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Properties

Label 24T65
24T65 1 22 1->22 23 1->23 2 21 2->21 24 2->24 3 16 3->16 3->21 4 15 4->15 4->22 5 5->2 5->3 6 6->1 6->4 7 7->2 20 7->20 8 8->1 19 8->19 9 9->5 9->7 10 10->6 10->8 11 11->6 11->24 12 12->5 12->23 13 13->9 13->12 14 14->10 14->11 15->3 15->10 16->4 16->9 17 17->13 17->16 18 18->14 18->15 19->7 19->14 20->8 20->13 21->18 21->19 22->17 22->20 23->11 23->17 24->12 24->18
Degree $24$
Order $72$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3\times C_{12}$

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Copy content magma:G := TransitiveGroup(24, 65);
 

Group invariants

Abstract group:  $S_3\times C_{12}$
Copy content magma:IdentifyGroup(G);
 
Order:  $72=2^{3} \cdot 3^{2}$
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
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Group action invariants

Degree $n$:  $24$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $65$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $12$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,22,17,13,9,5,2,21,18,14,10,6)(3,16,4,15)(7,20,8,19)(11,24,12,23)$, $(1,23,17,16,9,7,2,24,18,15,10,8)(3,21,19,14,11,6,4,22,20,13,12,5)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_4$ x 2, $C_2^2$
$6$:  $S_3$, $C_6$ x 3
$8$:  $C_4\times C_2$
$12$:  $D_{6}$, $C_{12}$ x 2, $C_6\times C_2$
$18$:  $S_3\times C_3$
$24$:  $S_3 \times C_4$, 24T2
$36$:  $C_6\times S_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: None

Degree 4: $C_4$ x 2, $C_2^2$

Degree 6: $S_3\times C_3$

Degree 8: $C_4\times C_2$

Degree 12: $C_6\times S_3$

Low degree siblings

36T27 x 2

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 $2^{12}$ $3$ $2$ $12$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)$
2C $2^{12}$ $3$ $2$ $12$ $( 1,19)( 2,20)( 3,10)( 4, 9)( 5,23)( 6,24)( 7,14)( 8,13)(11,17)(12,18)(15,22)(16,21)$
3A1 $3^{8}$ $1$ $3$ $16$ $( 1,18, 9)( 2,17,10)( 3,20,11)( 4,19,12)( 5,22,14)( 6,21,13)( 7,23,15)( 8,24,16)$
3A-1 $3^{8}$ $1$ $3$ $16$ $( 1, 9,18)( 2,10,17)( 3,11,20)( 4,12,19)( 5,14,22)( 6,13,21)( 7,15,23)( 8,16,24)$
3B $3^{8}$ $2$ $3$ $16$ $( 1,18, 9)( 2,17,10)( 3,11,20)( 4,12,19)( 5,22,14)( 6,21,13)( 7,15,23)( 8,16,24)$
3C1 $3^{4},1^{12}$ $2$ $3$ $8$ $( 1, 9,18)( 2,10,17)( 5,14,22)( 6,13,21)$
3C-1 $3^{4},1^{12}$ $2$ $3$ $8$ $( 1,18, 9)( 2,17,10)( 5,22,14)( 6,21,13)$
4A1 $4^{6}$ $1$ $4$ $18$ $( 1,13, 2,14)( 3,15, 4,16)( 5,18, 6,17)( 7,19, 8,20)( 9,21,10,22)(11,23,12,24)$
4A-1 $4^{6}$ $1$ $4$ $18$ $( 1,14, 2,13)( 3,16, 4,15)( 5,17, 6,18)( 7,20, 8,19)( 9,22,10,21)(11,24,12,23)$
4B1 $4^{6}$ $3$ $4$ $18$ $( 1,24, 2,23)( 3, 5, 4, 6)( 7, 9, 8,10)(11,14,12,13)(15,18,16,17)(19,21,20,22)$
4B-1 $4^{6}$ $3$ $4$ $18$ $( 1,23, 2,24)( 3, 6, 4, 5)( 7,10, 8, 9)(11,13,12,14)(15,17,16,18)(19,22,20,21)$
6A1 $6^{4}$ $1$ $6$ $20$ $( 1,10,18, 2, 9,17)( 3,12,20, 4,11,19)( 5,13,22, 6,14,21)( 7,16,23, 8,15,24)$
6A-1 $6^{4}$ $1$ $6$ $20$ $( 1,17, 9, 2,18,10)( 3,19,11, 4,20,12)( 5,21,14, 6,22,13)( 7,24,15, 8,23,16)$
6B $6^{4}$ $2$ $6$ $20$ $( 1,10,18, 2, 9,17)( 3,19,11, 4,20,12)( 5,13,22, 6,14,21)( 7,24,15, 8,23,16)$
6C1 $6^{2},2^{6}$ $2$ $6$ $16$ $( 1,17, 9, 2,18,10)( 3, 4)( 5,21,14, 6,22,13)( 7, 8)(11,12)(15,16)(19,20)(23,24)$
6C-1 $6^{2},2^{6}$ $2$ $6$ $16$ $( 1,10,18, 2, 9,17)( 3, 4)( 5,13,22, 6,14,21)( 7, 8)(11,12)(15,16)(19,20)(23,24)$
6D1 $6^{4}$ $3$ $6$ $20$ $( 1,11,18, 3, 9,20)( 2,12,17, 4,10,19)( 5,16,22, 8,14,24)( 6,15,21, 7,13,23)$
6D-1 $6^{4}$ $3$ $6$ $20$ $( 1,20, 9, 3,18,11)( 2,19,10, 4,17,12)( 5,24,14, 8,22,16)( 6,23,13, 7,21,15)$
6E1 $6^{4}$ $3$ $6$ $20$ $( 1, 4,18,19, 9,12)( 2, 3,17,20,10,11)( 5, 7,22,23,14,15)( 6, 8,21,24,13,16)$
6E-1 $6^{4}$ $3$ $6$ $20$ $( 1,12, 9,19,18, 4)( 2,11,10,20,17, 3)( 5,15,14,23,22, 7)( 6,16,13,24,21, 8)$
12A1 $12^{2}$ $1$ $12$ $22$ $( 1, 5,10,13,18,22, 2, 6, 9,14,17,21)( 3, 8,12,15,20,24, 4, 7,11,16,19,23)$
12A-1 $12^{2}$ $1$ $12$ $22$ $( 1,21,17,14, 9, 6, 2,22,18,13,10, 5)( 3,23,19,16,11, 7, 4,24,20,15,12, 8)$
12A5 $12^{2}$ $1$ $12$ $22$ $( 1,22,17,13, 9, 5, 2,21,18,14,10, 6)( 3,24,19,15,11, 8, 4,23,20,16,12, 7)$
12A-5 $12^{2}$ $1$ $12$ $22$ $( 1, 6,10,14,18,21, 2, 5, 9,13,17,22)( 3, 7,12,16,20,23, 4, 8,11,15,19,24)$
12B1 $12^{2}$ $2$ $12$ $22$ $( 1, 6,10,14,18,21, 2, 5, 9,13,17,22)( 3,23,19,16,11, 7, 4,24,20,15,12, 8)$
12B-1 $12^{2}$ $2$ $12$ $22$ $( 1, 5,10,13,18,22, 2, 6, 9,14,17,21)( 3,24,19,15,11, 8, 4,23,20,16,12, 7)$
12C1 $12,4^{3}$ $2$ $12$ $20$ $( 1,21,17,14, 9, 6, 2,22,18,13,10, 5)( 3,15, 4,16)( 7,19, 8,20)(11,23,12,24)$
12C-1 $12,4^{3}$ $2$ $12$ $20$ $( 1,14, 2,13)( 3, 8,12,15,20,24, 4, 7,11,16,19,23)( 5,17, 6,18)( 9,22,10,21)$
12C5 $12,4^{3}$ $2$ $12$ $20$ $( 1,13, 2,14)( 3, 7,12,16,20,23, 4, 8,11,15,19,24)( 5,18, 6,17)( 9,21,10,22)$
12C-5 $12,4^{3}$ $2$ $12$ $20$ $( 1,22,17,13, 9, 5, 2,21,18,14,10, 6)( 3,16, 4,15)( 7,20, 8,19)(11,24,12,23)$
12D1 $12^{2}$ $3$ $12$ $22$ $( 1, 7,17,24, 9,15, 2, 8,18,23,10,16)( 3,13,19, 5,11,21, 4,14,20, 6,12,22)$
12D-1 $12^{2}$ $3$ $12$ $22$ $( 1, 8,10,15,18,24, 2, 7, 9,16,17,23)( 3, 5,12,13,20,22, 4, 6,11,14,19,21)$
12D5 $12^{2}$ $3$ $12$ $22$ $( 1,15,10,24,18, 7, 2,16, 9,23,17, 8)( 3,21,12, 5,20,13, 4,22,11, 6,19,14)$
12D-5 $12^{2}$ $3$ $12$ $22$ $( 1,24,17,15, 9, 8, 2,23,18,16,10, 7)( 3,22,19,13,11, 5, 4,21,20,14,12, 6)$

Malle's constant $a(G)$:     $1/8$

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Character table

36 x 36 character table

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Regular extensions

Data not computed