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Properties

Label 24T4
Degree $24$
Order $24$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3\times Q_8$

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Show commands: Magma

magma: G := TransitiveGroup(24, 4);
 

Group action invariants

Degree $n$:  $24$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $4$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3\times Q_8$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $24$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,4,18,19,9,12,2,3,17,20,10,11)(5,8,21,24,14,16,6,7,22,23,13,15), (1,6,10,14,17,21,2,5,9,13,18,22)(3,7,12,16,19,24,4,8,11,15,20,23)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_2^2$
$6$:  $C_6$ x 3
$8$:  $Q_8$
$12$:  $C_6\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: $C_3$

Degree 4: $C_2^2$

Degree 6: $C_6$ x 3

Degree 8: $Q_8$

Degree 12: $C_6\times C_2$

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

LabelCycle 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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)$
$ 12, 12 $ $2$ $12$ $( 1, 3,18,20, 9,11, 2, 4,17,19,10,12)( 5, 7,21,23,14,15, 6, 8,22,24,13,16)$
$ 12, 12 $ $2$ $12$ $( 1, 5,10,13,17,22, 2, 6, 9,14,18,21)( 3, 8,12,15,19,23, 4, 7,11,16,20,24)$
$ 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1, 7, 2, 8)( 3,22, 4,21)( 5,12, 6,11)( 9,15,10,16)(13,19,14,20)(17,24,18,23)$
$ 3, 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 9,17)( 2,10,18)( 3,11,19)( 4,12,20)( 5,14,22)( 6,13,21)( 7,15,24) ( 8,16,23)$
$ 6, 6, 6, 6 $ $1$ $6$ $( 1,10,17, 2, 9,18)( 3,12,19, 4,11,20)( 5,13,22, 6,14,21)( 7,16,24, 8,15,23)$
$ 12, 12 $ $2$ $12$ $( 1,11,10,20,17, 3, 2,12, 9,19,18, 4)( 5,15,13,23,22, 7, 6,16,14,24,21, 8)$
$ 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,13, 2,14)( 3,15, 4,16)( 5,17, 6,18)( 7,20, 8,19)( 9,21,10,22)(11,24,12,23)$
$ 12, 12 $ $2$ $12$ $( 1,15,18, 8, 9,24, 2,16,17, 7,10,23)( 3, 5,20,21,11,14, 4, 6,19,22,12,13)$
$ 3, 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1,17, 9)( 2,18,10)( 3,19,11)( 4,20,12)( 5,22,14)( 6,21,13)( 7,24,15) ( 8,23,16)$
$ 6, 6, 6, 6 $ $1$ $6$ $( 1,18, 9, 2,17,10)( 3,20,11, 4,19,12)( 5,21,14, 6,22,13)( 7,23,15, 8,24,16)$
$ 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,19, 2,20)( 3,10, 4, 9)( 5,24, 6,23)( 7,13, 8,14)(11,18,12,17)(15,21,16,22)$
$ 12, 12 $ $2$ $12$ $( 1,21,18,14, 9, 6, 2,22,17,13,10, 5)( 3,24,20,16,11, 7, 4,23,19,15,12, 8)$
$ 12, 12 $ $2$ $12$ $( 1,23,10, 7,17,16, 2,24, 9, 8,18,15)( 3,13,12,22,19, 6, 4,14,11,21,20, 5)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $24=2^{3} \cdot 3$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $2$
Label:  24.11
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A1 3A-1 4A 4B 4C 6A1 6A-1 12A1 12A-1 12B1 12B-1 12C1 12C-1
Size 1 1 1 1 2 2 2 1 1 2 2 2 2 2 2
2 P 1A 1A 3A-1 3A1 2A 2A 2A 3A1 3A-1 6A1 6A-1 6A1 6A-1 6A1 6A-1
3 P 1A 2A 1A 1A 4C 4A 4B 2A 2A 4A 4A 4B 4B 4C 4C
Type
24.11.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
24.11.1b R 1 1 1 1 −1 −1 1 1 1 −1 −1 −1 −1 1 1
24.11.1c R 1 1 1 1 −1 1 −1 1 1 −1 −1 1 1 −1 −1
24.11.1d R 1 1 1 1 1 −1 −1 1 1 1 1 −1 −1 −1 −1
24.11.1e1 C 1 1 ζ3−1 ζ3 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3
24.11.1e2 C 1 1 ζ3 ζ3−1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1
24.11.1f1 C 1 1 ζ3−1 ζ3 −1 −1 1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 ζ3−1 ζ3
24.11.1f2 C 1 1 ζ3 ζ3−1 −1 −1 1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3 −ζ3−1 ζ3 ζ3−1
24.11.1g1 C 1 1 ζ3−1 ζ3 −1 1 −1 ζ3 ζ3−1 −ζ3−1 −ζ3 ζ3−1 ζ3 −ζ3−1 −ζ3
24.11.1g2 C 1 1 ζ3 ζ3−1 −1 1 −1 ζ3−1 ζ3 −ζ3 −ζ3−1 ζ3 ζ3−1 −ζ3 −ζ3−1
24.11.1h1 C 1 1 ζ3−1 ζ3 1 −1 −1 ζ3 ζ3−1 ζ3−1 ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3
24.11.1h2 C 1 1 ζ3 ζ3−1 1 −1 −1 ζ3−1 ζ3 ζ3 ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1
24.11.2a S 2 −2 2 2 0 0 0 −2 −2 0 0 0 0 0 0
24.11.2b1 C 2 −2 2ζ3−1 2ζ3 0 0 0 −2ζ3 −2ζ3−1 0 0 0 0 0 0
24.11.2b2 C 2 −2 2ζ3 2ζ3−1 0 0 0 −2ζ3−1 −2ζ3 0 0 0 0 0 0

magma: CharacterTable(G);