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Properties

Label 12T158
Degree $12$
Order $576$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $A_4^2:C_2^2$

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magma: G := TransitiveGroup(12, 158);
 

Group action invariants

Degree $n$:  $12$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $158$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $A_4^2:C_2^2$
CHM label:   $[2^{5}]F_{18}(6)$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,12)(2,3), (1,5,9)(4,8,12), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_2^2$
$6$:  $S_3$, $C_6$ x 3
$12$:  $D_{6}$, $C_6\times C_2$
$18$:  $S_3\times C_3$
$36$:  $C_6\times S_3$
$288$:  $A_4\wr C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: None

Degree 6: $S_3\times C_3$

Low degree siblings

16T1028, 18T175 x 2, 24T1481 x 2, 24T1484 x 2, 24T1485, 24T1486, 32T34599 x 2, 36T720, 36T723, 36T743 x 2, 36T744 x 2, 36T948 x 2, 36T949, 36T950, 36T951 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle 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 $ $9$ $2$ $( 1,12)( 2, 3)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $9$ $2$ $( 1,12)( 2, 3)( 6, 7)( 8, 9)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $2$ $( 1,12)( 4, 5)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $6$ $2$ $( 1,12)( 4, 5)( 6, 7)( 8, 9)$
$ 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,12)( 2, 3)( 4, 5)( 6, 7)( 8, 9)(10,11)$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $8$ $3$ $( 1, 5, 9)( 4, 8,12)$
$ 6, 2, 1, 1, 1, 1 $ $24$ $6$ $( 1, 5, 9,12, 4, 8)( 2, 3)$
$ 3, 3, 2, 2, 1, 1 $ $24$ $6$ $( 1, 5, 8)( 2, 3)( 4, 9,12)( 6, 7)$
$ 6, 2, 2, 2 $ $8$ $6$ $( 1, 5, 9,12, 4, 8)( 2, 3)( 6, 7)(10,11)$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $8$ $3$ $( 1, 9, 5)( 4,12, 8)$
$ 6, 2, 1, 1, 1, 1 $ $24$ $6$ $( 1, 9, 5,12, 8, 4)( 2, 3)$
$ 3, 3, 2, 2, 1, 1 $ $24$ $6$ $( 1, 8, 4)( 2, 3)( 5,12, 9)( 6, 7)$
$ 6, 2, 2, 2 $ $8$ $6$ $( 1, 9, 5,12, 8, 4)( 2, 3)( 6, 7)(10,11)$
$ 3, 3, 3, 3 $ $16$ $3$ $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$
$ 6, 6 $ $16$ $6$ $( 1, 5, 9,12, 4, 8)( 2, 6,10, 3, 7,11)$
$ 3, 3, 3, 3 $ $32$ $3$ $( 1, 9, 5)( 2, 6,10)( 3, 7,11)( 4,12, 8)$
$ 6, 6 $ $32$ $6$ $( 1, 9, 5,12, 8, 4)( 2, 6,10, 3, 7,11)$
$ 3, 3, 3, 3 $ $16$ $3$ $( 1, 9, 5)( 2,10, 6)( 3,11, 7)( 4,12, 8)$
$ 6, 6 $ $16$ $6$ $( 1, 9, 5,12, 8, 4)( 2,10, 6, 3,11, 7)$
$ 2, 2, 2, 2, 2, 2 $ $12$ $2$ $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$
$ 4, 4, 2, 2 $ $36$ $4$ $( 1, 7,12, 6)( 2, 8, 3, 9)( 4,10)( 5,11)$
$ 4, 4, 2, 2 $ $36$ $4$ $( 1, 7,12, 6)( 2, 8)( 3, 9)( 4,10, 5,11)$
$ 2, 2, 2, 2, 2, 2 $ $12$ $2$ $( 1, 7)( 2, 8)( 3, 9)( 4,11)( 5,10)( 6,12)$
$ 6, 6 $ $48$ $6$ $( 1, 7, 5,11, 9, 3)( 2,12, 6, 4,10, 8)$
$ 6, 6 $ $48$ $6$ $( 1, 7, 5,11, 9, 2)( 3,12, 6, 4,10, 8)$
$ 6, 6 $ $48$ $6$ $( 1, 7, 9, 3, 5,11)( 2, 4,10,12, 6, 8)$
$ 6, 6 $ $48$ $6$ $( 1, 7, 9, 2, 4,10)( 3, 5,11,12, 6, 8)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $576=2^{6} \cdot 3^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  576.8656
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 2F 2G 3A1 3A-1 3B1 3B-1 3C 4A 4B 6A1 6A-1 6B1 6B-1 6C1 6C-1 6D1 6D-1 6E 6F1 6F-1 6G1 6G-1
Size 1 1 6 6 9 9 12 12 8 8 16 16 32 36 36 8 8 16 16 24 24 24 24 32 48 48 48 48
2 P 1A 1A 1A 1A 1A 1A 1A 1A 3A-1 3A1 3B-1 3B1 3C 2D 2D 3A1 3A-1 3B-1 3B1 3A1 3A-1 3A1 3A-1 3C 3B1 3B-1 3B1 3B-1
3 P 1A 2A 2B 2C 2D 2E 2F 2G 1A 1A 1A 1A 1A 4A 4B 2A 2A 2A 2A 2B 2B 2C 2C 2A 2F 2F 2G 2G
Type
576.8656.1a R 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
576.8656.1b R 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
576.8656.1c R 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
576.8656.1d R 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
576.8656.1e1 C 1 1 1 1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 1 ζ3−1 ζ3 ζ3−1 ζ3
576.8656.1e2 C 1 1 1 1 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 1 ζ3 ζ3−1 ζ3 ζ3−1
576.8656.1f1 C 1 −1 −1 1 1 −1 −1 1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 −1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 ζ3 ζ3−1 −1 −ζ3−1 −ζ3 ζ3−1 ζ3
576.8656.1f2 C 1 −1 −1 1 1 −1 −1 1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 −1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3 ζ3−1 ζ3 −1 −ζ3 −ζ3−1 ζ3 ζ3−1
576.8656.1g1 C 1 −1 −1 1 1 −1 1 −1 ζ3−1 ζ3 ζ3 ζ3−1 1 −1 1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 ζ3 ζ3−1 −1 ζ3−1 ζ3 −ζ3−1 −ζ3
576.8656.1g2 C 1 −1 −1 1 1 −1 1 −1 ζ3 ζ3−1 ζ3−1 ζ3 1 −1 1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3 ζ3−1 ζ3 −1 ζ3 ζ3−1 −ζ3 −ζ3−1
576.8656.1h1 C 1 1 1 1 1 1 −1 −1 ζ3−1 ζ3 ζ3 ζ3−1 1 −1 −1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 1 −ζ3−1 −ζ3 −ζ3−1 −ζ3
576.8656.1h2 C 1 1 1 1 1 1 −1 −1 ζ3 ζ3−1 ζ3−1 ζ3 1 −1 −1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 1 −ζ3 −ζ3−1 −ζ3 −ζ3−1
576.8656.2a R 2 2 2 2 2 2 0 0 −1 −1 2 2 −1 0 0 −1 −1 2 2 −1 −1 −1 −1 −1 0 0 0 0
576.8656.2b R 2 −2 −2 2 2 −2 0 0 −1 −1 2 2 −1 0 0 1 1 −2 −2 1 1 −1 −1 1 0 0 0 0
576.8656.2c1 C 2 2 2 2 2 2 0 0 −ζ3 −ζ3−1 2ζ3−1 2ζ3 −1 0 0 −ζ3−1 −ζ3 2ζ3−1 2ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −1 0 0 0 0
576.8656.2c2 C 2 2 2 2 2 2 0 0 −ζ3−1 −ζ3 2ζ3 2ζ3−1 −1 0 0 −ζ3 −ζ3−1 2ζ3 2ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −1 0 0 0 0
576.8656.2d1 C 2 −2 −2 2 2 −2 0 0 −ζ3 −ζ3−1 2ζ3−1 2ζ3 −1 0 0 ζ3−1 ζ3 −2ζ3−1 −2ζ3 ζ3−1 ζ3 −ζ3−1 −ζ3 1 0 0 0 0
576.8656.2d2 C 2 −2 −2 2 2 −2 0 0 −ζ3−1 −ζ3 2ζ3 2ζ3−1 −1 0 0 ζ3 ζ3−1 −2ζ3 −2ζ3−1 ζ3 ζ3−1 −ζ3 −ζ3−1 1 0 0 0 0
576.8656.6a R 6 6 2 2 −2 −2 0 0 3 3 0 0 0 0 0 3 3 0 0 −1 −1 −1 −1 0 0 0 0 0
576.8656.6b R 6 −6 −2 2 −2 2 0 0 3 3 0 0 0 0 0 −3 −3 0 0 1 1 −1 −1 0 0 0 0 0
576.8656.6c1 C 6 6 2 2 −2 −2 0 0 3ζ3−1 3ζ3 0 0 0 0 0 3ζ3 3ζ3−1 0 0 −ζ3 −ζ3−1 −ζ3 −ζ3−1 0 0 0 0 0
576.8656.6c2 C 6 6 2 2 −2 −2 0 0 3ζ3 3ζ3−1 0 0 0 0 0 3ζ3−1 3ζ3 0 0 −ζ3−1 −ζ3 −ζ3−1 −ζ3 0 0 0 0 0
576.8656.6d1 C 6 −6 −2 2 −2 2 0 0 3ζ3−1 3ζ3 0 0 0 0 0 −3ζ3 −3ζ3−1 0 0 ζ3 ζ3−1 −ζ3 −ζ3−1 0 0 0 0 0
576.8656.6d2 C 6 −6 −2 2 −2 2 0 0 3ζ3 3ζ3−1 0 0 0 0 0 −3ζ3−1 −3ζ3 0 0 ζ3−1 ζ3 −ζ3−1 −ζ3 0 0 0 0 0
576.8656.9a R 9 9 −3 −3 1 1 3 3 0 0 0 0 0 −1 −1 0 0 0 0 0 0 0 0 0 0 0 0 0
576.8656.9b R 9 9 −3 −3 1 1 −3 −3 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0
576.8656.9c R 9 −9 3 −3 1 −1 −3 3 0 0 0 0 0 −1 1 0 0 0 0 0 0 0 0 0 0 0 0 0
576.8656.9d R 9 −9 3 −3 1 −1 3 −3 0 0 0 0 0 1 −1 0 0 0 0 0 0 0 0 0 0 0 0 0

magma: CharacterTable(G);