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Properties

Label 42T8
Degree $42$
Order $84$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_7:A_4$

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Show commands: Magma

magma: G := TransitiveGroup(42, 8);
 

Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
$3$:  $C_3$
$12$:  $A_4$
$21$:  $C_7:C_3$

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: 21T2

Low degree siblings

28T16

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $84=2^{2} \cdot 3 \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  84.11
magma: IdentifyGroup(G);
 
Character table:    
      2  2  2  .  .  2   2   2   2   2   2  2   2
      3  1  .  1  1  .   .   .   .   .   .  .   .
      7  1  1  .  .  1   1   1   1   1   1  1   1

        1a 2a 3a 3b 7a 14a 14b 14c 14d 14e 7b 14f
     2P 1a 1a 3b 3a 7a  7a  7a  7a  7b  7b 7b  7b
     3P 1a 2a 1a 1a 7b 14f 14d 14e 14a 14b 7a 14c
     5P 1a 2a 3b 3a 7b 14d 14e 14f 14b 14c 7a 14a
     7P 1a 2a 3a 3b 1a  2a  2a  2a  2a  2a 1a  2a
    11P 1a 2a 3b 3a 7a 14b 14c 14a 14e 14f 7b 14d
    13P 1a 2a 3a 3b 7b 14e 14f 14d 14c 14a 7a 14b

X.1      1  1  1  1  1   1   1   1   1   1  1   1
X.2      1  1  A /A  1   1   1   1   1   1  1   1
X.3      1  1 /A  A  1   1   1   1   1   1  1   1
X.4      3 -1  .  .  3  -1  -1  -1  -1  -1  3  -1
X.5      3  3  .  .  B   B   B   B  /B  /B /B  /B
X.6      3  3  .  . /B  /B  /B  /B   B   B  B   B
X.7      3 -1  .  .  B   C   D   E  /E  /C /B  /D
X.8      3 -1  .  .  B   D   E   C  /C  /D /B  /E
X.9      3 -1  .  .  B   E   C   D  /D  /E /B  /C
X.10     3 -1  .  . /B  /E  /C  /D   D   E  B   C
X.11     3 -1  .  . /B  /C  /D  /E   E   C  B   D
X.12     3 -1  .  . /B  /D  /E  /C   C   D  B   E

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = E(7)^3+E(7)^5+E(7)^6
  = (-1-Sqrt(-7))/2 = -1-b7
C = -E(7)^3-E(7)^5+E(7)^6
D = E(7)^3-E(7)^5-E(7)^6
E = -E(7)^3+E(7)^5-E(7)^6

magma: CharacterTable(G);