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Properties

Label 24T229
24T229 1 6 1->6 7 1->7 18 1->18 2 5 2->5 8 2->8 17 2->17 3 13 3->13 15 3->15 20 3->20 4 14 4->14 16 4->16 19 4->19 5->4 9 5->9 5->13 6->3 10 6->10 6->14 7->9 7->16 7->19 8->10 8->15 8->20 9->13 23 9->23 10->14 24 10->24 11 11->6 11->23 12 12->5 12->24 13->17 13->20 14->18 14->19 15->1 15->4 16->2 16->3 17->16 22 17->22 18->15 21 18->21 19->8 19->22 20->7 20->21 21->2 21->11 21->22 22->1 22->12 23->12 23->18 23->24 24->11 24->17
Degree $24$
Order $144$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3\times D_{12}$

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Copy content magma:G := TransitiveGroup(24, 229);
 

Group invariants

Abstract group:  $S_3\times D_{12}$
Copy content magma:IdentifyGroup(G);
 
Order:  $144=2^{4} \cdot 3^{2}$
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:   not nilpotent
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $24$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $229$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $2$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,18)(2,17)(3,20)(4,19)(5,13)(6,14)(7,16)(8,15)(21,22)(23,24)$, $(1,6,10,14,18,21,2,5,9,13,17,22)(3,15,4,16)(7,19,8,20)(11,23,12,24)$, $(1,7,9,23,18,15)(2,8,10,24,17,16)(3,13,20,21,11,6)(4,14,19,22,12,5)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$6$:  $S_3$ x 2
$8$:  $D_{4}$ x 2, $C_2^3$
$12$:  $D_{6}$ x 6
$16$:  $D_4\times C_2$
$24$:  $S_3 \times C_2^2$ x 2, $D_{12}$ x 2
$36$:  $S_3^2$
$48$:  12T28, 24T29
$72$:  12T37

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: None

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

Degree 6: $S_3^2$

Degree 8: $D_4\times C_2$

Degree 12: 12T37

Low degree siblings

24T229, 36T150 x 4

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 $2^{12}$ $3$ $2$ $12$ $( 1,12)( 2,11)( 3,17)( 4,18)( 5,15)( 6,16)( 7,22)( 8,21)( 9,19)(10,20)(13,24)(14,23)$
2C $2^{12}$ $3$ $2$ $12$ $( 1,11)( 2,12)( 3,18)( 4,17)( 5,16)( 6,15)( 7,21)( 8,22)( 9,20)(10,19)(13,23)(14,24)$
2D $2^{12}$ $6$ $2$ $12$ $( 1,19)( 2,20)( 3,17)( 4,18)( 5,16)( 6,15)( 7,13)( 8,14)( 9,12)(10,11)(21,23)(22,24)$
2E $2^{12}$ $6$ $2$ $12$ $( 1,16)( 2,15)( 3,14)( 4,13)( 5,11)( 6,12)( 7,10)( 8, 9)(17,23)(18,24)(19,21)(20,22)$
2F $2^{12}$ $18$ $2$ $12$ $( 1, 6)( 2, 5)( 3, 7)( 4, 8)( 9,21)(10,22)(11,23)(12,24)(13,18)(14,17)(15,20)(16,19)$
2G $2^{10},1^{4}$ $18$ $2$ $10$ $( 1, 9)( 2,10)( 3,20)( 4,19)( 5, 6)( 7,16)( 8,15)(13,22)(14,21)(23,24)$
3A $3^{8}$ $2$ $3$ $16$ $( 1,18, 9)( 2,17,10)( 3,20,11)( 4,19,12)( 5,22,14)( 6,21,13)( 7,23,15)( 8,24,16)$
3B $3^{8}$ $2$ $3$ $16$ $( 1,18, 9)( 2,17,10)( 3,11,20)( 4,12,19)( 5,22,14)( 6,21,13)( 7,15,23)( 8,16,24)$
3C $3^{4},1^{12}$ $4$ $3$ $8$ $( 3,20,11)( 4,19,12)( 7,23,15)( 8,24,16)$
4A $4^{6}$ $2$ $4$ $18$ $( 1,14, 2,13)( 3,16, 4,15)( 5,17, 6,18)( 7,20, 8,19)( 9,22,10,21)(11,24,12,23)$
4B $4^{6}$ $6$ $4$ $18$ $( 1,23, 2,24)( 3, 6, 4, 5)( 7,10, 8, 9)(11,13,12,14)(15,17,16,18)(19,22,20,21)$
6A $6^{4}$ $2$ $6$ $20$ $( 1,17, 9, 2,18,10)( 3,19,11, 4,20,12)( 5,21,14, 6,22,13)( 7,24,15, 8,23,16)$
6B $6^{4}$ $2$ $6$ $20$ $( 1,10,18, 2, 9,17)( 3,19,11, 4,20,12)( 5,13,22, 6,14,21)( 7,24,15, 8,23,16)$
6C $6^{2},2^{6}$ $4$ $6$ $16$ $( 1, 2)( 3,12,20, 4,11,19)( 5, 6)( 7,16,23, 8,15,24)( 9,10)(13,14)(17,18)(21,22)$
6D $6^{4}$ $6$ $6$ $20$ $( 1,19,18,12, 9, 4)( 2,20,17,11,10, 3)( 5,23,22,15,14, 7)( 6,24,21,16,13, 8)$
6E $6^{4}$ $6$ $6$ $20$ $( 1, 3, 9,11,18,20)( 2, 4,10,12,17,19)( 5, 8,14,16,22,24)( 6, 7,13,15,21,23)$
6F $6^{4}$ $12$ $6$ $20$ $( 1,12,18,19, 9, 4)( 2,11,17,20,10, 3)( 5, 8,22,16,14,24)( 6, 7,21,15,13,23)$
6G $6^{4}$ $12$ $6$ $20$ $( 1, 8,18,16, 9,24)( 2, 7,17,15,10,23)( 3,22,11,14,20, 5)( 4,21,12,13,19, 6)$
12A1 $12^{2}$ $2$ $12$ $22$ $( 1,21,17,14, 9, 6, 2,22,18,13,10, 5)( 3,23,19,16,11, 7, 4,24,20,15,12, 8)$
12A5 $12^{2}$ $2$ $12$ $22$ $( 1, 6,10,14,18,21, 2, 5, 9,13,17,22)( 3, 7,12,16,20,23, 4, 8,11,15,19,24)$
12B $12^{2}$ $4$ $12$ $22$ $( 1, 5,10,13,18,22, 2, 6, 9,14,17,21)( 3,24,19,15,11, 8, 4,23,20,16,12, 7)$
12C1 $12,4^{3}$ $4$ $12$ $20$ $( 1,13, 2,14)( 3, 7,12,16,20,23, 4, 8,11,15,19,24)( 5,18, 6,17)( 9,21,10,22)$
12C5 $12,4^{3}$ $4$ $12$ $20$ $( 1,21,17,14, 9, 6, 2,22,18,13,10, 5)( 3,15, 4,16)( 7,19, 8,20)(11,23,12,24)$
12D1 $12^{2}$ $6$ $12$ $22$ $( 1, 8,17,23, 9,16, 2, 7,18,24,10,15)( 3,14,19, 6,11,22, 4,13,20, 5,12,21)$
12D5 $12^{2}$ $6$ $12$ $22$ $( 1,16,10,23,18, 8, 2,15, 9,24,17, 7)( 3,22,12, 6,20,14, 4,21,11, 5,19,13)$

Malle's constant $a(G)$:     $1/8$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 4A 4B 6A 6B 6C 6D 6E 6F 6G 12A1 12A5 12B 12C1 12C5 12D1 12D5
Size 1 1 3 3 6 6 18 18 2 2 4 2 6 2 2 4 6 6 12 12 2 2 4 4 4 6 6
2 P 1A 1A 1A 1A 1A 1A 1A 1A 3A 3B 3C 2A 2A 3A 3B 3C 3A 3A 3B 3B 6A 6A 6B 6C 6C 6A 6A
3 P 1A 2A 2B 2C 2D 2E 2F 2G 1A 1A 1A 4A 4B 2A 2A 2A 2B 2C 2D 2E 4A 4A 4A 4A 4A 4B 4B
Type
144.144.1a 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 1 1 1
144.144.1b 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 1 −1 −1
144.144.1c 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 −1 1 1
144.144.1d 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 −1 1 1
144.144.1e 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 1 −1 −1
144.144.1f 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 1 1 1
144.144.1g 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 −1 −1 −1
144.144.1h 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 −1 −1 −1
144.144.2a R 2 2 0 0 2 2 0 0 2 −1 −1 2 0 2 −1 −1 0 0 −1 −1 2 2 −1 −1 −1 0 0
144.144.2b R 2 2 2 2 0 0 0 0 −1 2 −1 2 2 −1 2 −1 −1 −1 0 0 −1 −1 2 −1 −1 −1 −1
144.144.2c R 2 −2 −2 2 0 0 0 0 2 2 2 0 0 −2 −2 −2 −2 2 0 0 0 0 0 0 0 0 0
144.144.2d R 2 −2 2 −2 0 0 0 0 2 2 2 0 0 −2 −2 −2 2 −2 0 0 0 0 0 0 0 0 0
144.144.2e R 2 2 −2 −2 0 0 0 0 −1 2 −1 −2 2 −1 2 −1 1 1 0 0 1 1 −2 1 1 −1 −1
144.144.2f R 2 2 −2 −2 0 0 0 0 −1 2 −1 2 −2 −1 2 −1 1 1 0 0 −1 −1 2 −1 −1 1 1
144.144.2g R 2 2 0 0 −2 −2 0 0 2 −1 −1 2 0 2 −1 −1 0 0 1 1 2 2 −1 −1 −1 0 0
144.144.2h R 2 2 0 0 −2 2 0 0 2 −1 −1 −2 0 2 −1 −1 0 0 1 −1 −2 −2 1 1 1 0 0
144.144.2i R 2 2 0 0 2 −2 0 0 2 −1 −1 −2 0 2 −1 −1 0 0 −1 1 −2 −2 1 1 1 0 0
144.144.2j R 2 2 2 2 0 0 0 0 −1 2 −1 −2 −2 −1 2 −1 −1 −1 0 0 1 1 −2 1 1 1 1
144.144.2k1 R 2 −2 −2 2 0 0 0 0 −1 2 −1 0 0 1 −2 1 1 −1 0 0 −ζ12−1−ζ12 ζ12−1+ζ12 0 −ζ12−1−ζ12 ζ12−1+ζ12 ζ12−1+ζ12 −ζ12−1−ζ12
144.144.2k2 R 2 −2 −2 2 0 0 0 0 −1 2 −1 0 0 1 −2 1 1 −1 0 0 ζ12−1+ζ12 −ζ12−1−ζ12 0 ζ12−1+ζ12 −ζ12−1−ζ12 −ζ12−1−ζ12 ζ12−1+ζ12
144.144.2l1 R 2 −2 2 −2 0 0 0 0 −1 2 −1 0 0 1 −2 1 −1 1 0 0 −ζ12−1−ζ12 ζ12−1+ζ12 0 −ζ12−1−ζ12 ζ12−1+ζ12 −ζ12−1−ζ12 ζ12−1+ζ12
144.144.2l2 R 2 −2 2 −2 0 0 0 0 −1 2 −1 0 0 1 −2 1 −1 1 0 0 ζ12−1+ζ12 −ζ12−1−ζ12 0 ζ12−1+ζ12 −ζ12−1−ζ12 ζ12−1+ζ12 −ζ12−1−ζ12
144.144.4a R 4 4 0 0 0 0 0 0 −2 −2 1 4 0 −2 −2 1 0 0 0 0 −2 −2 −2 1 1 0 0
144.144.4b R 4 −4 0 0 0 0 0 0 4 −2 −2 0 0 −4 2 2 0 0 0 0 0 0 0 0 0 0 0
144.144.4c R 4 4 0 0 0 0 0 0 −2 −2 1 −4 0 −2 −2 1 0 0 0 0 2 2 2 −1 −1 0 0
144.144.4d1 R 4 −4 0 0 0 0 0 0 −2 −2 1 0 0 2 2 −1 0 0 0 0 −2ζ12−1−2ζ12 2ζ12−1+2ζ12 0 ζ12−1+ζ12 −ζ12−1−ζ12 0 0
144.144.4d2 R 4 −4 0 0 0 0 0 0 −2 −2 1 0 0 2 2 −1 0 0 0 0 2ζ12−1+2ζ12 −2ζ12−1−2ζ12 0 −ζ12−1−ζ12 ζ12−1+ζ12 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed