Properties

Label 28T3
Order \(28\)
n \(28\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_7:C_4$

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Group action invariants

Degree $n$ :  $28$
Transitive number $t$ :  $3$
Group :  $C_7:C_4$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,3,2,4)(5,27,6,28)(7,25,8,26)(9,24,10,23)(11,22,12,21)(13,19,14,20)(15,18,16,17), (1,8,2,7)(3,5,4,6)(9,28,10,27)(11,26,12,25)(13,23,14,24)(15,22,16,21)(17,19,18,20)
$|\Aut(F/K)|$:  $28$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
14:  $D_{7}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 7: $D_{7}$

Degree 14: $D_{7}$

Low degree siblings

There are no siblings 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, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)$
$ 4, 4, 4, 4, 4, 4, 4 $ $7$ $4$ $( 1, 3, 2, 4)( 5,27, 6,28)( 7,25, 8,26)( 9,24,10,23)(11,22,12,21)(13,19,14,20) (15,18,16,17)$
$ 4, 4, 4, 4, 4, 4, 4 $ $7$ $4$ $( 1, 4, 2, 3)( 5,28, 6,27)( 7,26, 8,25)( 9,23,10,24)(11,21,12,22)(13,20,14,19) (15,17,16,18)$
$ 7, 7, 7, 7 $ $2$ $7$ $( 1, 5, 9,13,18,21,26)( 2, 6,10,14,17,22,25)( 3, 7,11,16,19,24,27) ( 4, 8,12,15,20,23,28)$
$ 14, 14 $ $2$ $14$ $( 1, 6, 9,14,18,22,26, 2, 5,10,13,17,21,25)( 3, 8,11,15,19,23,27, 4, 7,12,16, 20,24,28)$
$ 7, 7, 7, 7 $ $2$ $7$ $( 1, 9,18,26, 5,13,21)( 2,10,17,25, 6,14,22)( 3,11,19,27, 7,16,24) ( 4,12,20,28, 8,15,23)$
$ 14, 14 $ $2$ $14$ $( 1,10,18,25, 5,14,21, 2, 9,17,26, 6,13,22)( 3,12,19,28, 7,15,24, 4,11,20,27, 8,16,23)$
$ 7, 7, 7, 7 $ $2$ $7$ $( 1,13,26, 9,21, 5,18)( 2,14,25,10,22, 6,17)( 3,16,27,11,24, 7,19) ( 4,15,28,12,23, 8,20)$
$ 14, 14 $ $2$ $14$ $( 1,14,26,10,21, 6,18, 2,13,25, 9,22, 5,17)( 3,15,27,12,24, 8,19, 4,16,28,11, 23, 7,20)$

Group invariants

Order:  $28=2^{2} \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [28, 1]
Character table:   
      2  2  2  2  2  1   1  1   1  1   1
      7  1  1  .  .  1   1  1   1  1   1

        1a 2a 4a 4b 7a 14a 7b 14b 7c 14c
     2P 1a 1a 2a 2a 7b  7b 7c  7c 7a  7a
     3P 1a 2a 4b 4a 7c 14c 7a 14a 7b 14b
     5P 1a 2a 4a 4b 7b 14b 7c 14c 7a 14a
     7P 1a 2a 4b 4a 1a  2a 1a  2a 1a  2a
    11P 1a 2a 4b 4a 7c 14c 7a 14a 7b 14b
    13P 1a 2a 4a 4b 7a 14a 7b 14b 7c 14c

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

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