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Properties

Label 28T18
Degree $28$
Order $112$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_4\times D_7$

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Show commands: Magma

magma: G := TransitiveGroup(28, 18);
 

Group action invariants

Degree $n$:  $28$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $18$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_4\times D_7$
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:  (1,25,2,26)(3,23,4,24)(5,21,6,22)(7,20,8,19)(9,18,10,17)(11,16,12,15)(13,27,14,28), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,14)(15,24)(16,23)(17,22)(18,21)(19,20)(25,28)(26,27), (1,14,11,9,7,6,4,2,13,12,10,8,5,3)(15,28,26,23,22,19,18)(16,27,25,24,21,20,17)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$8$:  $D_{4}$ x 2, $C_2^3$
$14$:  $D_{7}$
$16$:  $D_4\times C_2$
$28$:  $D_{14}$ x 3
$56$:  28T9

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 7: $D_{7}$

Degree 14: $D_{14}$

Low degree siblings

28T18 x 3

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $112=2^{4} \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  112.31
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 2F 2G 4A 4B 7A1 7A2 7A3 14A1 14A3 14A5 14B1 14B3 14B5 14C1 14C3 14C5 28A1 28A3 28A5
Size 1 1 2 2 7 7 14 14 2 14 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4
2 P 1A 1A 1A 1A 1A 1A 1A 1A 2A 2A 7A2 7A3 7A1 7A1 7A3 7A2 7A1 7A3 7A2 7A1 7A3 7A2 14A1 14A3 14A5
7 P 1A 2A 2B 2C 2D 2E 2F 2G 4A 4B 7A3 7A1 7A2 14A3 14A5 14A1 14B3 14B5 14B1 14C3 14C5 14C1 28A3 28A5 28A1
Type
112.31.1a 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
112.31.1b 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
112.31.1c 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
112.31.1d 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
112.31.1e 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
112.31.1f 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
112.31.1g 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
112.31.1h 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
112.31.2a R 2 −2 0 0 −2 2 0 0 0 0 2 2 2 −2 −2 −2 0 0 0 0 0 0 0 0 0
112.31.2b R 2 −2 0 0 2 −2 0 0 0 0 2 2 2 −2 −2 −2 0 0 0 0 0 0 0 0 0
112.31.2c1 R 2 2 2 2 0 0 0 0 2 0 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72
112.31.2c2 R 2 2 2 2 0 0 0 0 2 0 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7
112.31.2c3 R 2 2 2 2 0 0 0 0 2 0 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73
112.31.2d1 R 2 2 −2 −2 0 0 0 0 2 0 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 −ζ7−2−ζ72 −ζ7−1−ζ7 −ζ7−3−ζ73 −ζ7−2−ζ72 −ζ7−1−ζ7 −ζ7−3−ζ73 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72
112.31.2d2 R 2 2 −2 −2 0 0 0 0 2 0 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 −ζ7−1−ζ7 −ζ7−3−ζ73 −ζ7−2−ζ72 −ζ7−1−ζ7 −ζ7−3−ζ73 −ζ7−2−ζ72 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7
112.31.2d3 R 2 2 −2 −2 0 0 0 0 2 0 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 −ζ7−3−ζ73 −ζ7−2−ζ72 −ζ7−1−ζ7 −ζ7−3−ζ73 −ζ7−2−ζ72 −ζ7−1−ζ7 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73
112.31.2e1 R 2 2 −2 2 0 0 0 0 −2 0 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 −ζ7−2−ζ72 −ζ7−1−ζ7 −ζ7−3−ζ73 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 −ζ7−1−ζ7 −ζ7−3−ζ73 −ζ7−2−ζ72
112.31.2e2 R 2 2 −2 2 0 0 0 0 −2 0 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 −ζ7−1−ζ7 −ζ7−3−ζ73 −ζ7−2−ζ72 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 −ζ7−3−ζ73 −ζ7−2−ζ72 −ζ7−1−ζ7
112.31.2e3 R 2 2 −2 2 0 0 0 0 −2 0 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 −ζ7−3−ζ73 −ζ7−2−ζ72 −ζ7−1−ζ7 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 −ζ7−2−ζ72 −ζ7−1−ζ7 −ζ7−3−ζ73
112.31.2f1 R 2 2 2 −2 0 0 0 0 −2 0 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 −ζ7−2−ζ72 −ζ7−1−ζ7 −ζ7−3−ζ73 −ζ7−1−ζ7 −ζ7−3−ζ73 −ζ7−2−ζ72
112.31.2f2 R 2 2 2 −2 0 0 0 0 −2 0 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 −ζ7−1−ζ7 −ζ7−3−ζ73 −ζ7−2−ζ72 −ζ7−3−ζ73 −ζ7−2−ζ72 −ζ7−1−ζ7
112.31.2f3 R 2 2 2 −2 0 0 0 0 −2 0 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 −ζ7−3−ζ73 −ζ7−2−ζ72 −ζ7−1−ζ7 −ζ7−2−ζ72 −ζ7−1−ζ7 −ζ7−3−ζ73
112.31.4a1 R 4 −4 0 0 0 0 0 0 0 0 2ζ7−3+2ζ73 2ζ7−1+2ζ7 2ζ7−2+2ζ72 −2ζ7−2−2ζ72 −2ζ7−1−2ζ7 −2ζ7−3−2ζ73 0 0 0 0 0 0 0 0 0
112.31.4a2 R 4 −4 0 0 0 0 0 0 0 0 2ζ7−2+2ζ72 2ζ7−3+2ζ73 2ζ7−1+2ζ7 −2ζ7−1−2ζ7 −2ζ7−3−2ζ73 −2ζ7−2−2ζ72 0 0 0 0 0 0 0 0 0
112.31.4a3 R 4 −4 0 0 0 0 0 0 0 0 2ζ7−1+2ζ7 2ζ7−2+2ζ72 2ζ7−3+2ζ73 −2ζ7−3−2ζ73 −2ζ7−2−2ζ72 −2ζ7−1−2ζ7 0 0 0 0 0 0 0 0 0

magma: CharacterTable(G);