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Properties

Label 24T40
24T40 1 11 1->11 14 1->14 2 12 2->12 13 2->13 3 3->1 4 3->4 4->2 5 15 5->15 17 5->17 6 16 6->16 18 6->18 7 7->5 8 8->6 9 19 9->19 22 9->22 10 20 10->20 21 10->21 11->9 11->12 12->10 23 13->23 24 14->24 15->14 16->13 17->3 18->4 19->17 19->20 20->18 21->8 22->7 23->21 24->22
Degree $24$
Order $48$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3\times D_8$

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Copy content magma:G := TransitiveGroup(24, 40);
 

Group invariants

Abstract group:  $C_3\times D_8$
Copy content magma:IdentifyGroup(G);
 
Order:  $48=2^{4} \cdot 3$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  yes
Copy content magma:IsSolvable(G);
 
Nilpotency class:  $3$
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $24$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $40$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $6$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,14)(2,13)(3,4)(5,17)(6,18)(9,22)(10,21)(11,12)(19,20)$, $(1,11,9,19,17,3)(2,12,10,20,18,4)(5,15,14,24,22,7)(6,16,13,23,21,8)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_2^2$
$6$:  $C_6$ x 3
$8$:  $D_{4}$
$12$:  $C_6\times C_2$
$16$:  $D_{8}$
$24$:  $D_4 \times C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 4: $D_{4}$

Degree 6: $C_6$

Degree 8: $D_{8}$

Degree 12: $D_4 \times C_3$

Low degree siblings

24T40

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{24}$ $1$ $1$ $0$ $()$
2A $2^{12}$ $1$ $2$ $12$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)$
2B $2^{9},1^{6}$ $4$ $2$ $9$ $( 1,14)( 2,13)( 3, 4)( 5,17)( 6,18)( 9,22)(10,21)(11,12)(19,20)$
2C $2^{12}$ $4$ $2$ $12$ $( 1, 7)( 2, 8)( 3,21)( 4,22)( 5,12)( 6,11)( 9,15)(10,16)(13,19)(14,20)(17,24)(18,23)$
3A1 $3^{8}$ $1$ $3$ $16$ $( 1,17, 9)( 2,18,10)( 3,19,11)( 4,20,12)( 5,22,14)( 6,21,13)( 7,24,15)( 8,23,16)$
3A-1 $3^{8}$ $1$ $3$ $16$ $( 1, 9,17)( 2,10,18)( 3,11,19)( 4,12,20)( 5,14,22)( 6,13,21)( 7,15,24)( 8,16,23)$
4A $4^{6}$ $2$ $4$ $18$ $( 1,14, 2,13)( 3,16, 4,15)( 5,18, 6,17)( 7,19, 8,20)( 9,22,10,21)(11,23,12,24)$
6A1 $6^{4}$ $1$ $6$ $20$ $( 1,18, 9, 2,17,10)( 3,20,11, 4,19,12)( 5,21,14, 6,22,13)( 7,23,15, 8,24,16)$
6A-1 $6^{4}$ $1$ $6$ $20$ $( 1,10,17, 2, 9,18)( 3,12,19, 4,11,20)( 5,13,22, 6,14,21)( 7,16,24, 8,15,23)$
6B1 $6^{3},3^{2}$ $4$ $6$ $19$ $( 1,22,17,14, 9, 5)( 2,21,18,13,10, 6)( 3,12,19, 4,11,20)( 7,15,24)( 8,16,23)$
6B-1 $6^{3},3^{2}$ $4$ $6$ $19$ $( 1, 6, 9,13,17,21)( 2, 5,10,14,18,22)( 3,19,11)( 4,20,12)( 7,23,15, 8,24,16)$
6C1 $6^{4}$ $4$ $6$ $20$ $( 1,24, 9, 7,17,15)( 2,23,10, 8,18,16)( 3,13,11,21,19, 6)( 4,14,12,22,20, 5)$
6C-1 $6^{4}$ $4$ $6$ $20$ $( 1,16,17, 8, 9,23)( 2,15,18, 7,10,24)( 3, 5,19,22,11,14)( 4, 6,20,21,12,13)$
8A1 $8^{3}$ $2$ $8$ $21$ $( 1, 8,14,20, 2, 7,13,19)( 3, 9,16,22, 4,10,15,21)( 5,12,18,24, 6,11,17,23)$
8A3 $8^{3}$ $2$ $8$ $21$ $( 1,20,13, 8, 2,19,14, 7)( 3,22,15, 9, 4,21,16,10)( 5,24,17,12, 6,23,18,11)$
12A1 $12^{2}$ $2$ $12$ $22$ $( 1,21,18,14, 9, 6, 2,22,17,13,10, 5)( 3,24,20,16,11, 7, 4,23,19,15,12, 8)$
12A-1 $12^{2}$ $2$ $12$ $22$ $( 1, 5,10,13,17,22, 2, 6, 9,14,18,21)( 3, 8,12,15,19,23, 4, 7,11,16,20,24)$
24A1 $24$ $2$ $24$ $23$ $( 1,12,21, 8,18, 3,14,24, 9,20, 6,16, 2,11,22, 7,17, 4,13,23,10,19, 5,15)$
24A-1 $24$ $2$ $24$ $23$ $( 1, 4, 6, 8,10,11,14,15,17,20,21,23, 2, 3, 5, 7, 9,12,13,16,18,19,22,24)$
24A5 $24$ $2$ $24$ $23$ $( 1, 3, 6, 7,10,12,14,16,17,19,21,24, 2, 4, 5, 8, 9,11,13,15,18,20,22,23)$
24A-5 $24$ $2$ $24$ $23$ $( 1,11,21, 7,18, 4,14,23, 9,19, 6,15, 2,12,22, 8,17, 3,13,24,10,20, 5,16)$

Malle's constant $a(G)$:     $1/9$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 3A1 3A-1 4A 6A1 6A-1 6B1 6B-1 6C1 6C-1 8A1 8A3 12A1 12A-1 24A1 24A-1 24A5 24A-5
Size 1 1 4 4 1 1 2 1 1 4 4 4 4 2 2 2 2 2 2 2 2
2 P 1A 1A 1A 1A 3A-1 3A1 2A 3A-1 3A1 3A1 3A-1 3A-1 3A1 4A 4A 6A1 6A-1 12A1 12A-1 12A-1 12A1
3 P 1A 2A 2B 2C 1A 1A 4A 2A 2A 2B 2B 2C 2C 8A3 8A1 4A 4A 8A1 8A1 8A3 8A3
Type
48.25.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
48.25.1b R 1 1 −1 −1 1 1 1 1 1 −1 −1 −1 −1 1 1 1 1 1 1 1 1
48.25.1c R 1 1 −1 1 1 1 1 1 1 −1 −1 1 1 −1 −1 1 1 −1 −1 −1 −1
48.25.1d R 1 1 1 −1 1 1 1 1 1 1 1 −1 −1 −1 −1 1 1 −1 −1 −1 −1
48.25.1e1 C 1 1 1 1 ζ3−1 ζ3 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3
48.25.1e2 C 1 1 1 1 ζ3 ζ3−1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1
48.25.1f1 C 1 1 −1 −1 ζ3−1 ζ3 1 ζ3−1 ζ3 −ζ3−1 −ζ3 −ζ3 −ζ3−1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3
48.25.1f2 C 1 1 −1 −1 ζ3 ζ3−1 1 ζ3 ζ3−1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1
48.25.1g1 C 1 1 −1 1 ζ3−1 ζ3 1 ζ3−1 ζ3 −ζ3−1 −ζ3 ζ3 ζ3−1 −1 −1 ζ3 ζ3−1 −ζ3 −ζ3−1 −ζ3−1 −ζ3
48.25.1g2 C 1 1 −1 1 ζ3 ζ3−1 1 ζ3 ζ3−1 −ζ3 −ζ3−1 ζ3−1 ζ3 −1 −1 ζ3−1 ζ3 −ζ3−1 −ζ3 −ζ3 −ζ3−1
48.25.1h1 C 1 1 1 −1 ζ3−1 ζ3 1 ζ3−1 ζ3 ζ3−1 ζ3 −ζ3 −ζ3−1 −1 −1 ζ3 ζ3−1 −ζ3 −ζ3−1 −ζ3−1 −ζ3
48.25.1h2 C 1 1 1 −1 ζ3 ζ3−1 1 ζ3 ζ3−1 ζ3 ζ3−1 −ζ3−1 −ζ3 −1 −1 ζ3−1 ζ3 −ζ3−1 −ζ3 −ζ3 −ζ3−1
48.25.2a R 2 2 0 0 2 2 −2 2 2 0 0 0 0 0 0 −2 −2 0 0 0 0
48.25.2b1 R 2 −2 0 0 2 2 0 −2 −2 0 0 0 0 −ζ8−1−ζ8 ζ8−1+ζ8 0 0 −ζ8−1−ζ8 −ζ8−1−ζ8 ζ8−1+ζ8 ζ8−1+ζ8
48.25.2b2 R 2 −2 0 0 2 2 0 −2 −2 0 0 0 0 ζ8−1+ζ8 −ζ8−1−ζ8 0 0 ζ8−1+ζ8 ζ8−1+ζ8 −ζ8−1−ζ8 −ζ8−1−ζ8
48.25.2c1 C 2 2 0 0 2ζ3−1 2ζ3 −2 2ζ3−1 2ζ3 0 0 0 0 0 0 −2ζ3 −2ζ3−1 0 0 0 0
48.25.2c2 C 2 2 0 0 2ζ3 2ζ3−1 −2 2ζ3 2ζ3−1 0 0 0 0 0 0 −2ζ3−1 −2ζ3 0 0 0 0
48.25.2d1 C 2 −2 0 0 −2ζ244 2ζ248 0 2ζ244 −2ζ248 0 0 0 0 −ζ24−3−ζ243 ζ24−3+ζ243 0 0 ζ243−ζ245−ζ247 ζ24+ζ247 −ζ24−ζ247 −ζ243+ζ245+ζ247
48.25.2d2 C 2 −2 0 0 2ζ248 −2ζ244 0 −2ζ248 2ζ244 0 0 0 0 −ζ24−3−ζ243 ζ24−3+ζ243 0 0 ζ24+ζ247 ζ243−ζ245−ζ247 −ζ243+ζ245+ζ247 −ζ24−ζ247
48.25.2d3 C 2 −2 0 0 −2ζ244 2ζ248 0 2ζ244 −2ζ248 0 0 0 0 ζ24−3+ζ243 −ζ24−3−ζ243 0 0 −ζ243+ζ245+ζ247 −ζ24−ζ247 ζ24+ζ247 ζ243−ζ245−ζ247
48.25.2d4 C 2 −2 0 0 2ζ248 −2ζ244 0 −2ζ248 2ζ244 0 0 0 0 ζ24−3+ζ243 −ζ24−3−ζ243 0 0 −ζ24−ζ247 −ζ243+ζ245+ζ247 ζ243−ζ245−ζ247 ζ24+ζ247

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Regular extensions

Data not computed