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Properties

Label 42T14
Degree $42$
Order $84$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3\times D_7$

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Show commands: Magma

magma: G := TransitiveGroup(42, 14);
 

Group action invariants

Degree $n$:  $42$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $14$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_3\times D_7$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $6$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,25,8,32,13,37,20,2,26,7,31,14,38,19)(3,29,10,36,15,42,21,6,27,11,33,18,39,24)(4,30,9,35,16,41,22,5,28,12,34,17,40,23), (1,6)(2,5)(3,4)(7,41)(8,42)(9,39)(10,40)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,25)(24,26)
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$
$12$:  $D_{6}$
$14$:  $D_{7}$
$28$:  $D_{14}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 6: $S_3$

Degree 7: $D_{7}$

Degree 14: $D_{14}$

Degree 21: 21T8

Low degree siblings

21T8, 42T13, 42T15

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{42}$ $1$ $1$ $0$ $()$
2A $2^{21}$ $3$ $2$ $21$ $( 1, 2)( 3, 6)( 4, 5)( 7, 8)( 9,12)(10,11)(13,14)(15,18)(16,17)(19,20)(21,24)(22,23)(25,26)(27,29)(28,30)(31,32)(33,36)(34,35)(37,38)(39,42)(40,41)$
2B $2^{18},1^{6}$ $7$ $2$ $18$ $( 7,37)( 8,38)( 9,40)(10,39)(11,42)(12,41)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,25)(20,26)(21,27)(22,28)(23,30)(24,29)$
2C $2^{21}$ $21$ $2$ $21$ $( 1, 2)( 3, 6)( 4, 5)( 7,38)( 8,37)( 9,41)(10,42)(11,39)(12,40)(13,32)(14,31)(15,36)(16,35)(17,34)(18,33)(19,26)(20,25)(21,29)(22,30)(23,28)(24,27)$
3A $3^{14}$ $2$ $3$ $28$ $( 1, 3, 5)( 2, 4, 6)( 7, 9,11)( 8,10,12)(13,15,17)(14,16,18)(19,22,24)(20,21,23)(25,28,29)(26,27,30)(31,33,35)(32,34,36)(37,40,42)(38,39,41)$
6A $6^{6},3^{2}$ $14$ $6$ $34$ $( 1,10, 5, 8, 3,12)( 2, 9, 6, 7, 4,11)(13,39,17,38,15,41)(14,40,18,37,16,42)(19,34,24,32,22,36)(20,33,23,31,21,35)(25,28,29)(26,27,30)$
7A1 $7^{6}$ $2$ $7$ $36$ $( 1,31,20, 8,38,26,13)( 2,32,19, 7,37,25,14)( 3,33,21,10,39,27,15)( 4,34,22, 9,40,28,16)( 5,35,23,12,41,30,17)( 6,36,24,11,42,29,18)$
7A2 $7^{6}$ $2$ $7$ $36$ $( 1,26, 8,31,13,38,20)( 2,25, 7,32,14,37,19)( 3,27,10,33,15,39,21)( 4,28, 9,34,16,40,22)( 5,30,12,35,17,41,23)( 6,29,11,36,18,42,24)$
7A3 $7^{6}$ $2$ $7$ $36$ $( 1, 8,13,20,26,31,38)( 2, 7,14,19,25,32,37)( 3,10,15,21,27,33,39)( 4, 9,16,22,28,34,40)( 5,12,17,23,30,35,41)( 6,11,18,24,29,36,42)$
14A1 $14^{3}$ $6$ $14$ $39$ $( 1,25, 8,32,13,37,20, 2,26, 7,31,14,38,19)( 3,29,10,36,15,42,21, 6,27,11,33,18,39,24)( 4,30, 9,35,16,41,22, 5,28,12,34,17,40,23)$
14A3 $14^{3}$ $6$ $14$ $39$ $( 1,32,20, 7,38,25,13, 2,31,19, 8,37,26,14)( 3,36,21,11,39,29,15, 6,33,24,10,42,27,18)( 4,35,22,12,40,30,16, 5,34,23, 9,41,28,17)$
14A5 $14^{3}$ $6$ $14$ $39$ $( 1, 7,13,19,26,32,38, 2, 8,14,20,25,31,37)( 3,11,15,24,27,36,39, 6,10,18,21,29,33,42)( 4,12,16,23,28,35,40, 5, 9,17,22,30,34,41)$
21A1 $21^{2}$ $4$ $21$ $40$ $( 1,39,35,26,21,17, 8, 3,41,31,27,23,13,10, 5,38,33,30,20,15,12)( 2,40,36,25,22,18, 7, 4,42,32,28,24,14, 9, 6,37,34,29,19,16,11)$
21A2 $21^{2}$ $4$ $21$ $40$ $( 1,33,23, 8,39,30,13, 3,35,20,10,41,26,15, 5,31,21,12,38,27,17)( 2,34,24, 7,40,29,14, 4,36,19, 9,42,25,16, 6,32,22,11,37,28,18)$
21A4 $21^{2}$ $4$ $21$ $40$ $( 1,21,41,13,33,12,26, 3,23,38,15,35, 8,27, 5,20,39,17,31,10,30)( 2,22,42,14,34,11,25, 4,24,37,16,36, 7,28, 6,19,40,18,32, 9,29)$

Malle's constant $a(G)$:     $1/18$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $84=2^{2} \cdot 3 \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  84.8
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 6A 7A1 7A2 7A3 14A1 14A3 14A5 21A1 21A2 21A4
Size 1 3 7 21 2 14 2 2 2 6 6 6 4 4 4
2 P 1A 1A 1A 1A 3A 3A 7A3 7A1 7A2 7A1 7A3 7A2 21A2 21A4 21A1
3 P 1A 2A 2B 2C 1A 2B 7A1 7A2 7A3 14A3 14A5 14A1 7A3 7A1 7A2
7 P 1A 2A 2B 2C 3A 6A 1A 1A 1A 2A 2A 2A 3A 3A 3A
Type
84.8.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
84.8.1b R 1 −1 −1 1 1 −1 1 1 1 −1 −1 −1 1 1 1
84.8.1c R 1 −1 1 −1 1 1 1 1 1 −1 −1 −1 1 1 1
84.8.1d R 1 1 −1 −1 1 −1 1 1 1 1 1 1 1 1 1
84.8.2a R 2 0 2 0 −1 −1 2 2 2 0 0 0 −1 −1 −1
84.8.2b R 2 0 −2 0 −1 1 2 2 2 0 0 0 −1 −1 −1
84.8.2c1 R 2 2 0 0 2 0 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73
84.8.2c2 R 2 2 0 0 2 0 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72
84.8.2c3 R 2 2 0 0 2 0 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7
84.8.2d1 R 2 −2 0 0 2 0 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 −ζ7−2−ζ72 −ζ7−1−ζ7 −ζ7−3−ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73
84.8.2d2 R 2 −2 0 0 2 0 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 −ζ7−1−ζ7 −ζ7−3−ζ73 −ζ7−2−ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72
84.8.2d3 R 2 −2 0 0 2 0 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 −ζ7−3−ζ73 −ζ7−2−ζ72 −ζ7−1−ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7
84.8.4a1 R 4 0 0 0 −2 0 2ζ7−3+2ζ73 2ζ7−1+2ζ7 2ζ7−2+2ζ72 0 0 0 −ζ7−1−ζ7 −ζ7−2−ζ72 −ζ7−3−ζ73
84.8.4a2 R 4 0 0 0 −2 0 2ζ7−2+2ζ72 2ζ7−3+2ζ73 2ζ7−1+2ζ7 0 0 0 −ζ7−3−ζ73 −ζ7−1−ζ7 −ζ7−2−ζ72
84.8.4a3 R 4 0 0 0 −2 0 2ζ7−1+2ζ7 2ζ7−2+2ζ72 2ζ7−3+2ζ73 0 0 0 −ζ7−2−ζ72 −ζ7−3−ζ73 −ζ7−1−ζ7

magma: CharacterTable(G);