Properties

Label 28T40
Order \(252\)
n \(28\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $A_4\times C_7:C_3$

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Group action invariants

Degree $n$ :  $28$
Transitive number $t$ :  $40$
Group :  $A_4\times C_7:C_3$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,21,13,5,25,17,9)(2,24,15,6,28,19,10,4,23,14,8,27,18,12,3,22,16,7,26,20,11), (1,28,19)(2,26,18)(3,25,20)(4,27,17)(5,8,7)(9,16,23)(10,14,22)(11,13,24)(12,15,21)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
3:  $C_3$ x 4
9:  $C_3^2$
12:  $A_4$
21:  $C_7:C_3$
36:  $C_3\times A_4$
63:  21T7

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: $A_4$

Degree 7: $C_7:C_3$

Degree 14: None

Low degree siblings

42T39

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

Group invariants

Order:  $252=2^{2} \cdot 3^{2} \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [252, 27]
Character table:   
      2  2  2  2  .  .  .  .  .  .  2   2   2  2   .   .   2  2   .   .   2
      3  2  2  2  2  2  2  2  2  2  1   1   1  1   1   1   .  1   1   1   .
      7  1  .  .  1  .  .  1  .  .  1   .   .  1   1   1   1  1   1   1   1

        1a 3a 3b 3c 3d 3e 3f 3g 3h 2a  6a  6b 7a 21a 21b 14a 7b 21c 21d 14b
     2P 1a 3b 3a 3f 3h 3g 3c 3e 3d 1a  3b  3a 7a 21b 21a  7a 7b 21d 21c  7b
     3P 1a 1a 1a 1a 1a 1a 1a 1a 1a 2a  2a  2a 7b  7b  7b 14b 7a  7a  7a 14a
     5P 1a 3b 3a 3f 3h 3g 3c 3e 3d 2a  6b  6a 7b 21d 21c 14b 7a 21b 21a 14a
     7P 1a 3a 3b 3c 3d 3e 3f 3g 3h 2a  6a  6b 1a  3c  3f  2a 1a  3c  3f  2a
    11P 1a 3b 3a 3f 3h 3g 3c 3e 3d 2a  6b  6a 7a 21b 21a 14a 7b 21d 21c 14b
    13P 1a 3a 3b 3c 3d 3e 3f 3g 3h 2a  6a  6b 7b 21c 21d 14b 7a 21a 21b 14a
    17P 1a 3b 3a 3f 3h 3g 3c 3e 3d 2a  6b  6a 7b 21d 21c 14b 7a 21b 21a 14a
    19P 1a 3a 3b 3c 3d 3e 3f 3g 3h 2a  6a  6b 7b 21c 21d 14b 7a 21a 21b 14a

X.1      1  1  1  1  1  1  1  1  1  1   1   1  1   1   1   1  1   1   1   1
X.2      1  1  1  A  A  A /A /A /A  1   1   1  1   A  /A   1  1   A  /A   1
X.3      1  1  1 /A /A /A  A  A  A  1   1   1  1  /A   A   1  1  /A   A   1
X.4      1  A /A  1  A /A  1  A /A  1   A  /A  1   1   1   1  1   1   1   1
X.5      1 /A  A  1 /A  A  1 /A  A  1  /A   A  1   1   1   1  1   1   1   1
X.6      1  A /A  A /A  1 /A  1  A  1   A  /A  1   A  /A   1  1   A  /A   1
X.7      1 /A  A /A  A  1  A  1 /A  1  /A   A  1  /A   A   1  1  /A   A   1
X.8      1  A /A /A  1  A  A /A  1  1   A  /A  1  /A   A   1  1  /A   A   1
X.9      1 /A  A  A  1 /A /A  A  1  1  /A   A  1   A  /A   1  1   A  /A   1
X.10     3  3  3  .  .  .  .  .  . -1  -1  -1  3   .   .  -1  3   .   .  -1
X.11     3  .  .  3  .  .  3  .  .  3   .   .  C   C   C   C /C  /C  /C  /C
X.12     3  .  .  3  .  .  3  .  .  3   .   . /C  /C  /C  /C  C   C   C   C
X.13     3  .  .  B  .  . /B  .  .  3   .   .  C   E   F   C /C  /F  /E  /C
X.14     3  .  . /B  .  .  B  .  .  3   .   .  C   F   E   C /C  /E  /F  /C
X.15     3  .  .  B  .  . /B  .  .  3   .   . /C  /F  /E  /C  C   E   F   C
X.16     3  .  . /B  .  .  B  .  .  3   .   . /C  /E  /F  /C  C   F   E   C
X.17     3  B /B  .  .  .  .  .  . -1  -A -/A  3   .   .  -1  3   .   .  -1
X.18     3 /B  B  .  .  .  .  .  . -1 -/A  -A  3   .   .  -1  3   .   .  -1
X.19     9  .  .  .  .  .  .  .  . -3   .   .  D   .   .  -C /D   .   . -/C
X.20     9  .  .  .  .  .  .  .  . -3   .   . /D   .   . -/C  D   .   .  -C

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = 3*E(3)^2
  = (-3-3*Sqrt(-3))/2 = -3-3b3
C = E(7)^3+E(7)^5+E(7)^6
  = (-1-Sqrt(-7))/2 = -1-b7
D = 3*E(7)^3+3*E(7)^5+3*E(7)^6
  = (-3-3*Sqrt(-7))/2 = -3-3b7
E = E(21)^2+E(21)^8+E(21)^11
F = E(21)+E(21)^4+E(21)^16