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Properties

Label 42T121
Degree $42$
Order $588$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_7:F_7$

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Show commands: Magma

magma: G := TransitiveGroup(42, 121);
 

Group action invariants

Degree $n$:  $42$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $121$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_7:F_7$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $3$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,3,2)(4,34,42,18,29,22)(5,35,40,16,30,23)(6,36,41,17,28,24)(7,21,14,39,26,31)(8,19,15,37,27,32)(9,20,13,38,25,33)(10,12,11), (1,28,19,35,9,17)(2,29,20,36,7,18)(3,30,21,34,8,16)(4,26,23,31,12,13)(5,27,24,32,10,14)(6,25,22,33,11,15)(37,42,39,41,38,40)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_2^2$
$6$:  $C_6$ x 3
$12$:  $C_6\times C_2$
$42$:  $F_7$ x 2
$84$:  $F_7 \times C_2$ x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 6: $C_6$

Degree 7: None

Degree 14: 14T24

Degree 21: None

Low degree siblings

14T24 x 3, 28T77 x 3, 42T121 x 2

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 $ $1$ $1$ $()$
$ 7, 7, 7, 7, 7, 7 $ $12$ $7$ $( 1,31,20, 8,37,25,14)( 2,32,21, 9,38,26,15)( 3,33,19, 7,39,27,13) ( 4,30,11,35,18,41,24)( 5,28,12,36,16,42,22)( 6,29,10,34,17,40,23)$
$ 7, 7, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $12$ $7$ $( 4,18,30,41,11,24,35)( 5,16,28,42,12,22,36)( 6,17,29,40,10,23,34)$
$ 7, 7, 7, 7, 7, 7 $ $12$ $7$ $( 1,31,20, 8,37,25,14)( 2,32,21, 9,38,26,15)( 3,33,19, 7,39,27,13) ( 4,41,35,30,24,18,11)( 5,42,36,28,22,16,12)( 6,40,34,29,23,17,10)$
$ 7, 7, 7, 7, 7, 7 $ $6$ $7$ $( 1,20,37,14,31, 8,25)( 2,21,38,15,32, 9,26)( 3,19,39,13,33, 7,27) ( 4,24,41,18,35,11,30)( 5,22,42,16,36,12,28)( 6,23,40,17,34,10,29)$
$ 7, 7, 7, 7, 7, 7 $ $6$ $7$ $( 1,14,25,37, 8,20,31)( 2,15,26,38, 9,21,32)( 3,13,27,39, 7,19,33) ( 4,35,24,11,41,30,18)( 5,36,22,12,42,28,16)( 6,34,23,10,40,29,17)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $49$ $3$ $( 1, 2, 3)( 4,42,29)( 5,40,30)( 6,41,28)( 7,14,26)( 8,15,27)( 9,13,25) (10,11,12)(16,23,35)(17,24,36)(18,22,34)(19,37,32)(20,38,33)(21,39,31)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $49$ $3$ $( 1, 3, 2)( 4,29,42)( 5,30,40)( 6,28,41)( 7,26,14)( 8,27,15)( 9,25,13) (10,12,11)(16,35,23)(17,36,24)(18,34,22)(19,32,37)(20,33,38)(21,31,39)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $49$ $2$ $( 7,39)( 8,37)( 9,38)(10,40)(11,41)(12,42)(13,33)(14,31)(15,32)(16,36)(17,34) (18,35)(19,27)(20,25)(21,26)(22,28)(23,29)(24,30)$
$ 6, 6, 6, 6, 6, 6, 3, 3 $ $49$ $6$ $( 1, 2, 3)( 4,42,10,30,36,23)( 5,40,11,28,34,24)( 6,41,12,29,35,22) ( 7,31,26,39,14,21)( 8,32,27,37,15,19)( 9,33,25,38,13,20)(16,17,18)$
$ 6, 6, 6, 6, 6, 6, 3, 3 $ $49$ $6$ $( 1, 3, 2)( 4,29,16,24,40,12)( 5,30,17,22,41,10)( 6,28,18,23,42,11) ( 7,21,14,39,26,31)( 8,19,15,37,27,32)( 9,20,13,38,25,33)(34,36,35)$
$ 6, 6, 6, 6, 6, 6, 6 $ $49$ $6$ $( 1,28,19,35, 9,17)( 2,29,20,36, 7,18)( 3,30,21,34, 8,16)( 4,26,23,31,12,13) ( 5,27,24,32,10,14)( 6,25,22,33,11,15)(37,42,39,41,38,40)$
$ 6, 6, 6, 6, 6, 6, 6 $ $49$ $6$ $( 1, 6, 9,24,19,16)( 2, 4, 7,22,20,17)( 3, 5, 8,23,21,18)(10,26,35,13,42,31) (11,27,36,14,40,32)(12,25,34,15,41,33)(28,37,29,38,30,39)$
$ 14, 14, 14 $ $42$ $14$ $( 1,41,20,24,37, 4,14,30,31,11, 8,35,25,18)( 2,42,21,22,38, 5,15,28,32,12, 9, 36,26,16)( 3,40,19,23,39, 6,13,29,33,10, 7,34,27,17)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $7$ $2$ $( 1,11)( 2,12)( 3,10)( 4, 8)( 5, 9)( 6, 7)(13,40)(14,41)(15,42)(16,38)(17,39) (18,37)(19,34)(20,35)(21,36)(22,32)(23,33)(24,31)(25,30)(26,28)(27,29)$
$ 6, 6, 6, 6, 6, 6, 6 $ $49$ $6$ $( 1,22,13, 4,21,17)( 2,23,14, 5,19,18)( 3,24,15, 6,20,16)( 7,35,38,10,31,42) ( 8,36,39,11,32,40)( 9,34,37,12,33,41)(25,28,27,30,26,29)$
$ 6, 6, 6, 6, 6, 6, 6 $ $49$ $6$ $( 1,34,32,30, 7,16)( 2,35,33,28, 8,17)( 3,36,31,29, 9,18)( 4,39,12,20,23,26) ( 5,37,10,21,24,27)( 6,38,11,19,22,25)(13,42,14,40,15,41)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $7$ $2$ $( 1,18)( 2,16)( 3,17)( 4,31)( 5,32)( 6,33)( 7,23)( 8,24)( 9,22)(10,39)(11,37) (12,38)(13,29)(14,30)(15,28)(19,34)(20,35)(21,36)(25,41)(26,42)(27,40)$
$ 14, 14, 14 $ $42$ $14$ $( 1,41,14,11,25,24,37,35, 8, 4,20,18,31,30)( 2,42,15,12,26,22,38,36, 9, 5,21, 16,32,28)( 3,40,13,10,27,23,39,34, 7, 6,19,17,33,29)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $588=2^{2} \cdot 3 \cdot 7^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  588.37
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A1 3A-1 6A1 6A-1 6B1 6B-1 6C1 6C-1 7A 7B 7C 7D 7E 14A 14B
Size 1 7 7 49 49 49 49 49 49 49 49 49 6 6 12 12 12 42 42
2 P 1A 1A 1A 1A 3A-1 3A1 3A1 3A-1 3A1 3A-1 3A-1 3A1 7A 7B 7E 7C 7D 7A 7B
3 P 1A 2A 2B 2C 1A 1A 2B 2B 2C 2C 2A 2A 7A 7B 7E 7C 7D 14A 14B
7 P 1A 2A 2B 2C 3A1 3A-1 6B-1 6B1 6C1 6C-1 6A-1 6A1 1A 1A 1A 1A 1A 2B 2A
Type
588.37.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
588.37.1b R 1 −1 −1 1 1 1 −1 −1 −1 −1 1 1 1 1 1 1 1 −1 −1
588.37.1c R 1 −1 1 −1 1 1 −1 −1 1 1 −1 −1 1 1 1 1 1 1 −1
588.37.1d R 1 1 −1 −1 1 1 1 1 −1 −1 −1 −1 1 1 1 1 1 −1 1
588.37.1e1 C 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 1 1 1 1 1
588.37.1e2 C 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 1 1 1 1 1 1 1
588.37.1f1 C 1 −1 −1 1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3 ζ3 ζ3−1 1 1 1 1 1 −1 −1
588.37.1f2 C 1 −1 −1 1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 ζ3−1 ζ3 1 1 1 1 1 −1 −1
588.37.1g1 C 1 −1 1 −1 ζ3−1 ζ3 −ζ3 −ζ3−1 ζ3−1 ζ3 −ζ3 −ζ3−1 1 1 1 1 1 1 −1
588.37.1g2 C 1 −1 1 −1 ζ3 ζ3−1 −ζ3−1 −ζ3 ζ3 ζ3−1 −ζ3−1 −ζ3 1 1 1 1 1 1 −1
588.37.1h1 C 1 1 −1 −1 ζ3−1 ζ3 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 1 1 1 1 1 −1 1
588.37.1h2 C 1 1 −1 −1 ζ3 ζ3−1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3 1 1 1 1 1 −1 1
588.37.6a R 6 0 6 0 0 0 0 0 0 0 0 0 −1 6 −1 −1 −1 −1 0
588.37.6b R 6 6 0 0 0 0 0 0 0 0 0 0 6 −1 −1 −1 −1 0 −1
588.37.6c R 6 −6 0 0 0 0 0 0 0 0 0 0 6 −1 −1 −1 −1 0 1
588.37.6d R 6 0 −6 0 0 0 0 0 0 0 0 0 −1 6 −1 −1 −1 1 0
588.37.12a R 12 0 0 0 0 0 0 0 0 0 0 0 −2 −2 −2 −2 5 0 0
588.37.12b R 12 0 0 0 0 0 0 0 0 0 0 0 −2 −2 −2 5 −2 0 0
588.37.12c R 12 0 0 0 0 0 0 0 0 0 0 0 −2 −2 5 −2 −2 0 0

magma: CharacterTable(G);