# Properties

 Label 19T4 Order $$114$$ n $$19$$ Cyclic No Abelian No Solvable Yes Primitive Yes $p$-group No Group: $C_{19}:C_{6}$

# Related objects

## Group action invariants

 Degree $n$ : $19$ Transitive number $t$ : $4$ Group : $C_{19}:C_{6}$ Parity: $-1$ Primitive: Yes Nilpotency class: $-1$ (not nilpotent) Generators: (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19), (2,9,8,19,12,13)(3,17,15,18,4,6)(5,14,10,16,7,11) $|\Aut(F/K)|$: $1$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $C_6$

Resolvents shown for degrees $\leq 47$

## Subfields

Prime degree - none

## 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $3, 3, 3, 3, 3, 3, 1$ $19$ $3$ $( 2, 8,12)( 3,15, 4)( 5,10, 7)( 6,17,18)( 9,19,13)(11,14,16)$ $6, 6, 6, 1$ $19$ $6$ $( 2, 9, 8,19,12,13)( 3,17,15,18, 4, 6)( 5,14,10,16, 7,11)$ $3, 3, 3, 3, 3, 3, 1$ $19$ $3$ $( 2,12, 8)( 3, 4,15)( 5, 7,10)( 6,18,17)( 9,13,19)(11,16,14)$ $6, 6, 6, 1$ $19$ $6$ $( 2,13,12,19, 8, 9)( 3, 6, 4,18,15,17)( 5,11, 7,16,10,14)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 1$ $19$ $2$ $( 2,19)( 3,18)( 4,17)( 5,16)( 6,15)( 7,14)( 8,13)( 9,12)(10,11)$ $19$ $6$ $19$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19)$ $19$ $6$ $19$ $( 1, 3, 5, 7, 9,11,13,15,17,19, 2, 4, 6, 8,10,12,14,16,18)$ $19$ $6$ $19$ $( 1, 5, 9,13,17, 2, 6,10,14,18, 3, 7,11,15,19, 4, 8,12,16)$

## Group invariants

 Order: $114=2 \cdot 3 \cdot 19$ Cyclic: No Abelian: No Solvable: Yes GAP id: [114, 1]
 Character table:  2 1 1 1 1 1 1 . . . 3 1 1 1 1 1 1 . . . 19 1 . . . . . 1 1 1 1a 3a 6a 3b 6b 2a 19a 19b 19c 2P 1a 3b 3a 3a 3b 1a 19b 19c 19a 3P 1a 1a 2a 1a 2a 2a 19b 19c 19a 5P 1a 3b 6b 3a 6a 2a 19b 19c 19a 7P 1a 3a 6a 3b 6b 2a 19a 19b 19c 11P 1a 3b 6b 3a 6a 2a 19a 19b 19c 13P 1a 3a 6a 3b 6b 2a 19c 19a 19b 17P 1a 3b 6b 3a 6a 2a 19b 19c 19a 19P 1a 3a 6a 3b 6b 2a 1a 1a 1a X.1 1 1 1 1 1 1 1 1 1 X.2 1 1 -1 1 -1 -1 1 1 1 X.3 1 A -/A /A -A -1 1 1 1 X.4 1 /A -A A -/A -1 1 1 1 X.5 1 A /A /A A 1 1 1 1 X.6 1 /A A A /A 1 1 1 1 X.7 6 . . . . . B C D X.8 6 . . . . . C D B X.9 6 . . . . . D B C A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = E(19)^2+E(19)^3+E(19)^5+E(19)^14+E(19)^16+E(19)^17 C = E(19)^4+E(19)^6+E(19)^9+E(19)^10+E(19)^13+E(19)^15 D = E(19)+E(19)^7+E(19)^8+E(19)^11+E(19)^12+E(19)^18