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Properties

Label 14T21
Order \(448\)
n \(14\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2^3:F_8$

Related objects

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Group action invariants

Degree $n$ :  $14$
Transitive number $t$ :  $21$
Group :  $C_2^3:F_8$
CHM label :  $[2^{6}]7$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (2,9)(7,14), (1,3,5,7,9,11,13)(2,4,6,8,10,12,14)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
7:  $C_7$
56:  $C_2^3:C_7$ x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 7: $C_7$

Low degree siblings

14T21 x 6, 28T62 x 21, 28T63 x 14, 28T64 x 42, 28T65 x 7, 28T66 x 7

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 $ $1$ $1$ $()$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $7$ $2$ $( 6,13)( 7,14)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $7$ $2$ $( 5,12)( 7,14)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $7$ $2$ $( 4,11)( 7,14)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $7$ $2$ $( 4,11)( 5,12)( 6,13)( 7,14)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $7$ $2$ $( 3,10)( 5,12)( 6,13)( 7,14)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $7$ $2$ $( 3,10)( 4,11)( 6,13)( 7,14)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $7$ $2$ $( 3,10)( 4,11)( 5,12)( 7,14)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $7$ $2$ $( 2, 9)( 4,11)( 6,13)( 7,14)$
$ 2, 2, 2, 2, 2, 2, 1, 1 $ $7$ $2$ $( 2, 9)( 3,10)( 4,11)( 5,12)( 6,13)( 7,14)$
$ 7, 7 $ $64$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8, 9,10,11,12,13,14)$
$ 7, 7 $ $64$ $7$ $( 1, 3, 5, 7, 2, 4, 6)( 8,10,12,14, 9,11,13)$
$ 7, 7 $ $64$ $7$ $( 1, 4, 7, 3, 6, 2, 5)( 8,11,14,10,13, 9,12)$
$ 7, 7 $ $64$ $7$ $( 1, 5, 2, 6, 3, 7, 4)( 8,12, 9,13,10,14,11)$
$ 7, 7 $ $64$ $7$ $( 1, 6, 4, 2, 7, 5, 3)( 8,13,11, 9,14,12,10)$
$ 7, 7 $ $64$ $7$ $( 1, 7, 6, 5, 4, 3, 2)( 8,14,13,12,11,10, 9)$

Group invariants

Order:  $448=2^{6} \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [448, 1394]
Character table:    
      2  6  6  6  6  6  6  6  6  6  6  .  .  .  .  .  .
      7  1  .  .  .  .  .  .  .  .  .  1  1  1  1  1  1

        1a 2a 2b 2c 2d 2e 2f 2g 2h 2i 7a 7b 7c 7d 7e 7f
     2P 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 7b 7d 7f 7a 7c 7e
     3P 1a 2a 2b 2c 2d 2e 2f 2g 2h 2i 7c 7f 7b 7e 7a 7d
     5P 1a 2a 2b 2c 2d 2e 2f 2g 2h 2i 7e 7c 7a 7f 7d 7b
     7P 1a 2a 2b 2c 2d 2e 2f 2g 2h 2i 1a 1a 1a 1a 1a 1a

X.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  A  B  C /C /B /A
X.3      1  1  1  1  1  1  1  1  1  1  B /C /A  A  C /B
X.4      1  1  1  1  1  1  1  1  1  1  C /A  B /B  A /C
X.5      1  1  1  1  1  1  1  1  1  1 /C  A /B  B /A  C
X.6      1  1  1  1  1  1  1  1  1  1 /B  C  A /A /C  B
X.7      1  1  1  1  1  1  1  1  1  1 /A /B /C  C  B  A
X.8      7 -5  3 -1  3 -1  3 -1 -1 -1  .  .  .  .  .  .
X.9      7 -1  3 -1 -1 -1 -5 -1  3  3  .  .  .  .  .  .
X.10     7 -1 -5  3  3 -1 -1 -1  3 -1  .  .  .  .  .  .
X.11     7  3  3  3 -1 -1 -1 -1 -1 -5  .  .  .  .  .  .
X.12     7  3 -1 -5 -1 -1  3 -1  3 -1  .  .  .  .  .  .
X.13     7  3 -1 -1  3 -1 -1 -1 -5  3  .  .  .  .  .  .
X.14     7 -1 -1  3 -5 -1  3 -1 -1  3  .  .  .  .  .  .
X.15     7 -1 -1 -1 -1 -1 -1  7 -1 -1  .  .  .  .  .  .
X.16     7 -1 -1 -1 -1  7 -1 -1 -1 -1  .  .  .  .  .  .

A = E(7)^6
B = E(7)^5
C = E(7)^4