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Properties

Label 42T5
Degree $42$
Order $42$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_{21}$

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Show commands: Magma

magma: G := TransitiveGroup(42, 5);
 

Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$6$:  $S_3$
$14$:  $D_{7}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 6: $S_3$

Degree 7: $D_{7}$

Degree 14: $D_{7}$

Degree 21: $D_{21}$

Low degree siblings

21T5

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

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:  42.5
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A 7A1 7A2 7A3 21A1 21A2 21A4 21A5 21A8 21A10
Size 1 21 2 2 2 2 2 2 2 2 2 2
2 P 1A 1A 3A 7A3 7A1 7A2 21A8 21A1 21A5 21A4 21A2 21A10
3 P 1A 2A 1A 7A1 7A2 7A3 7A3 7A3 7A1 7A2 7A1 7A2
7 P 1A 2A 3A 1A 1A 1A 3A 3A 3A 3A 3A 3A
Type
42.5.1a R 1 1 1 1 1 1 1 1 1 1 1 1
42.5.1b R 1 −1 1 1 1 1 1 1 1 1 1 1
42.5.2a R 2 0 −1 2 2 2 −1 −1 −1 −1 −1 −1
42.5.2b1 R 2 0 2 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73
42.5.2b2 R 2 0 2 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72
42.5.2b3 R 2 0 2 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7
42.5.2c1 R 2 0 −1 ζ21−9+ζ219 ζ21−3+ζ213 ζ21−6+ζ216 ζ21−10+ζ2110 ζ21−1+ζ21 ζ21−2+ζ212 ζ21−8+ζ218 ζ21−4+ζ214 ζ21−5+ζ215
42.5.2c2 R 2 0 −1 ζ21−9+ζ219 ζ21−3+ζ213 ζ21−6+ζ216 ζ21−4+ζ214 ζ21−8+ζ218 ζ21−5+ζ215 ζ21−1+ζ21 ζ21−10+ζ2110 ζ21−2+ζ212
42.5.2c3 R 2 0 −1 ζ21−6+ζ216 ζ21−9+ζ219 ζ21−3+ζ213 ζ21−5+ζ215 ζ21−10+ζ2110 ζ21−1+ζ21 ζ21−4+ζ214 ζ21−2+ζ212 ζ21−8+ζ218
42.5.2c4 R 2 0 −1 ζ21−6+ζ216 ζ21−9+ζ219 ζ21−3+ζ213 ζ21−2+ζ212 ζ21−4+ζ214 ζ21−8+ζ218 ζ21−10+ζ2110 ζ21−5+ζ215 ζ21−1+ζ21
42.5.2c5 R 2 0 −1 ζ21−3+ζ213 ζ21−6+ζ216 ζ21−9+ζ219 ζ21−8+ζ218 ζ21−5+ζ215 ζ21−10+ζ2110 ζ21−2+ζ212 ζ21−1+ζ21 ζ21−4+ζ214
42.5.2c6 R 2 0 −1 ζ21−3+ζ213 ζ21−6+ζ216 ζ21−9+ζ219 ζ21−1+ζ21 ζ21−2+ζ212 ζ21−4+ζ214 ζ21−5+ζ215 ζ21−8+ζ218 ζ21−10+ζ2110

magma: CharacterTable(G);