↑

Properties

Label 42T31
Degree $42$
Order $168$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_7:A_4$

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(42, 31);
 

Group action invariants

Degree $n$:  $42$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $31$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_7:A_4$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,9,19)(2,10,20)(3,12,23)(4,11,24)(5,7,21)(6,8,22)(13,34,30)(14,33,29)(15,32,26)(16,31,25)(17,35,28)(18,36,27)(37,39,41)(38,40,42), (1,13,33,38,28,11)(2,14,34,37,27,12)(3,18,35,40,26,7)(4,17,36,39,25,8)(5,15,32,42,29,10)(6,16,31,41,30,9)(19,24,22)(20,23,21)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$12$:  $A_4$
$24$:  $A_4\times C_2$
$42$:  $F_7$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 6: $A_4$

Degree 7: $F_7$

Degree 14: None

Degree 21: 21T4

Low degree siblings

28T28, 42T30

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

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

magma: ConjugacyClasses(G);
 

Group invariants

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
Label:  168.49
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A1 3A-1 6A1 6A-1 7A 14A1 14A3 14A5
Size 1 3 7 21 28 28 28 28 6 6 6 6
2 P 1A 1A 1A 1A 3A-1 3A1 3A1 3A-1 7A 7A 7A 7A
3 P 1A 2A 2B 2C 1A 1A 2B 2B 7A 14A3 14A5 14A1
7 P 1A 2A 2B 2C 3A1 3A-1 6A1 6A-1 1A 2A 2A 2A
Type
168.49.1a R 1 1 1 1 1 1 1 1 1 1 1 1
168.49.1b R 1 1 −1 −1 1 1 −1 −1 1 1 1 1
168.49.1c1 C 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 1 1
168.49.1c2 C 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 1 1
168.49.1d1 C 1 1 −1 −1 ζ3−1 ζ3 −ζ3 −ζ3−1 1 1 1 1
168.49.1d2 C 1 1 −1 −1 ζ3 ζ3−1 −ζ3−1 −ζ3 1 1 1 1
168.49.3a R 3 −1 3 −1 0 0 0 0 3 −1 −1 −1
168.49.3b R 3 −1 −3 1 0 0 0 0 3 −1 −1 −1
168.49.6a R 6 6 0 0 0 0 0 0 −1 −1 −1 −1
168.49.6b1 R 6 −2 0 0 0 0 0 0 −1 2ζ7−3+1+2ζ73 2ζ7−2+1+2ζ72 2ζ7−1+1+2ζ7
168.49.6b2 R 6 −2 0 0 0 0 0 0 −1 2ζ7−2+1+2ζ72 2ζ7−1+1+2ζ7 2ζ7−3+1+2ζ73
168.49.6b3 R 6 −2 0 0 0 0 0 0 −1 2ζ7−1+1+2ζ7 2ζ7−3+1+2ζ73 2ζ7−2+1+2ζ72

magma: CharacterTable(G);