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Properties

Label 42T49
Degree $42$
Order $252$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{21}:D_6$

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Show commands: Magma

magma: G := TransitiveGroup(42, 49);
 

Group action invariants

Degree $n$:  $42$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $49$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_{21}:D_6$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $7$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,23,3,22,2,24)(4,19,5,20,6,21)(7,18,9,17,8,16)(10,15,12,13,11,14)(25,41,26,42,27,40)(28,37,29,39,30,38)(31,36,33,34,32,35), (1,19,37,13,31,7,26,2,20,38,14,32,9,25)(3,21,39,15,33,8,27)(4,22,40,16,34,12,30)(5,23,41,17,35,10,28,6,24,42,18,36,11,29)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$ x 2
$12$:  $D_{6}$ x 2
$14$:  $D_{7}$
$28$:  $D_{14}$
$36$:  $S_3^2$
$84$:  21T8 x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 6: $S_3^2$

Degree 7: $D_{7}$

Degree 14: $D_{7}$

Degree 21: None

Low degree siblings

There are no siblings 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $3$ $( 4, 5, 6)(10,12,11)(16,18,17)(22,24,23)(28,29,30)(34,35,36)(40,41,42)$
$ 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 $ $9$ $2$ $( 2, 3)( 5, 6)( 7, 8)(10,11)(13,15)(17,18)(19,21)(23,24)(25,27)(28,29)(32,33) (35,36)(38,39)(41,42)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,12,11)(13,15,14)(16,18,17)(19,21,20) (22,24,23)(25,27,26)(28,29,30)(31,32,33)(34,35,36)(37,38,39)(40,41,42)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 2, 3)( 4, 6, 5)( 7, 8, 9)(10,11,12)(13,15,14)(16,17,18)(19,21,20) (22,23,24)(25,27,26)(28,30,29)(31,32,33)(34,36,35)(37,38,39)(40,42,41)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $21$ $2$ $( 1, 4)( 2, 5)( 3, 6)( 7,41)( 8,42)( 9,40)(10,39)(11,38)(12,37)(13,35)(14,34) (15,36)(16,31)(17,33)(18,32)(19,28)(20,30)(21,29)(22,26)(23,27)(24,25)$
$ 6, 6, 6, 6, 6, 6, 6 $ $42$ $6$ $( 1, 4, 2, 5, 3, 6)( 7,41, 8,42, 9,40)(10,37,12,38,11,39)(13,35,15,36,14,34) (16,32,18,33,17,31)(19,28,21,29,20,30)(22,25,24,27,23,26)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $21$ $2$ $( 1, 4)( 2, 6)( 3, 5)( 7,42)( 8,41)( 9,40)(10,38)(11,39)(12,37)(13,36)(14,34) (15,35)(16,31)(17,32)(18,33)(19,29)(20,30)(21,28)(22,26)(23,25)(24,27)$
$ 6, 6, 6, 6, 6, 6, 6 $ $42$ $6$ $( 1, 4, 2, 6, 3, 5)( 7,42, 8,41, 9,40)(10,39,11,37,12,38)(13,36,15,35,14,34) (16,32,17,33,18,31)(19,29,21,28,20,30)(22,25,23,27,24,26)$
$ 21, 21 $ $4$ $21$ $( 1, 7,15,20,25,33,37, 2, 8,14,19,27,31,38, 3, 9,13,21,26,32,39) ( 4,10,18,22,29,35,40, 6,11,16,23,28,34,42, 5,12,17,24,30,36,41)$
$ 21, 21 $ $4$ $21$ $( 1, 7,15,20,25,33,37, 2, 8,14,19,27,31,38, 3, 9,13,21,26,32,39) ( 4,11,17,22,28,36,40, 5,10,16,24,29,34,41, 6,12,18,23,30,35,42)$
$ 21, 7, 7, 7 $ $4$ $21$ $( 1, 7,15,20,25,33,37, 2, 8,14,19,27,31,38, 3, 9,13,21,26,32,39) ( 4,12,16,22,30,34,40)( 5,11,18,24,28,35,41)( 6,10,17,23,29,36,42)$
$ 14, 14, 7, 7 $ $18$ $14$ $( 1, 7,14,19,26,32,37, 2, 9,13,20,25,31,38)( 3, 8,15,21,27,33,39) ( 4,10,16,23,30,36,40, 6,12,17,22,29,34,42)( 5,11,18,24,28,35,41)$
$ 21, 7, 7, 7 $ $4$ $21$ $( 1, 9,14,20,26,31,37)( 2, 7,13,19,25,32,38)( 3, 8,15,21,27,33,39) ( 4,10,18,22,29,35,40, 6,11,16,23,28,34,42, 5,12,17,24,30,36,41)$
$ 7, 7, 7, 7, 7, 7 $ $2$ $7$ $( 1, 9,14,20,26,31,37)( 2, 7,13,19,25,32,38)( 3, 8,15,21,27,33,39) ( 4,12,16,22,30,34,40)( 5,11,18,24,28,35,41)( 6,10,17,23,29,36,42)$
$ 14, 14, 7, 7 $ $18$ $14$ $( 1,13,26,38, 9,19,31, 2,14,25,37, 7,20,32)( 3,15,27,39, 8,21,33) ( 4,16,30,40,12,22,34)( 5,17,28,42,11,23,35, 6,18,29,41,10,24,36)$
$ 21, 7, 7, 7 $ $4$ $21$ $( 1,13,27,37, 7,21,31, 2,15,26,38, 8,20,32, 3,14,25,39, 9,19,33) ( 4,16,30,40,12,22,34)( 5,18,28,41,11,24,35)( 6,17,29,42,10,23,36)$
$ 21, 21 $ $4$ $21$ $( 1,13,27,37, 7,21,31, 2,15,26,38, 8,20,32, 3,14,25,39, 9,19,33) ( 4,17,28,40,10,24,34, 6,18,30,42,11,22,36, 5,16,29,41,12,23,35)$
$ 21, 21 $ $4$ $21$ $( 1,13,27,37, 7,21,31, 2,15,26,38, 8,20,32, 3,14,25,39, 9,19,33) ( 4,18,29,40,11,23,34, 5,17,30,41,10,22,35, 6,16,28,42,12,24,36)$
$ 7, 7, 7, 7, 7, 7 $ $2$ $7$ $( 1,14,26,37, 9,20,31)( 2,13,25,38, 7,19,32)( 3,15,27,39, 8,21,33) ( 4,16,30,40,12,22,34)( 5,18,28,41,11,24,35)( 6,17,29,42,10,23,36)$
$ 21, 7, 7, 7 $ $4$ $21$ $( 1,14,26,37, 9,20,31)( 2,13,25,38, 7,19,32)( 3,15,27,39, 8,21,33) ( 4,17,28,40,10,24,34, 6,18,30,42,11,22,36, 5,16,29,41,12,23,35)$
$ 14, 14, 7, 7 $ $18$ $14$ $( 1,19,37,13,31, 7,26, 2,20,38,14,32, 9,25)( 3,21,39,15,33, 8,27) ( 4,22,40,16,34,12,30)( 5,23,41,17,35,10,28, 6,24,42,18,36,11,29)$
$ 21, 7, 7, 7 $ $4$ $21$ $( 1,19,39,14,32, 8,26, 2,21,37,13,33, 9,25, 3,20,38,15,31, 7,27) ( 4,22,40,16,34,12,30)( 5,24,41,18,35,11,28)( 6,23,42,17,36,10,29)$
$ 21, 21 $ $4$ $21$ $( 1,19,39,14,32, 8,26, 2,21,37,13,33, 9,25, 3,20,38,15,31, 7,27) ( 4,23,41,16,36,11,30, 6,24,40,17,35,12,29, 5,22,42,18,34,10,28)$
$ 21, 21 $ $4$ $21$ $( 1,19,39,14,32, 8,26, 2,21,37,13,33, 9,25, 3,20,38,15,31, 7,27) ( 4,24,42,16,35,10,30, 5,23,40,18,36,12,28, 6,22,41,17,34,11,29)$
$ 7, 7, 7, 7, 7, 7 $ $2$ $7$ $( 1,20,37,14,31, 9,26)( 2,19,38,13,32, 7,25)( 3,21,39,15,33, 8,27) ( 4,22,40,16,34,12,30)( 5,24,41,18,35,11,28)( 6,23,42,17,36,10,29)$
$ 21, 7, 7, 7 $ $4$ $21$ $( 1,20,37,14,31, 9,26)( 2,19,38,13,32, 7,25)( 3,21,39,15,33, 8,27) ( 4,23,41,16,36,11,30, 6,24,40,17,35,12,29, 5,22,42,18,34,10,28)$

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:  252.37
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 3B 3C 6A 6B 7A1 7A2 7A3 14A1 14A3 14A5 21A1 21A2 21A4 21B1 21B2 21B4 21C1 21C-1 21C2 21C-2 21C4 21C-4
Size 1 9 21 21 2 2 4 42 42 2 2 2 18 18 18 4 4 4 4 4 4 4 4 4 4 4 4
2 P 1A 1A 1A 1A 3A 3B 3C 3A 3B 7A2 7A3 7A1 7A2 7A1 7A3 21A2 21B2 21C-1 21C1 21B4 21B1 21C2 21C4 21A1 21C-2 21A4 21C-4
3 P 1A 2A 2B 2C 1A 1A 1A 2B 2C 7A3 7A1 7A2 14A1 14A3 14A5 7A1 7A1 7A3 7A3 7A2 7A3 7A1 7A2 7A3 7A1 7A2 7A2
7 P 1A 2A 2B 2C 3A 3B 3C 6A 6B 1A 1A 1A 2A 2A 2A 3B 3A 3C 3C 3A 3A 3C 3C 3B 3C 3B 3C
Type
252.37.1a R 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
252.37.1b R 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
252.37.1c R 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
252.37.1d R 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
252.37.2a R 2 0 0 2 2 −1 −1 0 −1 2 2 2 0 0 0 −1 −1 −1 2 2 2 −1 −1 −1 −1 −1 −1
252.37.2b R 2 0 2 0 −1 2 −1 −1 0 2 2 2 0 0 0 2 2 2 −1 −1 −1 −1 −1 −1 −1 −1 −1
252.37.2c R 2 0 −2 0 −1 2 −1 1 0 2 2 2 0 0 0 2 2 2 −1 −1 −1 −1 −1 −1 −1 −1 −1
252.37.2d R 2 0 0 −2 2 −1 −1 0 1 2 2 2 0 0 0 −1 −1 −1 2 2 2 −1 −1 −1 −1 −1 −1
252.37.2e1 R 2 2 0 0 2 2 2 0 0 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73
252.37.2e2 R 2 2 0 0 2 2 2 0 0 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72
252.37.2e3 R 2 2 0 0 2 2 2 0 0 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7
252.37.2f1 R 2 −2 0 0 2 2 2 0 0 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 −ζ7−2−ζ72 −ζ7−1−ζ7 −ζ7−3−ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73
252.37.2f2 R 2 −2 0 0 2 2 2 0 0 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 −ζ7−1−ζ7 −ζ7−3−ζ73 −ζ7−2−ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72
252.37.2f3 R 2 −2 0 0 2 2 2 0 0 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 −ζ7−3−ζ73 −ζ7−2−ζ72 −ζ7−1−ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7
252.37.4a R 4 0 0 0 −2 −2 1 0 0 4 4 4 0 0 0 −2 −2 −2 −2 −2 −2 1 1 1 1 1 1
252.37.4b1 R 4 0 0 0 −2 4 −2 0 0 2ζ7−3+2ζ73 2ζ7−1+2ζ7 2ζ7−2+2ζ72 0 0 0 2ζ7−1+2ζ7 2ζ7−2+2ζ72 2ζ7−3+2ζ73 −ζ7−1−ζ7 −ζ7−2−ζ72 −ζ7−3−ζ73 −ζ7−1−ζ7 −ζ7−1−ζ7 −ζ7−2−ζ72 −ζ7−2−ζ72 −ζ7−3−ζ73 −ζ7−3−ζ73
252.37.4b2 R 4 0 0 0 −2 4 −2 0 0 2ζ7−2+2ζ72 2ζ7−3+2ζ73 2ζ7−1+2ζ7 0 0 0 2ζ7−3+2ζ73 2ζ7−1+2ζ7 2ζ7−2+2ζ72 −ζ7−3−ζ73 −ζ7−1−ζ7 −ζ7−2−ζ72 −ζ7−3−ζ73 −ζ7−3−ζ73 −ζ7−1−ζ7 −ζ7−1−ζ7 −ζ7−2−ζ72 −ζ7−2−ζ72
252.37.4b3 R 4 0 0 0 −2 4 −2 0 0 2ζ7−1+2ζ7 2ζ7−2+2ζ72 2ζ7−3+2ζ73 0 0 0 2ζ7−2+2ζ72 2ζ7−3+2ζ73 2ζ7−1+2ζ7 −ζ7−2−ζ72 −ζ7−3−ζ73 −ζ7−1−ζ7 −ζ7−2−ζ72 −ζ7−2−ζ72 −ζ7−3−ζ73 −ζ7−3−ζ73 −ζ7−1−ζ7 −ζ7−1−ζ7
252.37.4c1 R 4 0 0 0 4 −2 −2 0 0 2ζ7−3+2ζ73 2ζ7−1+2ζ7 2ζ7−2+2ζ72 0 0 0 −ζ7−1−ζ7 −ζ7−2−ζ72 −ζ7−3−ζ73 2ζ7−1+2ζ7 2ζ7−2+2ζ72 2ζ7−3+2ζ73 −ζ7−1−ζ7 −ζ7−1−ζ7 −ζ7−2−ζ72 −ζ7−2−ζ72 −ζ7−3−ζ73 −ζ7−3−ζ73
252.37.4c2 R 4 0 0 0 4 −2 −2 0 0 2ζ7−2+2ζ72 2ζ7−3+2ζ73 2ζ7−1+2ζ7 0 0 0 −ζ7−3−ζ73 −ζ7−1−ζ7 −ζ7−2−ζ72 2ζ7−3+2ζ73 2ζ7−1+2ζ7 2ζ7−2+2ζ72 −ζ7−3−ζ73 −ζ7−3−ζ73 −ζ7−1−ζ7 −ζ7−1−ζ7 −ζ7−2−ζ72 −ζ7−2−ζ72
252.37.4c3 R 4 0 0 0 4 −2 −2 0 0 2ζ7−1+2ζ7 2ζ7−2+2ζ72 2ζ7−3+2ζ73 0 0 0 −ζ7−2−ζ72 −ζ7−3−ζ73 −ζ7−1−ζ7 2ζ7−2+2ζ72 2ζ7−3+2ζ73 2ζ7−1+2ζ7 −ζ7−2−ζ72 −ζ7−2−ζ72 −ζ7−3−ζ73 −ζ7−3−ζ73 −ζ7−1−ζ7 −ζ7−1−ζ7
252.37.4d1 C 4 0 0 0 −2 −2 1 0 0 2ζ7−3+2ζ73 2ζ7−1+2ζ7 2ζ7−2+2ζ72 0 0 0 −ζ7−1−ζ7 −ζ7−2−ζ72 −ζ7−3−ζ73 −ζ7−1−ζ7 −ζ7−2−ζ72 −ζ7−3−ζ73 2ζ7−1−ζ7 −ζ7−1+2ζ7 2ζ7−2−ζ72 −ζ7−2+2ζ72 −ζ7−3+2ζ73 2ζ7−3−ζ73
252.37.4d2 C 4 0 0 0 −2 −2 1 0 0 2ζ7−3+2ζ73 2ζ7−1+2ζ7 2ζ7−2+2ζ72 0 0 0 −ζ7−1−ζ7 −ζ7−2−ζ72 −ζ7−3−ζ73 −ζ7−1−ζ7 −ζ7−2−ζ72 −ζ7−3−ζ73 −ζ7−1+2ζ7 2ζ7−1−ζ7 −ζ7−2+2ζ72 2ζ7−2−ζ72 2ζ7−3−ζ73 −ζ7−3+2ζ73
252.37.4d3 C 4 0 0 0 −2 −2 1 0 0 2ζ7−2+2ζ72 2ζ7−3+2ζ73 2ζ7−1+2ζ7 0 0 0 −ζ7−3−ζ73 −ζ7−1−ζ7 −ζ7−2−ζ72 −ζ7−3−ζ73 −ζ7−1−ζ7 −ζ7−2−ζ72 2ζ7−3−ζ73 −ζ7−3+2ζ73 −ζ7−1+2ζ7 2ζ7−1−ζ7 −ζ7−2+2ζ72 2ζ7−2−ζ72
252.37.4d4 C 4 0 0 0 −2 −2 1 0 0 2ζ7−2+2ζ72 2ζ7−3+2ζ73 2ζ7−1+2ζ7 0 0 0 −ζ7−3−ζ73 −ζ7−1−ζ7 −ζ7−2−ζ72 −ζ7−3−ζ73 −ζ7−1−ζ7 −ζ7−2−ζ72 −ζ7−3+2ζ73 2ζ7−3−ζ73 2ζ7−1−ζ7 −ζ7−1+2ζ7 2ζ7−2−ζ72 −ζ7−2+2ζ72
252.37.4d5 C 4 0 0 0 −2 −2 1 0 0 2ζ7−1+2ζ7 2ζ7−2+2ζ72 2ζ7−3+2ζ73 0 0 0 −ζ7−2−ζ72 −ζ7−3−ζ73 −ζ7−1−ζ7 −ζ7−2−ζ72 −ζ7−3−ζ73 −ζ7−1−ζ7 2ζ7−2−ζ72 −ζ7−2+2ζ72 −ζ7−3+2ζ73 2ζ7−3−ζ73 2ζ7−1−ζ7 −ζ7−1+2ζ7
252.37.4d6 C 4 0 0 0 −2 −2 1 0 0 2ζ7−1+2ζ7 2ζ7−2+2ζ72 2ζ7−3+2ζ73 0 0 0 −ζ7−2−ζ72 −ζ7−3−ζ73 −ζ7−1−ζ7 −ζ7−2−ζ72 −ζ7−3−ζ73 −ζ7−1−ζ7 −ζ7−2+2ζ72 2ζ7−2−ζ72 2ζ7−3−ζ73 −ζ7−3+2ζ73 −ζ7−1+2ζ7 2ζ7−1−ζ7

magma: CharacterTable(G);