Properties

Label 28T30
Order \(168\)
n \(28\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_7:S_4$

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Group action invariants

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

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 4: $S_4$

Degree 7: $D_{7}$

Degree 14: None

Low degree siblings

42T32, 42T33

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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $42$ $2$ $( 3, 4)( 5,25)( 6,26)( 7,28)( 8,27)( 9,21)(10,22)(11,24)(12,23)(13,17)(14,18) (15,20)(16,19)$
$ 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1 $ $8$ $3$ $( 2, 3, 4)( 6, 7, 8)(10,11,12)(14,15,16)(18,19,20)(22,23,24)(26,27,28)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)$
$ 4, 4, 4, 4, 4, 4, 4 $ $42$ $4$ $( 1, 2, 3, 4)( 5,26, 7,28)( 6,27, 8,25)( 9,22,11,24)(10,23,12,21)(13,18,15,20) (14,19,16,17)$
$ 7, 7, 7, 7 $ $2$ $7$ $( 1, 5, 9,13,17,21,25)( 2, 6,10,14,18,22,26)( 3, 7,11,15,19,23,27) ( 4, 8,12,16,20,24,28)$
$ 21, 7 $ $8$ $21$ $( 1, 5, 9,13,17,21,25)( 2, 7,12,14,19,24,26, 3, 8,10,15,20,22,27, 4, 6,11,16, 18,23,28)$
$ 21, 7 $ $8$ $21$ $( 1, 5, 9,13,17,21,25)( 2, 8,11,14,20,23,26, 4, 7,10,16,19,22,28, 3, 6,12,15, 18,24,27)$
$ 14, 14 $ $6$ $14$ $( 1, 6, 9,14,17,22,25, 2, 5,10,13,18,21,26)( 3, 8,11,16,19,24,27, 4, 7,12,15, 20,23,28)$
$ 7, 7, 7, 7 $ $2$ $7$ $( 1, 9,17,25, 5,13,21)( 2,10,18,26, 6,14,22)( 3,11,19,27, 7,15,23) ( 4,12,20,28, 8,16,24)$
$ 21, 7 $ $8$ $21$ $( 1, 9,17,25, 5,13,21)( 2,11,20,26, 7,16,22, 3,12,18,27, 8,14,23, 4,10,19,28, 6,15,24)$
$ 21, 7 $ $8$ $21$ $( 1, 9,17,25, 5,13,21)( 2,12,19,26, 8,15,22, 4,11,18,28, 7,14,24, 3,10,20,27, 6,16,23)$
$ 14, 14 $ $6$ $14$ $( 1,10,17,26, 5,14,21, 2, 9,18,25, 6,13,22)( 3,12,19,28, 7,16,23, 4,11,20,27, 8,15,24)$
$ 7, 7, 7, 7 $ $2$ $7$ $( 1,13,25, 9,21, 5,17)( 2,14,26,10,22, 6,18)( 3,15,27,11,23, 7,19) ( 4,16,28,12,24, 8,20)$
$ 21, 7 $ $8$ $21$ $( 1,13,25, 9,21, 5,17)( 2,15,28,10,23, 8,18, 3,16,26,11,24, 6,19, 4,14,27,12, 22, 7,20)$
$ 21, 7 $ $8$ $21$ $( 1,13,25, 9,21, 5,17)( 2,16,27,10,24, 7,18, 4,15,26,12,23, 6,20, 3,14,28,11, 22, 8,19)$
$ 14, 14 $ $6$ $14$ $( 1,14,25,10,21, 6,17, 2,13,26, 9,22, 5,18)( 3,16,27,12,23, 8,19, 4,15,28,11, 24, 7,20)$

Group invariants

Order:  $168=2^{3} \cdot 3 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [168, 46]
Character table:   
      2  3  2  .  3  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 3a 2b 4a 7a 21a 21b 14a 7b 21c 21d 14b 7c 21e 21f 14c
     2P 1a 1a 3a 1a 2b 7b 21d 21c  7b 7c 21e 21f  7c 7a 21a 21b  7a
     3P 1a 2a 1a 2b 4a 7c  7c  7c 14c 7a  7a  7a 14a 7b  7b  7b 14b
     5P 1a 2a 3a 2b 4a 7b 21c 21d 14b 7c 21f 21e 14c 7a 21b 21a 14a
     7P 1a 2a 3a 2b 4a 1a  3a  3a  2b 1a  3a  3a  2b 1a  3a  3a  2b
    11P 1a 2a 3a 2b 4a 7c 21e 21f 14c 7a 21b 21a 14a 7b 21c 21d 14b
    13P 1a 2a 3a 2b 4a 7a 21b 21a 14a 7b 21d 21c 14b 7c 21f 21e 14c
    17P 1a 2a 3a 2b 4a 7c 21f 21e 14c 7a 21a 21b 14a 7b 21d 21c 14b
    19P 1a 2a 3a 2b 4a 7b 21d 21c 14b 7c 21e 21f 14c 7a 21a 21b 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  . -1  2  .  2  -1  -1   2  2  -1  -1   2  2  -1  -1   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  . -1  2  .  A   G   H   A  B   L   K   B  C   I   J   C
X.8      2  . -1  2  .  A   H   G   A  B   K   L   B  C   J   I   C
X.9      2  . -1  2  .  C   I   J   C  A   H   G   A  B   L   K   B
X.10     2  . -1  2  .  C   J   I   C  A   G   H   A  B   K   L   B
X.11     2  . -1  2  .  B   K   L   B  C   I   J   C  A   G   H   A
X.12     2  . -1  2  .  B   L   K   B  C   J   I   C  A   H   G   A
X.13     3 -1  . -1  1  3   .   .  -1  3   .   .  -1  3   .   .  -1
X.14     3  1  . -1 -1  3   .   .  -1  3   .   .  -1  3   .   .  -1
X.15     6  .  . -2  .  D   .   .  -A  E   .   .  -B  F   .   .  -C
X.16     6  .  . -2  .  E   .   .  -B  F   .   .  -C  D   .   .  -A
X.17     6  .  . -2  .  F   .   .  -C  D   .   .  -A  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