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Properties

Label 28T6
Degree $28$
Order $56$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_7:D_4$

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Show commands: Magma

magma: G := TransitiveGroup(28, 6);
 

Group action invariants

Degree $n$:  $28$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $6$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_7:D_4$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $14$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,24)(2,23)(3,22)(4,21)(5,19)(6,20)(7,17)(8,18)(9,16)(10,15)(11,28)(12,27)(13,26)(14,25), (1,10,3,12,5,13,8)(2,9,4,11,6,14,7)(15,23,18,25,19,28,22,16,24,17,26,20,27,21)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$8$:  $D_{4}$
$14$:  $D_{7}$
$28$:  $D_{14}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 7: $D_{7}$

Degree 14: $D_{7}$

Low degree siblings

28T7

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $56=2^{3} \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  56.7
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 4A 7A1 7A2 7A3 14A1 14A3 14A5 14B1 14B-1 14B3 14B-3 14B5 14B-5
Size 1 1 2 14 14 2 2 2 2 2 2 2 2 2 2 2 2
2 P 1A 1A 1A 1A 2A 7A1 7A2 7A3 7A2 7A1 7A2 7A1 7A1 7A3 7A3 7A2 7A3
7 P 1A 2A 2B 2C 4A 7A2 7A3 7A1 14A3 14A5 14B-1 14B3 14B-3 14B-5 14A1 14B1 14B5
Type
56.7.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
56.7.1b R 1 1 −1 −1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1
56.7.1c R 1 1 −1 1 −1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1
56.7.1d R 1 1 1 −1 −1 1 1 1 1 1 1 1 1 1 1 1 1
56.7.2a R 2 −2 0 0 0 2 2 2 −2 −2 −2 0 0 0 0 0 0
56.7.2b1 R 2 2 2 0 0 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−3+ζ73
56.7.2b2 R 2 2 2 0 0 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−2+ζ72
56.7.2b3 R 2 2 2 0 0 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−1+ζ7
56.7.2c1 R 2 2 −2 0 0 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 −ζ7−2−ζ72 −ζ7−2−ζ72 −ζ7−1−ζ7 −ζ7−1−ζ7 −ζ7−3−ζ73 −ζ7−3−ζ73
56.7.2c2 R 2 2 −2 0 0 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 −ζ7−1−ζ7 −ζ7−1−ζ7 −ζ7−3−ζ73 −ζ7−3−ζ73 −ζ7−2−ζ72 −ζ7−2−ζ72
56.7.2c3 R 2 2 −2 0 0 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 −ζ7−3−ζ73 −ζ7−3−ζ73 −ζ7−2−ζ72 −ζ7−2−ζ72 −ζ7−1−ζ7 −ζ7−1−ζ7
56.7.2d1 C 2 −2 0 0 0 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 −ζ7−3−ζ73 −ζ7−2−ζ72 −ζ7−1−ζ7 ζ7−2−ζ72 −ζ7−2+ζ72 −ζ7−1+ζ7 ζ7−1−ζ7 ζ7−3−ζ73 −ζ7−3+ζ73
56.7.2d2 C 2 −2 0 0 0 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 −ζ7−3−ζ73 −ζ7−2−ζ72 −ζ7−1−ζ7 −ζ7−2+ζ72 ζ7−2−ζ72 ζ7−1−ζ7 −ζ7−1+ζ7 −ζ7−3+ζ73 ζ7−3−ζ73
56.7.2d3 C 2 −2 0 0 0 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 −ζ7−2−ζ72 −ζ7−1−ζ7 −ζ7−3−ζ73 ζ7−1−ζ7 −ζ7−1+ζ7 ζ7−3−ζ73 −ζ7−3+ζ73 −ζ7−2+ζ72 ζ7−2−ζ72
56.7.2d4 C 2 −2 0 0 0 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 −ζ7−2−ζ72 −ζ7−1−ζ7 −ζ7−3−ζ73 −ζ7−1+ζ7 ζ7−1−ζ7 −ζ7−3+ζ73 ζ7−3−ζ73 ζ7−2−ζ72 −ζ7−2+ζ72
56.7.2d5 C 2 −2 0 0 0 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 −ζ7−1−ζ7 −ζ7−3−ζ73 −ζ7−2−ζ72 ζ7−3−ζ73 −ζ7−3+ζ73 ζ7−2−ζ72 −ζ7−2+ζ72 ζ7−1−ζ7 −ζ7−1+ζ7
56.7.2d6 C 2 −2 0 0 0 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 −ζ7−1−ζ7 −ζ7−3−ζ73 −ζ7−2−ζ72 −ζ7−3+ζ73 ζ7−3−ζ73 −ζ7−2+ζ72 ζ7−2−ζ72 −ζ7−1+ζ7 ζ7−1−ζ7

magma: CharacterTable(G);