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Properties

Label 24T33
Degree $24$
Order $48$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_6:C_4$

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Show commands: Magma

magma: G := TransitiveGroup(24, 33);
 

Group action invariants

Degree $n$:  $24$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $33$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_6:C_4$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $4$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (3,12)(4,11)(5,22)(6,21)(7,8)(9,17)(10,18)(15,23)(16,24)(19,20), (1,15,6,19,9,24,13,3,17,7,21,11)(2,16,5,20,10,23,14,4,18,8,22,12)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$6$:  $S_3$
$8$:  $D_{4}$ x 2, $C_4\times C_2$
$12$:  $D_{6}$
$16$:  $C_2^2:C_4$
$24$:  $S_3 \times C_4$, $D_{12}$, $(C_6\times C_2):C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: $C_4$, $D_{4}$ x 2

Degree 6: $D_{6}$

Degree 8: $C_2^2:C_4$

Degree 12: $S_3 \times C_4$, $D_{12}$, $(C_6\times C_2):C_2$

Low degree siblings

24T33

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $48=2^{4} \cdot 3$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  48.14
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 3A 4A1 4A-1 4B1 4B-1 6A 6B 6C 12A1 12A-1 12A5 12A-5
Size 1 1 1 1 6 6 2 2 2 6 6 2 2 2 2 2 2 2
2 P 1A 1A 1A 1A 1A 1A 3A 2A 2A 2B 2B 3A 3A 3A 6A 6A 6A 6A
3 P 1A 2C 2A 2B 2D 2E 1A 4A-1 4A1 4B-1 4B1 2C 2B 2A 4A1 4A1 4A-1 4A-1
Type
48.14.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
48.14.1b R 1 1 1 1 −1 −1 1 −1 −1 1 1 1 1 1 −1 −1 −1 −1
48.14.1c R 1 1 1 1 −1 −1 1 1 1 −1 −1 1 1 1 1 1 1 1
48.14.1d R 1 1 1 1 1 1 1 −1 −1 −1 −1 1 1 1 −1 −1 −1 −1
48.14.1e1 C 1 −1 −1 1 −1 1 1 −i i i −i −1 1 −1 i −i i −i
48.14.1e2 C 1 −1 −1 1 −1 1 1 i −i −i i −1 1 −1 −i i −i i
48.14.1f1 C 1 −1 −1 1 1 −1 1 −i i −i i −1 1 −1 i −i i −i
48.14.1f2 C 1 −1 −1 1 1 −1 1 i −i i −i −1 1 −1 −i i −i i
48.14.2a R 2 2 2 2 0 0 −1 2 2 0 0 −1 −1 −1 −1 −1 −1 −1
48.14.2b R 2 −2 2 −2 0 0 2 0 0 0 0 −2 −2 2 0 0 0 0
48.14.2c R 2 2 −2 −2 0 0 2 0 0 0 0 2 −2 −2 0 0 0 0
48.14.2d R 2 2 2 2 0 0 −1 −2 −2 0 0 −1 −1 −1 1 1 1 1
48.14.2e1 C 2 −2 −2 2 0 0 −1 −2i 2i 0 0 1 −1 1 −i i −i i
48.14.2e2 C 2 −2 −2 2 0 0 −1 2i −2i 0 0 1 −1 1 i −i i −i
48.14.2f1 R 2 −2 2 −2 0 0 −1 0 0 0 0 1 1 −1 −ζ12−1−ζ12 −ζ12−1−ζ12 ζ12−1+ζ12 ζ12−1+ζ12
48.14.2f2 R 2 −2 2 −2 0 0 −1 0 0 0 0 1 1 −1 ζ12−1+ζ12 ζ12−1+ζ12 −ζ12−1−ζ12 −ζ12−1−ζ12
48.14.2g1 C 2 2 −2 −2 0 0 −1 0 0 0 0 −1 1 1 −1−2ζ3 1+2ζ3 1+2ζ3 −1−2ζ3
48.14.2g2 C 2 2 −2 −2 0 0 −1 0 0 0 0 −1 1 1 1+2ζ3 −1−2ζ3 −1−2ζ3 1+2ζ3

magma: CharacterTable(G);