Properties

Label 42T26
Degree $42$
Order $168$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $F_8:C_3$

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Show commands: Magma

magma: G := TransitiveGroup(42, 26);
 

Group invariants

Abstract group:  $F_8:C_3$
magma: IdentifyGroup(G);
 
Order:  $168=2^{3} \cdot 3 \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
magma: NilpotencyClass(G);
 

Group action invariants

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

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$3$:  $C_3$
$21$:  $C_7:C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 6: None

Degree 7: $C_7:C_3$

Degree 14: 14T11

Degree 21: 21T2

Low degree siblings

8T36, 14T11, 24T283, 28T27

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

Malle's constant $a(G)$:     $1/12$

magma: ConjugacyClasses(G);
 

Character table

1A 2A 3A1 3A-1 6A1 6A-1 7A1 7A-1
Size 1 7 28 28 28 28 24 24
2 P 1A 1A 3A-1 3A1 3A1 3A-1 7A1 7A-1
3 P 1A 2A 1A 1A 2A 2A 7A-1 7A1
7 P 1A 2A 3A1 3A-1 6A1 6A-1 1A 1A
Type
168.43.1a R 1 1 1 1 1 1 1 1
168.43.1b1 C 1 1 ζ31 ζ3 ζ3 ζ31 1 1
168.43.1b2 C 1 1 ζ3 ζ31 ζ31 ζ3 1 1
168.43.3a1 C 3 3 0 0 0 0 ζ731ζ7ζ72 ζ73+ζ7+ζ72
168.43.3a2 C 3 3 0 0 0 0 ζ73+ζ7+ζ72 ζ731ζ7ζ72
168.43.7a R 7 1 1 1 1 1 0 0
168.43.7b1 C 7 1 ζ31 ζ3 ζ3 ζ31 0 0
168.43.7b2 C 7 1 ζ3 ζ31 ζ31 ζ3 0 0

magma: CharacterTable(G);
 

Regular extensions

Data not computed