1 X.5 4 A C B . . . D G I I G E H H F X.6 4 B A C . . . F I H H I D G G E X.7 4 C B A . . . E H G G H F I I D X.8 4 D E F . . . C I H H I B G G A X.9 4 E F D . . . B G I I G A H H C X.10 4 F D E . . . A H G G H C I I B X.11 4 G H I . . . I C A E F G D B H X.12 4 H I G . . . G B C F D H E A I X.13 4 I G H . . . H A B D E I F C G X.14 4 G H I . . . I F E A C G B D H X.15 4 H I G . . . G D F C B H A E I X.16 4 I G H . . . H E D B A I C F G A = -2*E(7)-E(7)^2-2*E(7)^3-2*E(7)^4-E(7)^5-2*E(7)^6 B = -E(7)-2*E(7)^2-2*E(7)^3-2*E(7)^4-2*E(7)^5-E(7)^6 C = -2*E(7)-2*E(7)^2-E(7)^3-E(7)^4-2*E(7)^5-2*E(7)^6 D = 2*E(7)^2+2*E(7)^5 E = 2*E(7)^3+2*E(7)^4 F = 2*E(7)+2*E(7)^6 G = E(7)^2+E(7)^3+E(7)^4+E(7)^5 H = E(7)+E(7)^3+E(7)^4+E(7)^6 I = E(7)+E(7)^2+E(7)^5+E(7)^6 J = -E(4) = -Sqrt(-1) = -i