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Properties

Label 28T14
Degree $28$
Order $84$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{14}:C_6$

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Show commands: Magma

magma: G := TransitiveGroup(28, 14);
 

Group action invariants

Degree $n$:  $28$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $14$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_{14}:C_6$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $4$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,15)(2,16)(3,6,24,17,19,10)(4,5,23,18,20,9)(7,14,11,22,27,25)(8,13,12,21,28,26), (1,25,21,17,13,10,5,2,26,22,18,14,9,6)(3,28,24,20,16,12,7,4,27,23,19,15,11,8)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_2^2$
$6$:  $C_6$ x 3
$12$:  $C_6\times C_2$
$21$:  $C_7:C_3$
$42$:  $(C_7:C_3) \times C_2$ x 3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 7: $C_7:C_3$

Degree 14: $(C_7:C_3) \times C_2$ x 3

Low degree siblings

There are no siblings 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ $7$ $3$ $( 3,19,24)( 4,20,23)( 5, 9,18)( 6,10,17)( 7,27,11)( 8,28,12)(13,26,21) (14,25,22)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ $7$ $3$ $( 3,24,19)( 4,23,20)( 5,18, 9)( 6,17,10)( 7,11,27)( 8,12,28)(13,21,26) (14,22,25)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)$
$ 6, 6, 6, 6, 2, 2 $ $7$ $6$ $( 1, 2)( 3,20,24, 4,19,23)( 5,10,18, 6, 9,17)( 7,28,11, 8,27,12) (13,25,21,14,26,22)(15,16)$
$ 6, 6, 6, 6, 2, 2 $ $7$ $6$ $( 1, 2)( 3,23,19, 4,24,20)( 5,17, 9, 6,18,10)( 7,12,27, 8,11,28) (13,22,26,14,21,25)(15,16)$
$ 14, 14 $ $3$ $14$ $( 1, 3, 5, 7, 9,11,13,16,18,19,21,24,26,27)( 2, 4, 6, 8,10,12,14,15,17,20,22, 23,25,28)$
$ 6, 6, 6, 6, 2, 2 $ $7$ $6$ $( 1, 3,21,16,18, 7)( 2, 4,22,15,17, 8)( 5,11, 9,19,26,24)( 6,12,10,20,25,23) (13,27)(14,28)$
$ 6, 6, 6, 6, 2, 2 $ $7$ $6$ $( 1, 3,26,16,18,11)( 2, 4,25,15,17,12)( 5,19)( 6,20)( 7,13,24,21,27, 9) ( 8,14,23,22,28,10)$
$ 14, 14 $ $3$ $14$ $( 1, 4, 5, 8, 9,12,13,15,18,20,21,23,26,28)( 2, 3, 6, 7,10,11,14,16,17,19,22, 24,25,27)$
$ 6, 6, 6, 6, 2, 2 $ $7$ $6$ $( 1, 4,21,15,18, 8)( 2, 3,22,16,17, 7)( 5,12, 9,20,26,23)( 6,11,10,19,25,24) (13,28)(14,27)$
$ 6, 6, 6, 6, 2, 2 $ $7$ $6$ $( 1, 4,26,15,18,12)( 2, 3,25,16,17,11)( 5,20)( 6,19)( 7,14,24,22,27,10) ( 8,13,23,21,28, 9)$
$ 7, 7, 7, 7 $ $3$ $7$ $( 1, 5, 9,13,18,21,26)( 2, 6,10,14,17,22,25)( 3, 7,11,16,19,24,27) ( 4, 8,12,15,20,23,28)$
$ 14, 14 $ $3$ $14$ $( 1, 6, 9,14,18,22,26, 2, 5,10,13,17,21,25)( 3, 8,11,15,19,23,27, 4, 7,12,16, 20,24,28)$
$ 14, 14 $ $3$ $14$ $( 1, 7,13,19,26, 3, 9,16,21,27, 5,11,18,24)( 2, 8,14,20,25, 4,10,15,22,28, 6, 12,17,23)$
$ 14, 14 $ $3$ $14$ $( 1, 8,13,20,26, 4, 9,15,21,28, 5,12,18,23)( 2, 7,14,19,25, 3,10,16,22,27, 6, 11,17,24)$
$ 7, 7, 7, 7 $ $3$ $7$ $( 1,13,26, 9,21, 5,18)( 2,14,25,10,22, 6,17)( 3,16,27,11,24, 7,19) ( 4,15,28,12,23, 8,20)$
$ 14, 14 $ $3$ $14$ $( 1,14,26,10,21, 6,18, 2,13,25, 9,22, 5,17)( 3,15,27,12,24, 8,19, 4,16,28,11, 23, 7,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,15)( 2,16)( 3,17)( 4,18)( 5,20)( 6,19)( 7,22)( 8,21)( 9,23)(10,24)(11,25) (12,26)(13,28)(14,27)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,16)( 2,15)( 3,18)( 4,17)( 5,19)( 6,20)( 7,21)( 8,22)( 9,24)(10,23)(11,26) (12,25)(13,27)(14,28)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $84=2^{2} \cdot 3 \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  84.9
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A1 3A-1 6A1 6A-1 6B1 6B-1 6C1 6C-1 7A1 7A-1 14A1 14A-1 14B1 14B-1 14C1 14C-1
Size 1 1 1 1 7 7 7 7 7 7 7 7 3 3 3 3 3 3 3 3
2 P 1A 1A 1A 1A 3A-1 3A1 3A-1 3A1 3A1 3A-1 3A1 3A-1 7A1 7A-1 7A1 7A1 7A-1 7A1 7A-1 7A-1
3 P 1A 2A 2B 2C 1A 1A 2A 2A 2B 2B 2C 2C 7A-1 7A1 14A-1 14B1 14C-1 14C1 14A1 14B-1
7 P 1A 2A 2B 2C 3A1 3A-1 6C-1 6C1 6A1 6A-1 6B-1 6B1 1A 1A 2A 2C 2B 2B 2A 2C
Type
84.9.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
84.9.1b R 1 −1 −1 1 1 1 −1 −1 1 1 −1 −1 1 1 −1 −1 1 1 −1 −1
84.9.1c R 1 −1 1 −1 1 1 1 1 −1 −1 −1 −1 1 1 −1 −1 −1 −1 1 1
84.9.1d R 1 1 −1 −1 1 1 −1 −1 −1 −1 1 1 1 1 1 1 −1 −1 −1 −1
84.9.1e1 C 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 1 1 1 1 1 1
84.9.1e2 C 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 1 1 1 1 1 1 1 1
84.9.1f1 C 1 −1 −1 1 ζ3−1 ζ3 −ζ3 −ζ3−1 ζ3−1 ζ3 −ζ3 −ζ3−1 1 1 −1 −1 1 1 −1 −1
84.9.1f2 C 1 −1 −1 1 ζ3 ζ3−1 −ζ3−1 −ζ3 ζ3 ζ3−1 −ζ3−1 −ζ3 1 1 −1 −1 1 1 −1 −1
84.9.1g1 C 1 −1 1 −1 ζ3−1 ζ3 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 1 1 −1 −1 −1 −1 1 1
84.9.1g2 C 1 −1 1 −1 ζ3 ζ3−1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3 1 1 −1 −1 −1 −1 1 1
84.9.1h1 C 1 1 −1 −1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3 ζ3 ζ3−1 1 1 1 1 −1 −1 −1 −1
84.9.1h2 C 1 1 −1 −1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 ζ3−1 ζ3 1 1 1 1 −1 −1 −1 −1
84.9.3a1 C 3 3 3 3 0 0 0 0 0 0 0 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72
84.9.3a2 C 3 3 3 3 0 0 0 0 0 0 0 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72
84.9.3b1 C 3 −3 −3 3 0 0 0 0 0 0 0 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 ζ7−3+1+ζ7+ζ72 −ζ7−3−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 −ζ7−3−ζ7−ζ72 ζ7−3+1+ζ7+ζ72
84.9.3b2 C 3 −3 −3 3 0 0 0 0 0 0 0 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 −ζ7−3−ζ7−ζ72 ζ7−3+1+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 ζ7−3+1+ζ7+ζ72 −ζ7−3−ζ7−ζ72
84.9.3c1 C 3 −3 3 −3 0 0 0 0 0 0 0 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 ζ7−3+1+ζ7+ζ72 −ζ7−3−ζ7−ζ72 −ζ7−3−ζ7−ζ72 ζ7−3+1+ζ7+ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72
84.9.3c2 C 3 −3 3 −3 0 0 0 0 0 0 0 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 −ζ7−3−ζ7−ζ72 ζ7−3+1+ζ7+ζ72 ζ7−3+1+ζ7+ζ72 −ζ7−3−ζ7−ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72
84.9.3d1 C 3 3 −3 −3 0 0 0 0 0 0 0 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−ζ7−ζ72 ζ7−3+1+ζ7+ζ72 −ζ7−3−ζ7−ζ72 ζ7−3+1+ζ7+ζ72
84.9.3d2 C 3 3 −3 −3 0 0 0 0 0 0 0 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+1+ζ7+ζ72 −ζ7−3−ζ7−ζ72 ζ7−3+1+ζ7+ζ72 −ζ7−3−ζ7−ζ72

magma: CharacterTable(G);