Properties

Label 28T29
Order \(168\)
n \(28\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $A_4\times D_7$

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Group action invariants

Degree $n$ :  $28$
Transitive number $t$ :  $29$
Group :  $A_4\times D_7$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,22,15,5,26,19,9,2,23,13,6,27,17,10,3,21,14,7,25,18,11)(4,24,16,8,28,20,12), (1,28,3,25,4,27)(2,26)(5,24,7,21,8,23)(6,22)(9,20,11,17,12,19)(10,18)(13,16,15)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $C_6$
12:  $A_4$
14:  $D_{7}$
24:  $A_4\times C_2$
42:  21T3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: $A_4$

Degree 7: $D_{7}$

Degree 14: None

Low degree siblings

42T28, 42T29

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, 1, 1, 1, 1 $ $7$ $2$ $( 5,25)( 6,26)( 7,27)( 8,28)( 9,21)(10,22)(11,23)(12,24)(13,17)(14,18)(15,19) (16,20)$
$ 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1 $ $4$ $3$ $( 2, 3, 4)( 6, 7, 8)(10,11,12)(14,15,16)(18,19,20)(22,23,24)(26,27,28)$
$ 6, 6, 6, 3, 2, 2, 2, 1 $ $28$ $6$ $( 2, 3, 4)( 5,25)( 6,27, 8,26, 7,28)( 9,21)(10,23,12,22,11,24)(13,17) (14,19,16,18,15,20)$
$ 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1 $ $4$ $3$ $( 2, 4, 3)( 6, 8, 7)(10,12,11)(14,16,15)(18,20,19)(22,24,23)(26,28,27)$
$ 6, 6, 6, 3, 2, 2, 2, 1 $ $28$ $6$ $( 2, 4, 3)( 5,25)( 6,28, 7,26, 8,27)( 9,21)(10,24,11,22,12,23)(13,17) (14,20,15,18,16,19)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $21$ $2$ $( 1, 2)( 3, 4)( 5,26)( 6,25)( 7,28)( 8,27)( 9,22)(10,21)(11,24)(12,23)(13,18) (14,17)(15,20)(16,19)$
$ 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, 48]
Character table:   
      2  3  3  1   1  1   1  3  3  2   .   .   2  2   .   .   2  2   .   .   2
      3  1  1  1   1  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   1

        1a 2a 3a  6a 3b  6b 2b 2c 7a 21a 21b 14a 7b 21c 21d 14b 7c 21e 21f 14c
     2P 1a 1a 3b  3b 3a  3a 1a 1a 7b 21d 21c  7b 7c 21f 21e  7c 7a 21b 21a  7a
     3P 1a 2a 1a  2a 1a  2a 2b 2c 7c  7c  7c 14c 7a  7a  7a 14a 7b  7b  7b 14b
     5P 1a 2a 3b  6b 3a  6a 2b 2c 7b 21d 21c 14b 7c 21f 21e 14c 7a 21b 21a 14a
     7P 1a 2a 3a  6a 3b  6b 2b 2c 1a  3a  3b  2b 1a  3a  3b  2b 1a  3a  3b  2b
    11P 1a 2a 3b  6b 3a  6a 2b 2c 7c 21f 21e 14c 7a 21b 21a 14a 7b 21d 21c 14b
    13P 1a 2a 3a  6a 3b  6b 2b 2c 7a 21a 21b 14a 7b 21c 21d 14b 7c 21e 21f 14c
    17P 1a 2a 3b  6b 3a  6a 2b 2c 7c 21f 21e 14c 7a 21b 21a 14a 7b 21d 21c 14b
    19P 1a 2a 3a  6a 3b  6b 2b 2c 7b 21c 21d 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   1   1   1
X.2      1 -1  1  -1  1  -1  1 -1  1   1   1   1  1   1   1   1  1   1   1   1
X.3      1 -1  A  -A /A -/A  1 -1  1   A  /A   1  1   A  /A   1  1   A  /A   1
X.4      1 -1 /A -/A  A  -A  1 -1  1  /A   A   1  1  /A   A   1  1  /A   A   1
X.5      1  1  A   A /A  /A  1  1  1   A  /A   1  1   A  /A   1  1   A  /A   1
X.6      1  1 /A  /A  A   A  1  1  1  /A   A   1  1  /A   A   1  1  /A   A   1
X.7      2  .  2   .  2   .  2  .  C   C   C   C  E   E   E   E  D   D   D   D
X.8      2  .  2   .  2   .  2  .  D   D   D   D  C   C   C   C  E   E   E   E
X.9      2  .  2   .  2   .  2  .  E   E   E   E  D   D   D   D  C   C   C   C
X.10     2  .  B   . /B   .  2  .  C   I  /I   C  E   K  /K   E  D   J  /J   D
X.11     2  . /B   .  B   .  2  .  C  /I   I   C  E  /K   K   E  D  /J   J   D
X.12     2  .  B   . /B   .  2  .  D   J  /J   D  C   I  /I   C  E   K  /K   E
X.13     2  . /B   .  B   .  2  .  D  /J   J   D  C  /I   I   C  E  /K   K   E
X.14     2  .  B   . /B   .  2  .  E   K  /K   E  D   J  /J   D  C   I  /I   C
X.15     2  . /B   .  B   .  2  .  E  /K   K   E  D  /J   J   D  C  /I   I   C
X.16     3 -3  .   .  .   . -1  1  3   .   .  -1  3   .   .  -1  3   .   .  -1
X.17     3  3  .   .  .   . -1 -1  3   .   .  -1  3   .   .  -1  3   .   .  -1
X.18     6  .  .   .  .   . -2  .  F   .   .  -C  H   .   .  -E  G   .   .  -D
X.19     6  .  .   .  .   . -2  .  G   .   .  -D  F   .   .  -C  H   .   .  -E
X.20     6  .  .   .  .   . -2  .  H   .   .  -E  G   .   .  -D  F   .   .  -C

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = 2*E(3)^2
  = -1-Sqrt(-3) = -1-i3
C = E(7)^2+E(7)^5
D = E(7)+E(7)^6
E = E(7)^3+E(7)^4
F = 3*E(7)^2+3*E(7)^5
G = 3*E(7)+3*E(7)^6
H = 3*E(7)^3+3*E(7)^4
I = E(21)^8+E(21)^20
J = E(21)^11+E(21)^17
K = E(21)^2+E(21)^5