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Properties

Label 24T3
Degree $24$
Order $24$
Cyclic no
Abelian yes
Solvable yes
Primitive no
$p$-group no
Group: $C_2^2\times C_6$

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magma: G := TransitiveGroup(24, 3);
 

Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 7
$3$:  $C_3$
$4$:  $C_2^2$ x 7
$6$:  $C_6$ x 7
$8$:  $C_2^3$
$12$:  $C_6\times C_2$ x 7

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 7

Degree 3: $C_3$

Degree 4: $C_2^2$ x 7

Degree 6: $C_6$ x 7

Degree 8: $C_2^3$

Degree 12: $C_6\times C_2$ x 7

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $24=2^{3} \cdot 3$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  yes
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $1$
Label:  24.15
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 2F 2G 3A1 3A-1 6A1 6A-1 6B1 6B-1 6C1 6C-1 6D1 6D-1 6E1 6E-1 6F1 6F-1 6G1 6G-1
Size 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 P 1A 1A 1A 1A 1A 1A 1A 1A 3A-1 3A1 3A1 3A-1 3A-1 3A1 3A1 3A-1 3A1 3A1 3A1 3A-1 3A-1 3A-1 3A-1 3A1
3 P 1A 2D 2F 2E 2G 2A 2C 2B 1A 1A 2A 2A 2E 2E 2D 2F 2F 2G 2B 2C 2B 2D 2G 2C
Type
24.15.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
24.15.1b R 1 −1 −1 −1 −1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1 1 1 1 1 1 1
24.15.1c R 1 −1 −1 1 1 1 −1 −1 1 1 −1 −1 −1 −1 1 1 1 1 1 1 −1 −1 −1 −1
24.15.1d R 1 −1 1 −1 1 −1 −1 1 1 1 −1 −1 1 1 −1 −1 1 1 −1 −1 −1 −1 1 1
24.15.1e R 1 −1 1 1 −1 −1 1 −1 1 1 −1 −1 1 1 1 1 −1 −1 −1 −1 1 1 −1 −1
24.15.1f R 1 1 −1 −1 1 −1 1 −1 1 1 1 1 −1 −1 −1 −1 1 1 −1 −1 1 1 −1 −1
24.15.1g R 1 1 −1 1 −1 −1 −1 1 1 1 1 1 −1 −1 1 1 −1 −1 −1 −1 −1 −1 1 1
24.15.1h R 1 1 1 −1 −1 1 −1 −1 1 1 1 1 1 1 −1 −1 −1 −1 1 1 −1 −1 −1 −1
24.15.1i1 C 1 1 1 1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3
24.15.1i2 C 1 1 1 1 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1
24.15.1j1 C 1 −1 −1 −1 −1 1 1 1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3
24.15.1j2 C 1 −1 −1 −1 −1 1 1 1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1
24.15.1k1 C 1 −1 −1 1 1 1 −1 −1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3
24.15.1k2 C 1 −1 −1 1 1 1 −1 −1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1
24.15.1l1 C 1 −1 1 −1 1 −1 −1 1 ζ3−1 ζ3 −ζ3 −ζ3−1 ζ3−1 ζ3 −ζ3−1 −ζ3 ζ3−1 ζ3 −ζ3−1 −ζ3 −ζ3 −ζ3−1 ζ3−1 ζ3
24.15.1l2 C 1 −1 1 −1 1 −1 −1 1 ζ3 ζ3−1 −ζ3−1 −ζ3 ζ3 ζ3−1 −ζ3 −ζ3−1 ζ3 ζ3−1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 ζ3 ζ3−1
24.15.1m1 C 1 −1 1 1 −1 −1 1 −1 ζ3−1 ζ3 −ζ3 −ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3 ζ3 ζ3−1 −ζ3−1 −ζ3
24.15.1m2 C 1 −1 1 1 −1 −1 1 −1 ζ3 ζ3−1 −ζ3−1 −ζ3 ζ3 ζ3−1 ζ3 ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 ζ3−1 ζ3 −ζ3 −ζ3−1
24.15.1n1 C 1 1 −1 −1 1 −1 1 −1 ζ3−1 ζ3 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 ζ3−1 ζ3 −ζ3−1 −ζ3 ζ3 ζ3−1 −ζ3−1 −ζ3
24.15.1n2 C 1 1 −1 −1 1 −1 1 −1 ζ3 ζ3−1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3 −ζ3−1 ζ3 ζ3−1 −ζ3 −ζ3−1 ζ3−1 ζ3 −ζ3 −ζ3−1
24.15.1o1 C 1 1 −1 1 −1 −1 −1 1 ζ3−1 ζ3 ζ3 ζ3−1 −ζ3−1 −ζ3 ζ3−1 ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3 −ζ3−1 ζ3−1 ζ3
24.15.1o2 C 1 1 −1 1 −1 −1 −1 1 ζ3 ζ3−1 ζ3−1 ζ3 −ζ3 −ζ3−1 ζ3 ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 ζ3 ζ3−1
24.15.1p1 C 1 1 1 −1 −1 1 −1 −1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3
24.15.1p2 C 1 1 1 −1 −1 1 −1 −1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1

magma: CharacterTable(G);