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Properties

Label 14T33
Degree $14$
Order $1344$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $C_2^3.\GL(3,2)$

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Show commands: Magma

magma: G := TransitiveGroup(14, 33);
 

Group action invariants

Degree $n$:  $14$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $33$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2^3.\GL(3,2)$
CHM label:   $2^{3}`L_{7}(14)$
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:  (1,9,11)(2,4,8)(3,13,5)(6,12,10), (3,10)(5,12)(6,13)(7,14), (1,8)(2,11)(3,10)(4,9)(5,6)(12,13), (1,3,5,7,9,11,13)(2,4,6,8,10,12,14)
magma: Generators(G);
 

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

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

Low degree siblings

14T33, 28T152, 28T158 x 2, 42T208 x 2, 42T209 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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $7$ $2$ $( 3,10)( 5,12)( 6,13)( 7,14)$
$ 4, 4, 1, 1, 1, 1, 1, 1 $ $42$ $4$ $( 3,12,10, 5)( 6,14,13, 7)$
$ 2, 2, 2, 2, 2, 2, 1, 1 $ $84$ $2$ $( 1, 8)( 2, 9)( 3, 5)( 6, 7)(10,12)(13,14)$
$ 4, 4, 2, 2, 1, 1 $ $42$ $4$ $( 1, 8)( 3,12,10, 5)( 4,11)( 6, 7,13,14)$
$ 8, 4, 1, 1 $ $168$ $8$ $( 2, 3,11,14, 9,10, 4, 7)( 5, 6,12,13)$
$ 8, 2, 2, 2 $ $168$ $8$ $( 1, 8)( 2, 3,11, 7, 9,10, 4,14)( 5, 6)(12,13)$
$ 3, 3, 3, 3, 1, 1 $ $224$ $3$ $( 2, 3,12)( 4, 7, 6)( 5, 9,10)(11,14,13)$
$ 6, 3, 3, 2 $ $224$ $6$ $( 1, 8)( 2, 3, 5)( 4,14,13,11, 7, 6)( 9,10,12)$
$ 7, 7 $ $192$ $7$ $( 1, 2, 3, 4,12, 6, 7)( 5,13,14, 8, 9,10,11)$
$ 7, 7 $ $192$ $7$ $( 1, 2, 3,14,13,11, 5)( 4,12, 8, 9,10, 7, 6)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $1344=2^{6} \cdot 3 \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  1344.814
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 3A 4A 4B 6A 7A1 7A-1 8A 8B
Size 1 7 84 224 42 42 224 192 192 168 168
2 P 1A 1A 1A 3A 2A 2A 3A 7A1 7A-1 4A 4B
3 P 1A 2A 2B 1A 4A 4B 2A 7A-1 7A1 8A 8B
7 P 1A 2A 2B 3A 4A 4B 6A 1A 1A 8A 8B
Type
1344.814.1a R 1 1 1 1 1 1 1 1 1 1 1
1344.814.3a1 C 3 3 −1 0 −1 −1 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 1 1
1344.814.3a2 C 3 3 −1 0 −1 −1 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 1 1
1344.814.6a R 6 6 2 0 2 2 0 −1 −1 0 0
1344.814.7a R 7 7 −1 1 −1 −1 1 0 0 −1 −1
1344.814.7b R 7 −1 −1 1 −1 3 −1 0 0 1 −1
1344.814.7c R 7 −1 −1 1 3 −1 −1 0 0 −1 1
1344.814.8a R 8 8 0 −1 0 0 −1 1 1 0 0
1344.814.14a R 14 −2 −2 −1 2 2 1 0 0 0 0
1344.814.21a R 21 −3 1 0 −3 1 0 0 0 −1 1
1344.814.21b R 21 −3 1 0 1 −3 0 0 0 1 −1

magma: CharacterTable(G);