# Properties

 Label 14T12 Order $$196$$ n $$14$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $C_7:D_7.C_2$

# Related objects

## Group action invariants

 Degree $n$ : $14$ Transitive number $t$ : $12$ Group : $C_7:D_7.C_2$ CHM label : $1/2[D(7)^{2}]2$ Parity: $1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (2,4,6,8,10,12,14), (1,6,13,8)(2,9,12,5)(3,4,11,10)(7,14), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8) $|\Aut(F/K)|$: $1$

## 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 7: None

## Low degree siblings

14T12 x 3, 28T35 x 4

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 1, 1$ $49$ $2$ $( 3,13)( 4,14)( 5,11)( 6,12)( 7, 9)( 8,10)$ $7, 1, 1, 1, 1, 1, 1, 1$ $4$ $7$ $( 2, 4, 6, 8,10,12,14)$ $7, 1, 1, 1, 1, 1, 1, 1$ $4$ $7$ $( 2, 6,10,14, 4, 8,12)$ $7, 1, 1, 1, 1, 1, 1, 1$ $4$ $7$ $( 2, 8,14, 6,12, 4,10)$ $4, 4, 4, 2$ $49$ $4$ $( 1, 2)( 3, 4,13,14)( 5, 6,11,12)( 7, 8, 9,10)$ $4, 4, 4, 2$ $49$ $4$ $( 1, 2)( 3,14,13, 4)( 5,12,11, 6)( 7,10, 9, 8)$ $7, 7$ $4$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2, 4, 6, 8,10,12,14)$ $7, 7$ $4$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2, 6,10,14, 4, 8,12)$ $7, 7$ $4$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2, 8,14, 6,12, 4,10)$ $7, 7$ $4$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2,10, 4,12, 6,14, 8)$ $7, 7$ $4$ $7$ $( 1, 3, 5, 7, 9,11,13)( 2,12, 8, 4,14,10, 6)$ $7, 7$ $4$ $7$ $( 1, 5, 9,13, 3, 7,11)( 2, 6,10,14, 4, 8,12)$ $7, 7$ $4$ $7$ $( 1, 5, 9,13, 3, 7,11)( 2, 8,14, 6,12, 4,10)$ $7, 7$ $4$ $7$ $( 1, 5, 9,13, 3, 7,11)( 2,10, 4,12, 6,14, 8)$ $7, 7$ $4$ $7$ $( 1, 7,13, 5,11, 3, 9)( 2, 8,14, 6,12, 4,10)$

## 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 2a 7a 7b 7c 4a 4b 7d 7e 7f 7g 7h 7i 7j 7k 7l 2P 1a 1a 7b 7c 7a 2a 2a 7i 7k 7e 7h 7j 7l 7g 7f 7d 3P 1a 2a 7c 7a 7b 4b 4a 7l 7f 7k 7j 7g 7d 7h 7e 7i 5P 1a 2a 7b 7c 7a 4a 4b 7i 7k 7e 7h 7j 7l 7g 7f 7d 7P 1a 2a 1a 1a 1a 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