Properties

Label 42T15
Order \(84\)
n \(42\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $S_3\times D_7$

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Group action invariants

Degree $n$ :  $42$
Transitive number $t$ :  $15$
Group :  $S_3\times D_7$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,27,5,25,4,29)(2,28,6,26,3,30)(7,22,11,20,10,23)(8,21,12,19,9,24)(13,15,17,14,16,18)(31,39,35,37,34,42)(32,40,36,38,33,41), (1,17,26,41,8,23,31,5,13,30,38,12,20,35)(2,18,25,42,7,24,32,6,14,29,37,11,19,36)(3,15,27,39,10,21,33)(4,16,28,40,9,22,34)
$|\Aut(F/K)|$:  $14$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$
12:  $D_{6}$
14:  $D_{7}$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 6: $D_{6}$

Degree 7: $D_{7}$

Degree 14: $D_{7}$

Degree 21: 21T8

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

Group invariants

Order:  $84=2^{2} \cdot 3 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [84, 8]
Character table:   
      2  2  2  2  2  1  1  1   1   .  1   1   .  1   1   .
      3  1  .  1  .  1  1  1   .   1  1   .   1  1   .   1
      7  1  1  .  .  .  1  1   1   1  1   1   1  1   1   1

        1a 2a 2b 2c 6a 3a 7a 14a 21a 7b 14b 21b 7c 14c 21c
     2P 1a 1a 1a 1a 3a 3a 7b  7b 21b 7c  7c 21c 7a  7a 21a
     3P 1a 2a 2b 2c 2b 1a 7c 14c  7c 7a 14a  7a 7b 14b  7b
     5P 1a 2a 2b 2c 6a 3a 7b 14b 21b 7c 14c 21c 7a 14a 21a
     7P 1a 2a 2b 2c 6a 3a 1a  2a  3a 1a  2a  3a 1a  2a  3a
    11P 1a 2a 2b 2c 6a 3a 7c 14c 21c 7a 14a 21a 7b 14b 21b
    13P 1a 2a 2b 2c 6a 3a 7a 14a 21a 7b 14b 21b 7c 14c 21c
    17P 1a 2a 2b 2c 6a 3a 7c 14c 21c 7a 14a 21a 7b 14b 21b
    19P 1a 2a 2b 2c 6a 3a 7b 14b 21b 7c 14c 21c 7a 14a 21a

X.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
X.3      1 -1  1 -1  1  1  1  -1   1  1  -1   1  1  -1   1
X.4      1  1 -1 -1 -1  1  1   1   1  1   1   1  1   1   1
X.5      2  . -2  .  1 -1  2   .  -1  2   .  -1  2   .  -1
X.6      2  .  2  . -1 -1  2   .  -1  2   .  -1  2   .  -1
X.7      2 -2  .  .  .  2  A  -A   A  C  -C   C  B  -B   B
X.8      2 -2  .  .  .  2  B  -B   B  A  -A   A  C  -C   C
X.9      2 -2  .  .  .  2  C  -C   C  B  -B   B  A  -A   A
X.10     2  2  .  .  .  2  A   A   A  C   C   C  B   B   B
X.11     2  2  .  .  .  2  B   B   B  A   A   A  C   C   C
X.12     2  2  .  .  .  2  C   C   C  B   B   B  A   A   A
X.13     4  .  .  .  . -2  D   .  -B  F   .  -A  E   .  -C
X.14     4  .  .  .  . -2  E   .  -C  D   .  -B  F   .  -A
X.15     4  .  .  .  . -2  F   .  -A  E   .  -C  D   .  -B

A = E(7)^3+E(7)^4
B = E(7)^2+E(7)^5
C = E(7)+E(7)^6
D = 2*E(7)^2+2*E(7)^5
E = 2*E(7)+2*E(7)^6
F = 2*E(7)^3+2*E(7)^4