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Properties

Label 42T45
Degree $42$
Order $252$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3\times F_7$

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Show commands: Magma

magma: G := TransitiveGroup(42, 45);
 

Group action invariants

Degree $n$:  $42$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $45$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_3\times F_7$
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:  (1,33,41,14,27,24)(2,34,42,13,28,23)(3,36,37,15,29,20)(4,35,38,16,30,19)(5,31,39,18,26,22)(6,32,40,17,25,21)(7,10,12)(8,9,11), (1,26,7,31,14,38,20,2,25,8,32,13,37,19)(3,30,10,35,15,42,21,5,27,11,33,18,40,23)(4,29,9,36,16,41,22,6,28,12,34,17,39,24)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_2^2$
$6$:  $S_3$, $C_6$ x 3
$12$:  $D_{6}$, $C_6\times C_2$
$18$:  $S_3\times C_3$
$36$:  $C_6\times S_3$
$42$:  $F_7$
$84$:  $F_7 \times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 6: $S_3$

Degree 7: $F_7$

Degree 14: $F_7 \times C_2$

Degree 21: $S_3\times F_7$

Low degree siblings

21T15, 42T43, 42T44, 42T52

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $252=2^{2} \cdot 3^{2} \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  252.26
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 3B1 3B-1 3C1 3C-1 6A1 6A-1 6B 6C1 6C-1 6D1 6D-1 6E1 6E-1 7A 14A 21A
Size 1 3 7 21 2 7 7 14 14 7 7 14 14 14 21 21 21 21 6 18 12
2 P 1A 1A 1A 1A 3A 3B-1 3B1 3C-1 3C1 3B1 3B-1 3A 3C1 3C-1 3B1 3B1 3B-1 3B-1 7A 7A 21A
3 P 1A 2A 2B 2C 1A 1A 1A 1A 1A 2B 2B 2B 2B 2B 2C 2A 2A 2C 7A 14A 7A
7 P 1A 2A 2B 2C 3A 3B1 3B-1 3C1 3C-1 6A1 6A-1 6B 6C1 6C-1 6D1 6E1 6E-1 6D-1 1A 2A 3A
Type
252.26.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
252.26.1b R 1 −1 −1 1 1 1 1 1 1 −1 −1 −1 −1 −1 1 1 −1 −1 1 −1 1
252.26.1c R 1 −1 1 −1 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 1 −1 1
252.26.1d R 1 1 −1 −1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 1 1 1 1 1
252.26.1e1 C 1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 1 1 1
252.26.1e2 C 1 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 1 1 1
252.26.1f1 C 1 −1 −1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 −ζ3 −ζ3−1 −1 −ζ3−1 −ζ3 ζ3 ζ3−1 −ζ3 −ζ3−1 1 −1 1
252.26.1f2 C 1 −1 −1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 −ζ3−1 −ζ3 −1 −ζ3 −ζ3−1 ζ3−1 ζ3 −ζ3−1 −ζ3 1 −1 1
252.26.1g1 C 1 −1 1 −1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3 −ζ3−1 1 −1 1
252.26.1g2 C 1 −1 1 −1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 1 −1 1
252.26.1h1 C 1 1 −1 −1 1 ζ3−1 ζ3 ζ3 ζ3−1 −ζ3 −ζ3−1 −1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 ζ3 ζ3−1 1 1 1
252.26.1h2 C 1 1 −1 −1 1 ζ3 ζ3−1 ζ3−1 ζ3 −ζ3−1 −ζ3 −1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 ζ3−1 ζ3 1 1 1
252.26.2a R 2 0 2 0 −1 2 2 −1 −1 2 2 −1 −1 −1 0 0 0 0 2 0 −1
252.26.2b R 2 0 −2 0 −1 2 2 −1 −1 −2 −2 1 1 1 0 0 0 0 2 0 −1
252.26.2c1 C 2 0 2 0 −1 2ζ3−1 2ζ3 −ζ3 −ζ3−1 2ζ3 2ζ3−1 −1 −ζ3−1 −ζ3 0 0 0 0 2 0 −1
252.26.2c2 C 2 0 2 0 −1 2ζ3 2ζ3−1 −ζ3−1 −ζ3 2ζ3−1 2ζ3 −1 −ζ3 −ζ3−1 0 0 0 0 2 0 −1
252.26.2d1 C 2 0 −2 0 −1 2ζ3−1 2ζ3 −ζ3 −ζ3−1 −2ζ3 −2ζ3−1 1 ζ3−1 ζ3 0 0 0 0 2 0 −1
252.26.2d2 C 2 0 −2 0 −1 2ζ3 2ζ3−1 −ζ3−1 −ζ3 −2ζ3−1 −2ζ3 1 ζ3 ζ3−1 0 0 0 0 2 0 −1
252.26.6a R 6 6 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 −1 −1
252.26.6b R 6 −6 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 1 −1
252.26.12a R 12 0 0 0 −6 0 0 0 0 0 0 0 0 0 0 0 0 0 −2 0 1

magma: CharacterTable(G);