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Properties

Label 12T215
Degree $12$
Order $1296$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^4:\OD_{16}$

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Show commands: Magma

magma: G := TransitiveGroup(12, 215);
 

Group action invariants

Degree $n$:  $12$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $215$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3^4:\OD_{16}$
CHM label:   $1/2[F_{36}^{2}]2$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,12,7,6)(2,9,8,11,10,5,4,3), (4,8,12), (2,10)(4,8), (1,7)(2,8,10,4)(3,9,11,5)(6,12)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $C_4\times C_2$
$16$:  $C_8:C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: $C_4$

Degree 6: None

Low degree siblings

12T215, 18T288, 18T297 x 2, 24T2899 x 2, 24T2900 x 2, 24T2901 x 2, 24T2928 x 2, 24T2939 x 2, 36T2164, 36T2165, 36T2166, 36T2195 x 2, 36T2307 x 2, 36T2308 x 2, 36T2313, 36T2314, 36T2319

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 $ $1$ $1$ $()$
$ 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $3$ $( 4,12, 8)$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $16$ $3$ $( 3, 7,11)( 4,12, 8)$
$ 3, 3, 1, 1, 1, 1, 1, 1 $ $8$ $3$ $( 2, 6,10)( 4,12, 8)$
$ 3, 3, 3, 1, 1, 1 $ $16$ $3$ $( 2, 6,10)( 3, 7,11)( 4,12, 8)$
$ 3, 3, 3, 1, 1, 1 $ $16$ $3$ $( 2, 6,10)( 3, 7,11)( 4, 8,12)$
$ 3, 3, 3, 3 $ $16$ $3$ $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4,12, 8)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $81$ $2$ $( 5, 9)( 6,10)( 7,11)( 8,12)$
$ 4, 4, 2, 2 $ $81$ $4$ $( 1, 7)( 2, 8,10, 4)( 3, 9,11, 5)( 6,12)$
$ 4, 4, 2, 2 $ $81$ $4$ $( 1,11, 9, 7)( 2,12,10, 4)( 3, 5)( 6, 8)$
$ 8, 4 $ $162$ $8$ $( 1,12, 7, 6)( 2, 9, 8,11,10, 5, 4, 3)$
$ 8, 4 $ $162$ $8$ $( 1, 6, 7,12)( 2,11, 4, 9,10, 3, 8, 5)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $18$ $2$ $( 6,10)( 8,12)$
$ 3, 2, 2, 1, 1, 1, 1, 1 $ $72$ $6$ $( 3, 7,11)( 6,10)( 8,12)$
$ 3, 3, 2, 2, 1, 1 $ $72$ $6$ $( 1, 5, 9)( 3, 7,11)( 6,10)( 8,12)$
$ 4, 4, 2, 2 $ $162$ $4$ $( 1, 7)( 2, 8, 6, 4)( 3, 9,11, 5)(10,12)$
$ 8, 4 $ $162$ $8$ $( 1,12,11,10)( 2, 9, 8, 7, 6, 5, 4, 3)$
$ 8, 4 $ $162$ $8$ $( 1, 6, 3, 8)( 2,11, 4, 9,10, 7,12, 5)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $1296=2^{4} \cdot 3^{4}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  1296.3508
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 3A 3B 3C 3D 3E 3F 4A1 4A-1 4B 6A 6B 8A1 8A-1 8B1 8B-1
Size 1 18 81 8 8 16 16 16 16 81 81 162 72 72 162 162 162 162
2 P 1A 1A 1A 3A 3B 3C 3D 3E 3F 2B 2B 2B 3A 3B 4A1 4A-1 4A-1 4A1
3 P 1A 2A 2B 1A 1A 1A 1A 1A 1A 4A-1 4A1 4B 2A 2A 8A-1 8A1 8B-1 8B1
Type
1296.3508.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1296.3508.1b R 1 −1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 1 1
1296.3508.1c R 1 −1 1 1 1 1 1 1 1 1 1 −1 −1 −1 1 1 −1 −1
1296.3508.1d R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1
1296.3508.1e1 C 1 −1 1 1 1 1 1 1 1 −1 −1 1 −1 −1 −i i i −i
1296.3508.1e2 C 1 −1 1 1 1 1 1 1 1 −1 −1 1 −1 −1 i −i −i i
1296.3508.1f1 C 1 1 1 1 1 1 1 1 1 −1 −1 −1 1 1 −i i −i i
1296.3508.1f2 C 1 1 1 1 1 1 1 1 1 −1 −1 −1 1 1 i −i i −i
1296.3508.2a1 C 2 0 −2 2 2 2 2 2 2 −2i 2i 0 0 0 0 0 0 0
1296.3508.2a2 C 2 0 −2 2 2 2 2 2 2 2i −2i 0 0 0 0 0 0 0
1296.3508.8a R 8 4 0 2 5 −1 −1 2 −4 0 0 0 −2 1 0 0 0 0
1296.3508.8b R 8 4 0 5 2 −1 −1 −4 2 0 0 0 1 −2 0 0 0 0
1296.3508.8c R 8 −4 0 2 5 −1 −1 2 −4 0 0 0 2 −1 0 0 0 0
1296.3508.8d R 8 −4 0 5 2 −1 −1 −4 2 0 0 0 −1 2 0 0 0 0
1296.3508.16a R 16 0 0 −2 −2 −2 7 −2 −2 0 0 0 0 0 0 0 0 0
1296.3508.16b R 16 0 0 −2 −2 7 −2 −2 −2 0 0 0 0 0 0 0 0 0
1296.3508.16c R 16 0 0 −8 4 −2 −2 1 4 0 0 0 0 0 0 0 0 0
1296.3508.16d R 16 0 0 4 −8 −2 −2 4 1 0 0 0 0 0 0 0 0 0

magma: CharacterTable(G);