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Properties

Label 42T13
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, 13);
 

Group action invariants

Degree $n$:  $42$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $13$
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);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,27,7,34,13,39,19,4,25,10,31,16,37,21)(2,28,8,33,14,40,20,3,26,9,32,15,38,22)(5,30,11,36,17,41,23,6,29,12,35,18,42,24), (3,5)(4,6)(7,37)(8,38)(9,42)(10,41)(11,40)(12,39)(13,31)(14,32)(15,35)(16,36)(17,33)(18,34)(19,25)(20,26)(21,30)(22,29)(23,28)(24,27)
magma: Generators(G);
 

Low degree resolvents

|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: $D_{6}$

Degree 7: $D_{7}$

Degree 14: $D_{14}$

Degree 21: 21T8

Low degree siblings

21T8, 42T14, 42T15

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

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

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:    
      2  2  2  2  2  1  1  1   1   .  1   1   .  1   1   .
      3  1  .  1  .  1  1  1   .   1  1   .   1  1   .   1
      7  1  .  .  1  1  .  1   1   1  1   1   1  1   1   1

        1a 2a 2b 2c 3a 6a 7a 14a 21a 7b 14b 21b 7c 14c 21c
     2P 1a 1a 1a 1a 3a 3a 7b  7b 21b 7c  7c 21c 7a  7a 21a
     3P 1a 2a 2b 2c 1a 2b 7c 14c  7c 7a 14a  7a 7b 14b  7b
     5P 1a 2a 2b 2c 3a 6a 7b 14b 21b 7c 14c 21c 7a 14a 21a
     7P 1a 2a 2b 2c 3a 6a 1a  2c  3a 1a  2c  3a 1a  2c  3a
    11P 1a 2a 2b 2c 3a 6a 7c 14c 21c 7a 14a 21a 7b 14b 21b
    13P 1a 2a 2b 2c 3a 6a 7a 14a 21a 7b 14b 21b 7c 14c 21c
    17P 1a 2a 2b 2c 3a 6a 7c 14c 21c 7a 14a 21a 7b 14b 21b
    19P 1a 2a 2b 2c 3a 6a 7b 14b 21b 7c 14c 21c 7a 14a 21a

X.1      1  1  1  1  1  1  1   1   1  1   1   1  1   1   1
X.2      1 -1 -1  1  1 -1  1   1   1  1   1   1  1   1   1
X.3      1 -1  1 -1  1  1  1  -1   1  1  -1   1  1  -1   1
X.4      1  1 -1 -1  1 -1  1  -1   1  1  -1   1  1  -1   1
X.5      2  . -2  . -1  1  2   .  -1  2   .  -1  2   .  -1
X.6      2  .  2  . -1 -1  2   .  -1  2   .  -1  2   .  -1
X.7      2  .  . -2  2  .  A  -A   A  C  -C   C  B  -B   B
X.8      2  .  . -2  2  .  B  -B   B  A  -A   A  C  -C   C
X.9      2  .  . -2  2  .  C  -C   C  B  -B   B  A  -A   A
X.10     2  .  .  2  2  .  A   A   A  C   C   C  B   B   B
X.11     2  .  .  2  2  .  B   B   B  A   A   A  C   C   C
X.12     2  .  .  2  2  .  C   C   C  B   B   B  A   A   A
X.13     4  .  .  . -2  .  D   .  -B  F   .  -A  E   .  -C
X.14     4  .  .  . -2  .  E   .  -C  D   .  -B  F   .  -A
X.15     4  .  .  . -2  .  F   .  -A  E   .  -C  D   .  -B

A = E(7)^3+E(7)^4
B = E(7)^2+E(7)^5
C = E(7)+E(7)^6
D = 2*E(7)^2+2*E(7)^5
E = 2*E(7)+2*E(7)^6
F = 2*E(7)^3+2*E(7)^4

magma: CharacterTable(G);