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Properties

Label 28T28
Order \(168\)
n \(28\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_7:A_4$

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Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $C_6$
12:  $A_4$
24:  $A_4\times C_2$
42:  $F_7$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: $A_4$

Degree 7: $F_7$

Degree 14: None

Low degree siblings

42T30, 42T31

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

Group invariants

Order:  $168=2^{3} \cdot 3 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [168, 49]
Character table:    
      2  3  3  1   1   1  1  3  3   2  2   2   2
      3  1  1  1   1   1  1  .  .   .  .   .   .
      7  1  .  .   .   .  .  1  .   1  1   1   1

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

X.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
X.3      1 -1  A  -A -/A /A  1 -1   1  1   1   1
X.4      1 -1 /A -/A  -A  A  1 -1   1  1   1   1
X.5      1  1  A   A  /A /A  1  1   1  1   1   1
X.6      1  1 /A  /A   A  A  1  1   1  1   1   1
X.7      3 -3  .   .   .  . -1  1  -1  3  -1  -1
X.8      3  3  .   .   .  . -1 -1  -1  3  -1  -1
X.9      6  .  .   .   .  .  6  .  -1 -1  -1  -1
X.10     6  .  .   .   .  . -2  .   B -1   D   C
X.11     6  .  .   .   .  . -2  .   C -1   B   D
X.12     6  .  .   .   .  . -2  .   D -1   C   B

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = E(7)-E(7)^2-E(7)^3-E(7)^4-E(7)^5+E(7)^6
C = -E(7)-E(7)^2+E(7)^3+E(7)^4-E(7)^5-E(7)^6
D = -E(7)+E(7)^2-E(7)^3-E(7)^4+E(7)^5-E(7)^6