# Properties

 Label 11T3 Order $$55$$ n $$11$$ Cyclic No Abelian No Solvable Yes Primitive Yes $p$-group No Group: $C_{11}:C_5$

# Related objects

## Group action invariants

 Degree $n$ : $11$ Transitive number $t$ : $3$ Group : $C_{11}:C_5$ CHM label : $F_{55}(11)=11:5$ Parity: $1$ Primitive: Yes Nilpotency class: $-1$ (not nilpotent) Generators: (1,3,9,5,4)(2,6,7,10,8), (1,2,3,4,5,6,7,8,9,10,11) $|\Aut(F/K)|$: $1$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
5:  $C_5$

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $5, 5, 1$ $11$ $5$ $( 2, 4,10, 6, 5)( 3, 7, 8,11, 9)$ $5, 5, 1$ $11$ $5$ $( 2, 5, 6,10, 4)( 3, 9,11, 8, 7)$ $5, 5, 1$ $11$ $5$ $( 2, 6, 4, 5,10)( 3,11, 7, 9, 8)$ $5, 5, 1$ $11$ $5$ $( 2,10, 5, 4, 6)( 3, 8, 9, 7,11)$ $11$ $5$ $11$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11)$ $11$ $5$ $11$ $( 1, 3, 5, 7, 9,11, 2, 4, 6, 8,10)$

## Group invariants

 Order: $55=5 \cdot 11$ Cyclic: No Abelian: No Solvable: Yes GAP id: [55, 1]
 Character table:  5 1 1 1 1 1 . . 11 1 . . . . 1 1 1a 5a 5b 5c 5d 11a 11b 2P 1a 5d 5c 5a 5b 11b 11a 3P 1a 5c 5d 5b 5a 11a 11b 5P 1a 1a 1a 1a 1a 11a 11b 7P 1a 5d 5c 5a 5b 11b 11a 11P 1a 5a 5b 5c 5d 1a 1a X.1 1 1 1 1 1 1 1 X.2 1 A /A /B B 1 1 X.3 1 B /B A /A 1 1 X.4 1 /B B /A A 1 1 X.5 1 /A A B /B 1 1 X.6 5 . . . . C /C X.7 5 . . . . /C C A = E(5)^4 B = E(5)^3 C = E(11)^2+E(11)^6+E(11)^7+E(11)^8+E(11)^10 = (-1-Sqrt(-11))/2 = -1-b11