Properties

Label 42T32
Order \(168\)
n \(42\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_7:S_4$

Learn more about

Group action invariants

Degree $n$ :  $42$
Transitive number $t$ :  $32$
Group :  $C_7:S_4$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,21,2,22)(3,19)(4,20)(5,24,6,23)(7,15,8,16)(9,13)(10,14)(11,17,12,18)(25,41,26,42)(27,40,28,39)(29,37)(30,38)(31,34,32,33)(35,36), (1,35,22,8,37,26,17,3,32,23,9,39,27,14,6,33,20,11,42,30,15)(2,36,21,7,38,25,18,4,31,24,10,40,28,13,5,34,19,12,41,29,16)
$|\Aut(F/K)|$:  $2$

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

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

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  .  -1  3   .   .   .   .  -1  3   .   .  -1  3
X.14     3 -1  1 -1  .  -1  3   .   .   .   .  -1  3   .   .  -1  3
X.15     6 -2  .  .  .  -A  D   .   .   .   .  -B  E   .   .  -C  F
X.16     6 -2  .  .  .  -B  E   .   .   .   .  -C  F   .   .  -A  D
X.17     6 -2  .  .  .  -C  F   .   .   .   .  -A  D   .   .  -B  E

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