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Properties

Label 28T2
Degree $28$
Order $28$
Cyclic no
Abelian yes
Solvable yes
Primitive no
$p$-group no
Group: $C_2\times C_{14}$

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Show commands: Magma

magma: G := TransitiveGroup(28, 2);
 

Group action invariants

Degree $n$:  $28$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $2$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2\times C_{14}$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $28$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (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), (1,19,9,28,17,8,25,15,6,23,14,4,22,12)(2,20,10,27,18,7,26,16,5,24,13,3,21,11)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$7$:  $C_7$
$14$:  $C_{14}$ x 3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 7: $C_7$

Degree 14: $C_{14}$ 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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)$
$ 14, 14 $ $1$ $14$ $( 1, 3, 6, 7, 9,11,14,16,17,20,22,24,25,27)( 2, 4, 5, 8,10,12,13,15,18,19,21, 23,26,28)$
$ 14, 14 $ $1$ $14$ $( 1, 4, 6, 8, 9,12,14,15,17,19,22,23,25,28)( 2, 3, 5, 7,10,11,13,16,18,20,21, 24,26,27)$
$ 14, 14 $ $1$ $14$ $( 1, 5, 9,13,17,21,25, 2, 6,10,14,18,22,26)( 3, 8,11,15,20,23,27, 4, 7,12,16, 19,24,28)$
$ 7, 7, 7, 7 $ $1$ $7$ $( 1, 6, 9,14,17,22,25)( 2, 5,10,13,18,21,26)( 3, 7,11,16,20,24,27) ( 4, 8,12,15,19,23,28)$
$ 14, 14 $ $1$ $14$ $( 1, 7,14,20,25, 3, 9,16,22,27, 6,11,17,24)( 2, 8,13,19,26, 4,10,15,21,28, 5, 12,18,23)$
$ 14, 14 $ $1$ $14$ $( 1, 8,14,19,25, 4, 9,15,22,28, 6,12,17,23)( 2, 7,13,20,26, 3,10,16,21,27, 5, 11,18,24)$
$ 7, 7, 7, 7 $ $1$ $7$ $( 1, 9,17,25, 6,14,22)( 2,10,18,26, 5,13,21)( 3,11,20,27, 7,16,24) ( 4,12,19,28, 8,15,23)$
$ 14, 14 $ $1$ $14$ $( 1,10,17,26, 6,13,22, 2, 9,18,25, 5,14,21)( 3,12,20,28, 7,15,24, 4,11,19,27, 8,16,23)$
$ 14, 14 $ $1$ $14$ $( 1,11,22, 3,14,24, 6,16,25, 7,17,27, 9,20)( 2,12,21, 4,13,23, 5,15,26, 8,18, 28,10,19)$
$ 14, 14 $ $1$ $14$ $( 1,12,22, 4,14,23, 6,15,25, 8,17,28, 9,19)( 2,11,21, 3,13,24, 5,16,26, 7,18, 27,10,20)$
$ 14, 14 $ $1$ $14$ $( 1,13,25,10,22, 5,17, 2,14,26, 9,21, 6,18)( 3,15,27,12,24, 8,20, 4,16,28,11, 23, 7,19)$
$ 7, 7, 7, 7 $ $1$ $7$ $( 1,14,25, 9,22, 6,17)( 2,13,26,10,21, 5,18)( 3,16,27,11,24, 7,20) ( 4,15,28,12,23, 8,19)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,15)( 2,16)( 3,18)( 4,17)( 5,20)( 6,19)( 7,21)( 8,22)( 9,23)(10,24)(11,26) (12,25)(13,27)(14,28)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,16)( 2,15)( 3,17)( 4,18)( 5,19)( 6,20)( 7,22)( 8,21)( 9,24)(10,23)(11,25) (12,26)(13,28)(14,27)$
$ 7, 7, 7, 7 $ $1$ $7$ $( 1,17, 6,22, 9,25,14)( 2,18, 5,21,10,26,13)( 3,20, 7,24,11,27,16) ( 4,19, 8,23,12,28,15)$
$ 14, 14 $ $1$ $14$ $( 1,18, 6,21, 9,26,14, 2,17, 5,22,10,25,13)( 3,19, 7,23,11,28,16, 4,20, 8,24, 12,27,15)$
$ 14, 14 $ $1$ $14$ $( 1,19, 9,28,17, 8,25,15, 6,23,14, 4,22,12)( 2,20,10,27,18, 7,26,16, 5,24,13, 3,21,11)$
$ 14, 14 $ $1$ $14$ $( 1,20, 9,27,17, 7,25,16, 6,24,14, 3,22,11)( 2,19,10,28,18, 8,26,15, 5,23,13, 4,21,12)$
$ 14, 14 $ $1$ $14$ $( 1,21,14, 5,25,18, 9, 2,22,13, 6,26,17,10)( 3,23,16, 8,27,19,11, 4,24,15, 7, 28,20,12)$
$ 7, 7, 7, 7 $ $1$ $7$ $( 1,22,14, 6,25,17, 9)( 2,21,13, 5,26,18,10)( 3,24,16, 7,27,20,11) ( 4,23,15, 8,28,19,12)$
$ 14, 14 $ $1$ $14$ $( 1,23,17,12, 6,28,22,15, 9, 4,25,19,14, 8)( 2,24,18,11, 5,27,21,16,10, 3,26, 20,13, 7)$
$ 14, 14 $ $1$ $14$ $( 1,24,17,11, 6,27,22,16, 9, 3,25,20,14, 7)( 2,23,18,12, 5,28,21,15,10, 4,26, 19,13, 8)$
$ 7, 7, 7, 7 $ $1$ $7$ $( 1,25,22,17,14, 9, 6)( 2,26,21,18,13,10, 5)( 3,27,24,20,16,11, 7) ( 4,28,23,19,15,12, 8)$
$ 14, 14 $ $1$ $14$ $( 1,26,22,18,14,10, 6, 2,25,21,17,13, 9, 5)( 3,28,24,19,16,12, 7, 4,27,23,20, 15,11, 8)$
$ 14, 14 $ $1$ $14$ $( 1,27,25,24,22,20,17,16,14,11, 9, 7, 6, 3)( 2,28,26,23,21,19,18,15,13,12,10, 8, 5, 4)$
$ 14, 14 $ $1$ $14$ $( 1,28,25,23,22,19,17,15,14,12, 9, 8, 6, 4)( 2,27,26,24,21,20,18,16,13,11,10, 7, 5, 3)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $28=2^{2} \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  yes
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $1$
Label:  28.4
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 7A1 7A-1 7A2 7A-2 7A3 7A-3 14A1 14A-1 14A3 14A-3 14A5 14A-5 14B1 14B-1 14B3 14B-3 14B5 14B-5 14C1 14C-1 14C3 14C-3 14C5 14C-5
Size 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
2 P 1A 1A 1A 1A 7A-1 7A-2 7A2 7A3 7A-3 7A1 7A2 7A3 7A1 7A2 7A-2 7A-2 7A2 7A-1 7A-3 7A3 7A-3 7A-3 7A-1 7A-1 7A1 7A1 7A-2 7A3
7 P 1A 2A 2B 2C 7A2 7A-3 7A3 7A1 7A-1 7A-2 14C3 14C1 14B5 14A-1 14A1 14B-3 14B3 14B-5 14C-1 14B1 14B-1 14A5 14C-5 14A-3 14A3 14C5 14C-3 14A-5
Type
28.4.1a 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
28.4.1b 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
28.4.1c 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
28.4.1d 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
28.4.1e1 C 1 1 1 1 ζ7−3 ζ73 ζ7 ζ7−1 ζ7−2 ζ72 ζ72 ζ7−2 ζ7−1 ζ7 ζ73 ζ7−3 ζ7−3 ζ73 ζ7−2 ζ72 ζ7−1 ζ7 ζ7−3 ζ73 ζ7−2 ζ72 ζ7−1 ζ7
28.4.1e2 C 1 1 1 1 ζ73 ζ7−3 ζ7−1 ζ7 ζ72 ζ7−2 ζ7−2 ζ72 ζ7 ζ7−1 ζ7−3 ζ73 ζ73 ζ7−3 ζ72 ζ7−2 ζ7 ζ7−1 ζ73 ζ7−3 ζ72 ζ7−2 ζ7 ζ7−1
28.4.1e3 C 1 1 1 1 ζ7−2 ζ72 ζ73 ζ7−3 ζ7 ζ7−1 ζ7−1 ζ7 ζ7−3 ζ73 ζ72 ζ7−2 ζ7−2 ζ72 ζ7 ζ7−1 ζ7−3 ζ73 ζ7−2 ζ72 ζ7 ζ7−1 ζ7−3 ζ73
28.4.1e4 C 1 1 1 1 ζ72 ζ7−2 ζ7−3 ζ73 ζ7−1 ζ7 ζ7 ζ7−1 ζ73 ζ7−3 ζ7−2 ζ72 ζ72 ζ7−2 ζ7−1 ζ7 ζ73 ζ7−3 ζ72 ζ7−2 ζ7−1 ζ7 ζ73 ζ7−3
28.4.1e5 C 1 1 1 1 ζ7−1 ζ7 ζ7−2 ζ72 ζ7−3 ζ73 ζ73 ζ7−3 ζ72 ζ7−2 ζ7 ζ7−1 ζ7−1 ζ7 ζ7−3 ζ73 ζ72 ζ7−2 ζ7−1 ζ7 ζ7−3 ζ73 ζ72 ζ7−2
28.4.1e6 C 1 1 1 1 ζ7 ζ7−1 ζ72 ζ7−2 ζ73 ζ7−3 ζ7−3 ζ73 ζ7−2 ζ72 ζ7−1 ζ7 ζ7 ζ7−1 ζ73 ζ7−3 ζ7−2 ζ72 ζ7 ζ7−1 ζ73 ζ7−3 ζ7−2 ζ72
28.4.1f1 C 1 −1 −1 1 ζ7−3 ζ73 ζ7 ζ7−1 ζ7−2 ζ72 −ζ72 −ζ7−2 −ζ7−1 −ζ7 −ζ73 −ζ7−3 −ζ7−3 −ζ73 −ζ7−2 −ζ72 −ζ7−1 −ζ7 ζ7−3 ζ73 ζ7−2 ζ72 ζ7−1 ζ7
28.4.1f2 C 1 −1 −1 1 ζ73 ζ7−3 ζ7−1 ζ7 ζ72 ζ7−2 −ζ7−2 −ζ72 −ζ7 −ζ7−1 −ζ7−3 −ζ73 −ζ73 −ζ7−3 −ζ72 −ζ7−2 −ζ7 −ζ7−1 ζ73 ζ7−3 ζ72 ζ7−2 ζ7 ζ7−1
28.4.1f3 C 1 −1 −1 1 ζ7−2 ζ72 ζ73 ζ7−3 ζ7 ζ7−1 −ζ7−1 −ζ7 −ζ7−3 −ζ73 −ζ72 −ζ7−2 −ζ7−2 −ζ72 −ζ7 −ζ7−1 −ζ7−3 −ζ73 ζ7−2 ζ72 ζ7 ζ7−1 ζ7−3 ζ73
28.4.1f4 C 1 −1 −1 1 ζ72 ζ7−2 ζ7−3 ζ73 ζ7−1 ζ7 −ζ7 −ζ7−1 −ζ73 −ζ7−3 −ζ7−2 −ζ72 −ζ72 −ζ7−2 −ζ7−1 −ζ7 −ζ73 −ζ7−3 ζ72 ζ7−2 ζ7−1 ζ7 ζ73 ζ7−3
28.4.1f5 C 1 −1 −1 1 ζ7−1 ζ7 ζ7−2 ζ72 ζ7−3 ζ73 −ζ73 −ζ7−3 −ζ72 −ζ7−2 −ζ7 −ζ7−1 −ζ7−1 −ζ7 −ζ7−3 −ζ73 −ζ72 −ζ7−2 ζ7−1 ζ7 ζ7−3 ζ73 ζ72 ζ7−2
28.4.1f6 C 1 −1 −1 1 ζ7 ζ7−1 ζ72 ζ7−2 ζ73 ζ7−3 −ζ7−3 −ζ73 −ζ7−2 −ζ72 −ζ7−1 −ζ7 −ζ7 −ζ7−1 −ζ73 −ζ7−3 −ζ7−2 −ζ72 ζ7 ζ7−1 ζ73 ζ7−3 ζ7−2 ζ72
28.4.1g1 C 1 −1 1 −1 ζ7−3 ζ73 ζ7 ζ7−1 ζ7−2 ζ72 −ζ72 −ζ7−2 −ζ7−1 −ζ7 −ζ73 −ζ7−3 ζ7−3 ζ73 ζ7−2 ζ72 ζ7−1 ζ7 −ζ7−3 −ζ73 −ζ7−2 −ζ72 −ζ7−1 −ζ7
28.4.1g2 C 1 −1 1 −1 ζ73 ζ7−3 ζ7−1 ζ7 ζ72 ζ7−2 −ζ7−2 −ζ72 −ζ7 −ζ7−1 −ζ7−3 −ζ73 ζ73 ζ7−3 ζ72 ζ7−2 ζ7 ζ7−1 −ζ73 −ζ7−3 −ζ72 −ζ7−2 −ζ7 −ζ7−1
28.4.1g3 C 1 −1 1 −1 ζ7−2 ζ72 ζ73 ζ7−3 ζ7 ζ7−1 −ζ7−1 −ζ7 −ζ7−3 −ζ73 −ζ72 −ζ7−2 ζ7−2 ζ72 ζ7 ζ7−1 ζ7−3 ζ73 −ζ7−2 −ζ72 −ζ7 −ζ7−1 −ζ7−3 −ζ73
28.4.1g4 C 1 −1 1 −1 ζ72 ζ7−2 ζ7−3 ζ73 ζ7−1 ζ7 −ζ7 −ζ7−1 −ζ73 −ζ7−3 −ζ7−2 −ζ72 ζ72 ζ7−2 ζ7−1 ζ7 ζ73 ζ7−3 −ζ72 −ζ7−2 −ζ7−1 −ζ7 −ζ73 −ζ7−3
28.4.1g5 C 1 −1 1 −1 ζ7−1 ζ7 ζ7−2 ζ72 ζ7−3 ζ73 −ζ73 −ζ7−3 −ζ72 −ζ7−2 −ζ7 −ζ7−1 ζ7−1 ζ7 ζ7−3 ζ73 ζ72 ζ7−2 −ζ7−1 −ζ7 −ζ7−3 −ζ73 −ζ72 −ζ7−2
28.4.1g6 C 1 −1 1 −1 ζ7 ζ7−1 ζ72 ζ7−2 ζ73 ζ7−3 −ζ7−3 −ζ73 −ζ7−2 −ζ72 −ζ7−1 −ζ7 ζ7 ζ7−1 ζ73 ζ7−3 ζ7−2 ζ72 −ζ7 −ζ7−1 −ζ73 −ζ7−3 −ζ7−2 −ζ72
28.4.1h1 C 1 1 −1 −1 ζ7−3 ζ73 ζ7 ζ7−1 ζ7−2 ζ72 ζ72 ζ7−2 ζ7−1 ζ7 ζ73 ζ7−3 −ζ7−3 −ζ73 −ζ7−2 −ζ72 −ζ7−1 −ζ7 −ζ7−3 −ζ73 −ζ7−2 −ζ72 −ζ7−1 −ζ7
28.4.1h2 C 1 1 −1 −1 ζ73 ζ7−3 ζ7−1 ζ7 ζ72 ζ7−2 ζ7−2 ζ72 ζ7 ζ7−1 ζ7−3 ζ73 −ζ73 −ζ7−3 −ζ72 −ζ7−2 −ζ7 −ζ7−1 −ζ73 −ζ7−3 −ζ72 −ζ7−2 −ζ7 −ζ7−1
28.4.1h3 C 1 1 −1 −1 ζ7−2 ζ72 ζ73 ζ7−3 ζ7 ζ7−1 ζ7−1 ζ7 ζ7−3 ζ73 ζ72 ζ7−2 −ζ7−2 −ζ72 −ζ7 −ζ7−1 −ζ7−3 −ζ73 −ζ7−2 −ζ72 −ζ7 −ζ7−1 −ζ7−3 −ζ73
28.4.1h4 C 1 1 −1 −1 ζ72 ζ7−2 ζ7−3 ζ73 ζ7−1 ζ7 ζ7 ζ7−1 ζ73 ζ7−3 ζ7−2 ζ72 −ζ72 −ζ7−2 −ζ7−1 −ζ7 −ζ73 −ζ7−3 −ζ72 −ζ7−2 −ζ7−1 −ζ7 −ζ73 −ζ7−3
28.4.1h5 C 1 1 −1 −1 ζ7−1 ζ7 ζ7−2 ζ72 ζ7−3 ζ73 ζ73 ζ7−3 ζ72 ζ7−2 ζ7 ζ7−1 −ζ7−1 −ζ7 −ζ7−3 −ζ73 −ζ72 −ζ7−2 −ζ7−1 −ζ7 −ζ7−3 −ζ73 −ζ72 −ζ7−2
28.4.1h6 C 1 1 −1 −1 ζ7 ζ7−1 ζ72 ζ7−2 ζ73 ζ7−3 ζ7−3 ζ73 ζ7−2 ζ72 ζ7−1 ζ7 −ζ7 −ζ7−1 −ζ73 −ζ7−3 −ζ7−2 −ζ72 −ζ7 −ζ7−1 −ζ73 −ζ7−3 −ζ7−2 −ζ72

magma: CharacterTable(G);