Properties

Label 23T3
Order \(253\)
n \(23\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No
Group: $C_{23}:C_{11}$

Related objects

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Group action invariants

Degree $n$ :  $23$
Transitive number $t$ :  $3$
Group :  $C_{23}:C_{11}$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (2,3,5,9,17,10,19,14,4,7,13)(6,11,21,18,12,23,22,20,16,8,15), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
11:  $C_{11}$

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 TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 11, 11, 1 $ $23$ $11$ $( 2, 3, 5, 9,17,10,19,14, 4, 7,13)( 6,11,21,18,12,23,22,20,16, 8,15)$
$ 11, 11, 1 $ $23$ $11$ $( 2, 4,10, 5,13,14,17, 3, 7,19, 9)( 6,16,23,21,15,20,12,11, 8,22,18)$
$ 11, 11, 1 $ $23$ $11$ $( 2, 5,17,19, 4,13, 3, 9,10,14, 7)( 6,21,12,22,16,15,11,18,23,20, 8)$
$ 11, 11, 1 $ $23$ $11$ $( 2, 7,14,10, 9, 3,13, 4,19,17, 5)( 6, 8,20,23,18,11,15,16,22,12,21)$
$ 11, 11, 1 $ $23$ $11$ $( 2, 9,19, 7, 3,17,14,13, 5,10, 4)( 6,18,22, 8,11,12,20,15,21,23,16)$
$ 11, 11, 1 $ $23$ $11$ $( 2,10,13,17, 7, 9, 4, 5,14, 3,19)( 6,23,15,12, 8,18,16,21,20,11,22)$
$ 11, 11, 1 $ $23$ $11$ $( 2,13, 7, 4,14,19,10,17, 9, 5, 3)( 6,15, 8,16,20,22,23,12,18,21,11)$
$ 11, 11, 1 $ $23$ $11$ $( 2,14, 9,13,19, 5, 7,10, 3, 4,17)( 6,20,18,15,22,21, 8,23,11,16,12)$
$ 11, 11, 1 $ $23$ $11$ $( 2,17, 4, 3,10, 7, 5,19,13, 9,14)( 6,12,16,11,23, 8,21,22,15,18,20)$
$ 11, 11, 1 $ $23$ $11$ $( 2,19, 3,14, 5, 4, 9, 7,17,13,10)( 6,22,11,20,21,16,18, 8,12,15,23)$
$ 23 $ $11$ $23$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19,20,21,22,23)$
$ 23 $ $11$ $23$ $( 1, 6,11,16,21, 3, 8,13,18,23, 5,10,15,20, 2, 7,12,17,22, 4, 9,14,19)$

Group invariants

Order:  $253=11 \cdot 23$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [253, 1]
Character table:   
     11  1   1   1   1   1   1   1   1   1   1   1   .   .
     23  1   .   .   .   .   .   .   .   .   .   .   1   1

        1a 11a 11b 11c 11d 11e 11f 11g 11h 11i 11j 23a 23b
     2P 1a 11c 11f 11i 11h 11j 11g 11d 11e 11b 11a 23a 23b
     3P 1a 11e 11c 11j 11f 11d 11i 11b 11g 11a 11h 23a 23b
     5P 1a 11f 11h 11g 11a 11i 11e 11j 11c 11d 11b 23b 23a
     7P 1a 11h 11a 11e 11b 11g 11c 11i 11f 11j 11d 23b 23a
    11P 1a  1a  1a  1a  1a  1a  1a  1a  1a  1a  1a 23b 23a
    13P 1a 11c 11f 11i 11h 11j 11g 11d 11e 11b 11a 23a 23b
    17P 1a 11j 11i 11a 11g 11h 11b 11f 11d 11c 11e 23b 23a
    19P 1a 11b 11d 11f 11j 11c 11h 11e 11a 11g 11i 23b 23a
    23P 1a 11a 11b 11c 11d 11e 11f 11g 11h 11i 11j  1a  1a

X.1      1   1   1   1   1   1   1   1   1   1   1   1   1
X.2      1   A  /C   B  /B   C   E  /A  /D   D  /E   1   1
X.3      1   B   E   D  /D  /E  /A  /B   C  /C   A   1   1
X.4      1   C   B  /E   E  /B   D  /C  /A   A  /D   1   1
X.5      1   D  /A  /C   C   A  /B  /D  /E   E   B   1   1
X.6      1   E  /D  /A   A   D   C  /E   B  /B  /C   1   1
X.7      1  /E   D   A  /A  /D  /C   E  /B   B   C   1   1
X.8      1  /D   A   C  /C  /A   B   D   E  /E  /B   1   1
X.9      1  /C  /B   E  /E   B  /D   C   A  /A   D   1   1
X.10     1  /B  /E  /D   D   E   A   B  /C   C  /A   1   1
X.11     1  /A   C  /B   B  /C  /E   A   D  /D   E   1   1
X.12    11   .   .   .   .   .   .   .   .   .   .   F  /F
X.13    11   .   .   .   .   .   .   .   .   .   .  /F   F

A = E(11)^10
B = E(11)^9
C = E(11)^8
D = E(11)^7
E = E(11)^6
F = E(23)^5+E(23)^7+E(23)^10+E(23)^11+E(23)^14+E(23)^15+E(23)^17+E(23)^19+E(23)^20+E(23)^21+E(23)^22
  = (-1-Sqrt(-23))/2 = -1-b23