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Properties

Label 42T39
Degree $42$
Order $252$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $A_4\times C_7:C_3$

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Show commands: Magma

magma: G := TransitiveGroup(42, 39);
 

Group action invariants

Degree $n$:  $42$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $39$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $A_4\times C_7:C_3$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,14,25,39,9,23,36,3,16,28,41,12,20,32,5,18,30,38,7,21,33)(2,13,26,40,10,24,35,4,15,27,42,11,19,31,6,17,29,37,8,22,34), (1,21,15)(2,22,16)(3,24,18)(4,23,17)(5,19,13)(6,20,14)(7,32,37)(8,31,38)(9,34,39)(10,33,40)(11,36,41)(12,35,42)(25,27,29)(26,28,30)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$3$:  $C_3$ x 4
$9$:  $C_3^2$
$12$:  $A_4$
$21$:  $C_7:C_3$
$36$:  $C_3\times A_4$
$63$:  21T7

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 6: $A_4$

Degree 7: $C_7:C_3$

Degree 14: None

Degree 21: 21T7

Low degree siblings

28T40

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

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:  252.27
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A1 3A-1 3B1 3B-1 3C1 3C-1 3D1 3D-1 6A1 6A-1 7A1 7A-1 14A1 14A-1 21A1 21A-1 21A2 21A-2
Size 1 3 4 4 7 7 28 28 28 28 21 21 3 3 9 9 12 12 12 12
2 P 1A 1A 3A-1 3A1 3B-1 3B1 3C-1 3C1 3D-1 3D1 3B1 3B-1 7A1 7A-1 7A-1 7A1 21A2 21A1 21A-1 21A-2
3 P 1A 2A 1A 1A 1A 1A 1A 1A 1A 1A 2A 2A 7A-1 7A1 14A-1 14A1 7A1 7A1 7A-1 7A-1
7 P 1A 2A 3A1 3A-1 3B1 3B-1 3C1 3C-1 3D1 3D-1 6A1 6A-1 1A 1A 2A 2A 3A1 3A-1 3A1 3A-1
Type
252.27.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
252.27.1b1 C 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 1 1 ζ3−1 ζ3 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1
252.27.1b2 C 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 1 1 ζ3 ζ3−1 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3
252.27.1c1 C 1 1 ζ3−1 ζ3 ζ3−1 ζ3 1 1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1
252.27.1c2 C 1 1 ζ3 ζ3−1 ζ3 ζ3−1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3
252.27.1d1 C 1 1 ζ3−1 ζ3 1 1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1
252.27.1d2 C 1 1 ζ3 ζ3−1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3
252.27.1e1 C 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 1 1 1 1 1 1 1 1
252.27.1e2 C 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 1 1 1 1 1 1 1 1
252.27.3a R 3 −1 0 0 3 3 0 0 0 0 −1 −1 3 3 −1 −1 0 0 0 0
252.27.3b1 C 3 3 3 3 0 0 0 0 0 0 0 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72
252.27.3b2 C 3 3 3 3 0 0 0 0 0 0 0 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72
252.27.3c1 C 3 −1 0 0 3ζ3−1 3ζ3 0 0 0 0 −ζ3 −ζ3−1 3 3 −1 −1 0 0 0 0
252.27.3c2 C 3 −1 0 0 3ζ3 3ζ3−1 0 0 0 0 −ζ3−1 −ζ3 3 3 −1 −1 0 0 0 0
252.27.3d1 C 3 3 3ζ21−7 3ζ217 0 0 0 0 0 0 0 0 −ζ21−10−ζ21−ζ214−ζ218+ζ219 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 −ζ21−10−ζ21−ζ214−ζ218+ζ219 −ζ21−10+1−ζ212+ζ217−ζ218 ζ21−ζ212+ζ214−ζ219 −ζ21+ζ212−ζ214−ζ217+ζ219 ζ21−10+ζ212+ζ218
252.27.3d2 C 3 3 3ζ217 3ζ21−7 0 0 0 0 0 0 0 0 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 −ζ21−10−ζ21−ζ214−ζ218+ζ219 −ζ21−10−ζ21−ζ214−ζ218+ζ219 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 ζ21−ζ212+ζ214−ζ219 −ζ21−10+1−ζ212+ζ217−ζ218 ζ21−10+ζ212+ζ218 −ζ21+ζ212−ζ214−ζ217+ζ219
252.27.3d3 C 3 3 3ζ21−7 3ζ217 0 0 0 0 0 0 0 0 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 −ζ21−10−ζ21−ζ214−ζ218+ζ219 −ζ21−10−ζ21−ζ214−ζ218+ζ219 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 ζ21−10+ζ212+ζ218 −ζ21+ζ212−ζ214−ζ217+ζ219 ζ21−ζ212+ζ214−ζ219 −ζ21−10+1−ζ212+ζ217−ζ218
252.27.3d4 C 3 3 3ζ217 3ζ21−7 0 0 0 0 0 0 0 0 −ζ21−10−ζ21−ζ214−ζ218+ζ219 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 −ζ21−10−ζ21−ζ214−ζ218+ζ219 −ζ21+ζ212−ζ214−ζ217+ζ219 ζ21−10+ζ212+ζ218 −ζ21−10+1−ζ212+ζ217−ζ218 ζ21−ζ212+ζ214−ζ219
252.27.9a1 C 9 −3 0 0 0 0 0 0 0 0 0 0 −3ζ7−3−3−3ζ7−3ζ72 3ζ7−3+3ζ7+3ζ72 −ζ7−3−ζ7−ζ72 ζ7−3+1+ζ7+ζ72 0 0 0 0
252.27.9a2 C 9 −3 0 0 0 0 0 0 0 0 0 0 3ζ7−3+3ζ7+3ζ72 −3ζ7−3−3−3ζ7−3ζ72 ζ7−3+1+ζ7+ζ72 −ζ7−3−ζ7−ζ72 0 0 0 0

magma: CharacterTable(G);