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Properties

Label 42T33
Degree $42$
Order $168$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_7:S_4$

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Show commands: Magma

magma: G := TransitiveGroup(42, 33);
 

Group action invariants

Degree $n$:  $42$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $33$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_7:S_4$
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:  (1,25,10,33,15,38,23,6,29,7,31,14,42,21,4,27,11,36,18,39,20)(2,26,9,34,16,37,24,5,30,8,32,13,41,22,3,28,12,35,17,40,19), (1,39,2,40)(3,37)(4,38)(5,42,6,41)(7,31,8,32)(9,35)(10,36)(11,34,12,33)(13,30)(14,29)(15,28,16,27)(17,26,18,25)(21,24,22,23)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$6$:  $S_3$
$14$:  $D_{7}$
$24$:  $S_4$
$42$:  $D_{21}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 6: $S_4$

Degree 7: $D_{7}$

Degree 14: None

Degree 21: $D_{21}$

Low degree siblings

28T30, 42T32

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $168=2^{3} \cdot 3 \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  168.46
magma: IdentifyGroup(G);
 
Character table:    
      2  3  3  2  2  .  2   2   .   .   .   .   2  2   .   .  2   2
      3  1  .  .  .  1  1   .   1   1   1   1   .  1   1   1  1   .
      7  1  1  .  .  1  1   1   1   1   1   1   1  1   1   1  1   1

        1a 2a 4a 2b 3a 7a 14a 21a 21b 21c 21d 14b 7b 21e 21f 7c 14c
     2P 1a 1a 2a 1a 3a 7b  7b 21d 21c 21e 21f  7c 7c 21a 21b 7a  7a
     3P 1a 2a 4a 2b 1a 7c 14c  7c  7c  7a  7a 14a 7a  7b  7b 7b 14b
     5P 1a 2a 4a 2b 3a 7b 14b 21c 21d 21f 21e 14c 7c 21b 21a 7a 14a
     7P 1a 2a 4a 2b 3a 1a  2a  3a  3a  3a  3a  2a 1a  3a  3a 1a  2a
    11P 1a 2a 4a 2b 3a 7c 14c 21e 21f 21b 21a 14a 7a 21c 21d 7b 14b
    13P 1a 2a 4a 2b 3a 7a 14a 21b 21a 21d 21c 14b 7b 21f 21e 7c 14c
    17P 1a 2a 4a 2b 3a 7c 14c 21f 21e 21a 21b 14a 7a 21d 21c 7b 14b
    19P 1a 2a 4a 2b 3a 7b 14b 21d 21c 21e 21f 14c 7c 21a 21b 7a 14a

X.1      1  1  1  1  1  1   1   1   1   1   1   1  1   1   1  1   1
X.2      1  1 -1 -1  1  1   1   1   1   1   1   1  1   1   1  1   1
X.3      2  2  .  . -1  2   2  -1  -1  -1  -1   2  2  -1  -1  2   2
X.4      2  2  .  .  2  A   A   A   A   B   B   B  B   C   C  C   C
X.5      2  2  .  .  2  B   B   B   B   C   C   C  C   A   A  A   A
X.6      2  2  .  .  2  C   C   C   C   A   A   A  A   B   B  B   B
X.7      2  2  .  . -1  A   A   G   H   L   K   B  B   I   J  C   C
X.8      2  2  .  . -1  A   A   H   G   K   L   B  B   J   I  C   C
X.9      2  2  .  . -1  C   C   I   J   H   G   A  A   L   K  B   B
X.10     2  2  .  . -1  C   C   J   I   G   H   A  A   K   L  B   B
X.11     2  2  .  . -1  B   B   K   L   I   J   C  C   G   H  A   A
X.12     2  2  .  . -1  B   B   L   K   J   I   C  C   H   G  A   A
X.13     3 -1 -1  1  .  3  -1   .   .   .   .  -1  3   .   .  3  -1
X.14     3 -1  1 -1  .  3  -1   .   .   .   .  -1  3   .   .  3  -1
X.15     6 -2  .  .  .  D  -A   .   .   .   .  -B  E   .   .  F  -C
X.16     6 -2  .  .  .  E  -B   .   .   .   .  -C  F   .   .  D  -A
X.17     6 -2  .  .  .  F  -C   .   .   .   .  -A  D   .   .  E  -B

A = E(7)^3+E(7)^4
B = E(7)+E(7)^6
C = E(7)^2+E(7)^5
D = 3*E(7)^3+3*E(7)^4
E = 3*E(7)+3*E(7)^6
F = 3*E(7)^2+3*E(7)^5
G = E(21)^5+E(21)^16
H = E(21)^2+E(21)^19
I = E(21)^8+E(21)^13
J = E(21)+E(21)^20
K = E(21)^10+E(21)^11
L = E(21)^4+E(21)^17

magma: CharacterTable(G);