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Properties

Label 24T15
Degree $24$
Order $24$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3\times D_4$

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Show commands: Magma

magma: G := TransitiveGroup(24, 15);
 

Group action invariants

Degree $n$:  $24$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $15$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3\times D_4$
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:  (1,17,9,2,18,10)(3,8,12,15,20,24)(4,7,11,16,19,23)(5,21,14,6,22,13), (1,24,9,8,18,15)(2,23,10,7,17,16)(3,13,12,21,20,6)(4,14,11,22,19,5)
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
$8$:  $D_{4}$
$12$:  $C_6\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: $C_3$

Degree 4: $C_2^2$, $D_{4}$ x 2

Degree 6: $C_6$ x 3

Degree 8: $D_4$

Degree 12: $C_6\times C_2$, $D_4 \times C_3$ x 2

Low degree siblings

12T14 x 2

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

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:  24.10
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A1 3A-1 4A 6A1 6A-1 6B1 6B-1 6C1 6C-1 12A1 12A-1
Size 1 1 2 2 1 1 2 1 1 2 2 2 2 2 2
2 P 1A 1A 1A 1A 3A-1 3A1 2A 3A1 3A-1 3A-1 3A1 3A1 3A-1 6A1 6A-1
3 P 1A 2A 2B 2C 1A 1A 4A 2A 2A 2C 2B 2C 2B 4A 4A
Type
24.10.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
24.10.1b R 1 1 −1 −1 1 1 1 1 1 −1 −1 −1 −1 1 1
24.10.1c R 1 1 −1 1 1 1 −1 1 1 −1 −1 1 1 −1 −1
24.10.1d R 1 1 1 −1 1 1 −1 1 1 1 1 −1 −1 −1 −1
24.10.1e1 C 1 1 1 1 ζ3−1 ζ3 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3
24.10.1e2 C 1 1 1 1 ζ3 ζ3−1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1
24.10.1f1 C 1 1 −1 −1 ζ3−1 ζ3 1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 ζ3−1 ζ3
24.10.1f2 C 1 1 −1 −1 ζ3 ζ3−1 1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3 −ζ3−1 ζ3 ζ3−1
24.10.1g1 C 1 1 −1 1 ζ3−1 ζ3 −1 ζ3 ζ3−1 −ζ3−1 −ζ3 ζ3−1 ζ3 −ζ3−1 −ζ3
24.10.1g2 C 1 1 −1 1 ζ3 ζ3−1 −1 ζ3−1 ζ3 −ζ3 −ζ3−1 ζ3 ζ3−1 −ζ3 −ζ3−1
24.10.1h1 C 1 1 1 −1 ζ3−1 ζ3 −1 ζ3 ζ3−1 ζ3−1 ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3
24.10.1h2 C 1 1 1 −1 ζ3 ζ3−1 −1 ζ3−1 ζ3 ζ3 ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1
24.10.2a R 2 −2 0 0 2 2 0 −2 −2 0 0 0 0 0 0
24.10.2b1 C 2 −2 0 0 2ζ3−1 2ζ3 0 −2ζ3 −2ζ3−1 0 0 0 0 0 0
24.10.2b2 C 2 −2 0 0 2ζ3 2ζ3−1 0 −2ζ3−1 −2ζ3 0 0 0 0 0 0

magma: CharacterTable(G);