Properties

Label 42T33
Order \(168\)
n \(42\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_7:S_4$

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Group action invariants

Degree $n$ :  $42$
Transitive number $t$ :  $33$
Group :  $C_7:S_4$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
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)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

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

Resolvents shown for degrees $\leq 10$

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

There are no siblings with degree $\leq 10$
Data on whether or not a number field with this Galois group has arithmetically equivalent fields has not been computed.

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

Group invariants

Order:  $168=2^{3} \cdot 3 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [168, 46]
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