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Properties

Label 12T55
Order \(96\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times C_4^2:C_3$

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $55$
Group :  $C_2\times C_4^2:C_3$
CHM label :  $[1/2.4^{3}]3$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (3,9)(6,12), (1,10,7,4)(3,6,9,12), (1,5,9)(2,6,10)(3,7,11)(4,8,12)
$|\Aut(F/K)|$:  $4$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $C_6$
12:  $A_4$
24:  $A_4\times C_2$
48:  12T31

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 4: None

Degree 6: $A_4$

Low degree siblings

12T55, 24T173, 24T174 x 2, 24T175 x 2, 32T416

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 $ $1$ $1$ $()$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $3$ $2$ $( 3, 9)( 6,12)$
$ 4, 4, 1, 1, 1, 1 $ $3$ $4$ $( 2, 5, 8,11)( 3, 6, 9,12)$
$ 4, 4, 1, 1, 1, 1 $ $3$ $4$ $( 2, 5, 8,11)( 3,12, 9, 6)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $3$ $2$ $( 2, 8)( 3, 9)( 5,11)( 6,12)$
$ 4, 4, 1, 1, 1, 1 $ $3$ $4$ $( 2,11, 8, 5)( 3, 6, 9,12)$
$ 4, 4, 1, 1, 1, 1 $ $3$ $4$ $( 2,11, 8, 5)( 3,12, 9, 6)$
$ 3, 3, 3, 3 $ $16$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)$
$ 6, 6 $ $16$ $6$ $( 1, 2, 3, 7, 8, 9)( 4, 5, 6,10,11,12)$
$ 3, 3, 3, 3 $ $16$ $3$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)$
$ 6, 6 $ $16$ $6$ $( 1, 3, 8, 7, 9, 2)( 4, 6,11,10,12, 5)$
$ 4, 4, 2, 2 $ $3$ $4$ $( 1, 4, 7,10)( 2, 5, 8,11)( 3, 9)( 6,12)$
$ 4, 4, 2, 2 $ $3$ $4$ $( 1, 4, 7,10)( 2, 8)( 3,12, 9, 6)( 5,11)$
$ 4, 4, 2, 2 $ $3$ $4$ $( 1, 4, 7,10)( 2,11, 8, 5)( 3, 9)( 6,12)$
$ 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$
$ 4, 4, 2, 2 $ $3$ $4$ $( 1, 7)( 2,11, 8, 5)( 3,12, 9, 6)( 4,10)$

Group invariants

Order:  $96=2^{5} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [96, 68]
Character table:    
      2  5  5   5   5  5   5   5  1   1  1   1   5   5   5  5   5
      3  1  .   .   .  .   .   .  1   1  1   1   .   .   .  1   .

        1a 2a  4a  4b 2b  4c  4d 3a  6a 3b  6b  4e  4f  4g 2c  4h
     2P 1a 1a  2b  2b 1a  2b  2b 3b  3b 3a  3a  2b  2b  2b 1a  2b
     3P 1a 2a  4d  4c 2b  4b  4a 1a  2c 1a  2c  4h  4g  4f 2c  4e
     5P 1a 2a  4a  4b 2b  4c  4d 3b  6b 3a  6a  4e  4f  4g 2c  4h

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

A = 1+2*E(4)
  = 1+2*Sqrt(-1) = 1+2i
B = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3