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Properties

Label 23T2
Degree $23$
Order $46$
Cyclic no
Abelian no
Solvable yes
Primitive yes
$p$-group no
Group: $D_{23}$

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Show commands: Magma

magma: G := TransitiveGroup(23, 2);
 

Group action invariants

Degree $n$:  $23$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $2$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_{23}$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  yes
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23), (2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)
magma: Generators(G);
 

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

46T2

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $46=2 \cdot 23$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  46.1
magma: IdentifyGroup(G);
 
Character table:    
      2  1  1   .   .   .   .   .   .   .   .   .   .   .
     23  1  .   1   1   1   1   1   1   1   1   1   1   1

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

X.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
X.3      2  .   A   D   H   E   J   I   C   F   B   G   K
X.4      2  .   B   J   E   G   A   F   I   H   K   D   C
X.5      2  .   C   B   D   J   K   E   H   G   I   A   F
X.6      2  .   D   E   I   F   G   K   B   C   J   H   A
X.7      2  .   E   F   K   C   H   A   J   B   G   I   D
X.8      2  .   F   C   A   B   I   D   G   J   H   K   E
X.9      2  .   G   H   C   I   E   B   A   K   D   F   J
X.10     2  .   H   I   B   K   F   J   D   A   E   C   G
X.11     2  .   I   K   J   A   C   G   E   D   F   B   H
X.12     2  .   J   G   F   H   D   C   K   I   A   E   B
X.13     2  .   K   A   G   D   B   H   F   E   C   J   I

A = E(23)^3+E(23)^20
B = E(23)^4+E(23)^19
C = E(23)^2+E(23)^21
D = E(23)^6+E(23)^17
E = E(23)^11+E(23)^12
F = E(23)+E(23)^22
G = E(23)^7+E(23)^16
H = E(23)^9+E(23)^14
I = E(23)^5+E(23)^18
J = E(23)^8+E(23)^15
K = E(23)^10+E(23)^13

magma: CharacterTable(G);