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Properties

Label 42T3
Degree $42$
Order $42$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3\times D_7$

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Show commands: Magma

magma: G := TransitiveGroup(42, 3);
 

Group action invariants

Degree $n$:  $42$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $3$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3\times D_7$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $42$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,32,4,34,6,36)(2,31,3,33,5,35)(7,25,9,27,11,29)(8,26,10,28,12,30)(13,24,16,19,17,22)(14,23,15,20,18,21)(37,42,40,38,41,39), (1,11,4,7,6,9)(2,12,3,8,5,10)(13,41,16,37,17,40)(14,42,15,38,18,39)(19,34,22,36,24,32)(20,33,21,35,23,31)(25,30,27,26,29,28)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$14$:  $D_{7}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 6: $C_6$

Degree 7: $D_{7}$

Degree 14: $D_{7}$

Degree 21: 21T3

Low degree siblings

21T3

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $42=2 \cdot 3 \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  42.4
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A1 3A-1 6A1 6A-1 7A1 7A2 7A3 21A1 21A-1 21A2 21A-2 21A4 21A-4
Size 1 7 1 1 7 7 2 2 2 2 2 2 2 2 2
2 P 1A 1A 3A-1 3A1 3A1 3A-1 7A1 7A2 7A3 21A2 21A-2 21A-4 21A4 21A1 21A-1
3 P 1A 2A 1A 1A 2A 2A 7A2 7A3 7A1 7A1 7A1 7A2 7A2 7A3 7A3
7 P 1A 2A 3A1 3A-1 6A1 6A-1 1A 1A 1A 3A-1 3A1 3A-1 3A1 3A1 3A-1
Type
42.4.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
42.4.1b R 1 −1 1 1 −1 −1 1 1 1 1 1 1 1 1 1
42.4.1c1 C 1 1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1
42.4.1c2 C 1 1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3
42.4.1d1 C 1 −1 ζ3−1 ζ3 −ζ3 −ζ3−1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1
42.4.1d2 C 1 −1 ζ3 ζ3−1 −ζ3−1 −ζ3 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3
42.4.2a1 R 2 0 2 2 0 0 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73
42.4.2a2 R 2 0 2 2 0 0 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72
42.4.2a3 R 2 0 2 2 0 0 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7
42.4.2b1 C 2 0 2ζ21−7 2ζ217 0 0 ζ21−9+ζ219 ζ21−3+ζ213 ζ21−6+ζ216 ζ214+ζ2110 ζ21−10−ζ213−ζ2110 −ζ21−10+1−ζ212+ζ213−ζ215+ζ217+ζ2110 ζ21−10−1+ζ21+ζ212−ζ213+ζ215−ζ216−ζ217+ζ218−ζ2110 −ζ21−10+1−ζ21−ζ212+ζ213−ζ214−ζ215+ζ216−ζ218 ζ212+ζ215
42.4.2b2 C 2 0 2ζ217 2ζ21−7 0 0 ζ21−9+ζ219 ζ21−3+ζ213 ζ21−6+ζ216 ζ21−10−ζ213−ζ2110 ζ214+ζ2110 ζ21−10−1+ζ21+ζ212−ζ213+ζ215−ζ216−ζ217+ζ218−ζ2110 −ζ21−10+1−ζ212+ζ213−ζ215+ζ217+ζ2110 ζ212+ζ215 −ζ21−10+1−ζ21−ζ212+ζ213−ζ214−ζ215+ζ216−ζ218
42.4.2b3 C 2 0 2ζ21−7 2ζ217 0 0 ζ21−6+ζ216 ζ21−9+ζ219 ζ21−3+ζ213 −ζ21−10+1−ζ21−ζ212+ζ213−ζ214−ζ215+ζ216−ζ218 ζ212+ζ215 ζ21−10−ζ213−ζ2110 ζ214+ζ2110 ζ21−10−1+ζ21+ζ212−ζ213+ζ215−ζ216−ζ217+ζ218−ζ2110 −ζ21−10+1−ζ212+ζ213−ζ215+ζ217+ζ2110
42.4.2b4 C 2 0 2ζ217 2ζ21−7 0 0 ζ21−6+ζ216 ζ21−9+ζ219 ζ21−3+ζ213 ζ212+ζ215 −ζ21−10+1−ζ21−ζ212+ζ213−ζ214−ζ215+ζ216−ζ218 ζ214+ζ2110 ζ21−10−ζ213−ζ2110 −ζ21−10+1−ζ212+ζ213−ζ215+ζ217+ζ2110 ζ21−10−1+ζ21+ζ212−ζ213+ζ215−ζ216−ζ217+ζ218−ζ2110
42.4.2b5 C 2 0 2ζ21−7 2ζ217 0 0 ζ21−3+ζ213 ζ21−6+ζ216 ζ21−9+ζ219 ζ21−10−1+ζ21+ζ212−ζ213+ζ215−ζ216−ζ217+ζ218−ζ2110 −ζ21−10+1−ζ212+ζ213−ζ215+ζ217+ζ2110 ζ212+ζ215 −ζ21−10+1−ζ21−ζ212+ζ213−ζ214−ζ215+ζ216−ζ218 ζ214+ζ2110 ζ21−10−ζ213−ζ2110
42.4.2b6 C 2 0 2ζ217 2ζ21−7 0 0 ζ21−3+ζ213 ζ21−6+ζ216 ζ21−9+ζ219 −ζ21−10+1−ζ212+ζ213−ζ215+ζ217+ζ2110 ζ21−10−1+ζ21+ζ212−ζ213+ζ215−ζ216−ζ217+ζ218−ζ2110 −ζ21−10+1−ζ21−ζ212+ζ213−ζ214−ζ215+ζ216−ζ218 ζ212+ζ215 ζ21−10−ζ213−ζ2110 ζ214+ζ2110

magma: CharacterTable(G);