Properties

Label 28T43
Order \(336\)
n \(28\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No
Group: $C_2\times \PSL(2,7)$

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Group action invariants

Degree $n$ :  $28$
Transitive number $t$ :  $43$
Group :  $C_2\times \PSL(2,7)$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,4,11,6,9,18,25,8,14,16,20,22,28,24)(2,3,12,5,10,17,26,7,13,15,19,21,27,23), (1,18,3,8,28,13)(2,17,4,7,27,14)(5,16,21,19,11,10)(6,15,22,20,12,9)(23,26)(24,25)
$|\Aut(F/K)|$:  $4$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
168:  $\GL(3,2)$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: None

Degree 7: $\GL(3,2)$

Degree 14: $\PSL(2,7)$, 14T17, $\GL(3,2) \times C_2$

Low degree siblings

14T17 x 2, 14T19 x 2, 16T714, 28T43, 42T78, 42T79, 42T80 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

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 $ $56$ $3$ $( 3, 5,25)( 4, 6,26)( 9,17,15)(10,18,16)(11,21,28)(12,22,27)(13,19,24) (14,20,23)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $21$ $2$ $( 3,14)( 4,13)( 5,11)( 6,12)( 9,25)(10,26)(15,20)(16,19)(17,28)(18,27)(21,23) (22,24)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $21$ $2$ $( 1, 2)( 3, 4)( 5,26)( 6,25)( 7, 8)( 9,12)(10,11)(13,14)(15,22)(16,21)(17,27) (18,28)(19,23)(20,24)$
$ 4, 4, 4, 4, 4, 4, 2, 2 $ $42$ $4$ $( 1, 2)( 3,10,17,24)( 4, 9,18,23)( 5,12,20,16)( 6,11,19,15)( 7, 8) (13,21,27,25)(14,22,28,26)$
$ 7, 7, 7, 7 $ $24$ $7$ $( 1, 3, 5,15,25,17, 9)( 2, 4, 6,16,26,18,10)( 7,14,20,11,23,28,21) ( 8,13,19,12,24,27,22)$
$ 4, 4, 4, 4, 4, 4, 2, 2 $ $42$ $4$ $( 1, 3, 7,14)( 2, 4, 8,13)( 5, 9,23,15)( 6,10,24,16)(11,20,21,25)(12,19,22,26) (17,28)(18,27)$
$ 7, 7, 7, 7 $ $24$ $7$ $( 1, 3, 9,28,20,15,23)( 2, 4,10,27,19,16,24)( 5,11,25, 7,14,21,17) ( 6,12,26, 8,13,22,18)$
$ 14, 14 $ $24$ $14$ $( 1, 4, 5,18,21,26,15, 8,14,19,28,10,23,12)( 2, 3, 6,17,22,25,16, 7,13,20,27, 9,24,11)$
$ 14, 14 $ $24$ $14$ $( 1, 4, 9,26,11,27, 5, 8,14,22,23,16,17,19)( 2, 3,10,25,12,28, 6, 7,13,21,24, 15,18,20)$
$ 6, 6, 6, 6, 2, 2 $ $56$ $6$ $( 1, 4,17, 8,14,27)( 2, 3,18, 7,13,28)( 5,16,23,19,11,26)( 6,15,24,20,12,25) ( 9,22)(10,21)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 8)( 2, 7)( 3,13)( 4,14)( 5,19)( 6,20)( 9,22)(10,21)(11,16)(12,15)(17,27) (18,28)(23,26)(24,25)$

Group invariants

Order:  $336=2^{4} \cdot 3 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  [336, 209]
Character table:   
      2  4  1  4  4  3  1  3  1   1   1  1  4
      3  1  1  .  .  .  .  .  .   .   .  1  1
      7  1  .  .  .  .  1  .  1   1   1  .  1

        1a 3a 2a 2b 4a 7a 4b 7b 14a 14b 6a 2c
     2P 1a 3a 1a 1a 2a 7a 2a 7b  7b  7a 3a 1a
     3P 1a 1a 2a 2b 4a 7b 4b 7a 14b 14a 2c 2c
     5P 1a 3a 2a 2b 4a 7b 4b 7a 14b 14a 6a 2c
     7P 1a 3a 2a 2b 4a 1a 4b 1a  2c  2c 6a 2c
    11P 1a 3a 2a 2b 4a 7a 4b 7b 14a 14b 6a 2c
    13P 1a 3a 2a 2b 4a 7b 4b 7a 14b 14a 6a 2c

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

A = E(7)^3+E(7)^5+E(7)^6
  = (-1-Sqrt(-7))/2 = -1-b7