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Properties

Label 28T10
Degree $28$
Order $56$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_{28}$

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Show commands: Magma

magma: G := TransitiveGroup(28, 10);
 

Group action invariants

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

Low degree siblings

28T10

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

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:  56.5
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 4A 7A1 7A2 7A3 14A1 14A3 14A5 28A1 28A3 28A5 28A9 28A11 28A13
Size 1 1 14 14 2 2 2 2 2 2 2 2 2 2 2 2 2
2 P 1A 1A 1A 1A 2A 7A2 7A3 7A1 7A1 7A3 7A2 14A5 14A5 14A1 14A3 14A1 14A3
7 P 1A 2A 2B 2C 4A 7A3 7A1 7A2 14A3 14A5 14A1 28A1 28A13 28A11 28A5 28A3 28A9
Type
56.5.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
56.5.1b R 1 1 −1 −1 1 1 1 1 1 1 1 1 1 1 1 1 1
56.5.1c R 1 1 −1 1 −1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1
56.5.1d R 1 1 1 −1 −1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1
56.5.2a R 2 −2 0 0 0 2 2 2 −2 −2 −2 0 0 0 0 0 0
56.5.2b1 R 2 2 0 0 2 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7
56.5.2b2 R 2 2 0 0 2 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73
56.5.2b3 R 2 2 0 0 2 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72
56.5.2c1 R 2 2 0 0 −2 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 −ζ7−1−ζ7 −ζ7−3−ζ73 −ζ7−2−ζ72 −ζ7−2−ζ72 −ζ7−3−ζ73 −ζ7−1−ζ7
56.5.2c2 R 2 2 0 0 −2 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 −ζ7−3−ζ73 −ζ7−2−ζ72 −ζ7−1−ζ7 −ζ7−1−ζ7 −ζ7−2−ζ72 −ζ7−3−ζ73
56.5.2c3 R 2 2 0 0 −2 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 −ζ7−2−ζ72 −ζ7−1−ζ7 −ζ7−3−ζ73 −ζ7−3−ζ73 −ζ7−1−ζ7 −ζ7−2−ζ72
56.5.2d1 R 2 −2 0 0 0 −ζ28−2−ζ282 ζ28−4+ζ284 −ζ28−6−ζ286 ζ28−6+ζ286 −ζ28−4−ζ284 ζ28−2+ζ282 −ζ28−3−ζ283 ζ28−5+ζ285 ζ28−1+ζ28 −ζ28−1−ζ28 −ζ28−5−ζ285 ζ28−3+ζ283
56.5.2d2 R 2 −2 0 0 0 −ζ28−2−ζ282 ζ28−4+ζ284 −ζ28−6−ζ286 ζ28−6+ζ286 −ζ28−4−ζ284 ζ28−2+ζ282 ζ28−3+ζ283 −ζ28−5−ζ285 −ζ28−1−ζ28 ζ28−1+ζ28 ζ28−5+ζ285 −ζ28−3−ζ283
56.5.2d3 R 2 −2 0 0 0 −ζ28−6−ζ286 −ζ28−2−ζ282 ζ28−4+ζ284 −ζ28−4−ζ284 ζ28−2+ζ282 ζ28−6+ζ286 −ζ28−5−ζ285 ζ28−1+ζ28 −ζ28−3−ζ283 ζ28−3+ζ283 −ζ28−1−ζ28 ζ28−5+ζ285
56.5.2d4 R 2 −2 0 0 0 −ζ28−6−ζ286 −ζ28−2−ζ282 ζ28−4+ζ284 −ζ28−4−ζ284 ζ28−2+ζ282 ζ28−6+ζ286 ζ28−5+ζ285 −ζ28−1−ζ28 ζ28−3+ζ283 −ζ28−3−ζ283 ζ28−1+ζ28 −ζ28−5−ζ285
56.5.2d5 R 2 −2 0 0 0 ζ28−4+ζ284 −ζ28−6−ζ286 −ζ28−2−ζ282 ζ28−2+ζ282 ζ28−6+ζ286 −ζ28−4−ζ284 −ζ28−1−ζ28 −ζ28−3−ζ283 −ζ28−5−ζ285 ζ28−5+ζ285 ζ28−3+ζ283 ζ28−1+ζ28
56.5.2d6 R 2 −2 0 0 0 ζ28−4+ζ284 −ζ28−6−ζ286 −ζ28−2−ζ282 ζ28−2+ζ282 ζ28−6+ζ286 −ζ28−4−ζ284 ζ28−1+ζ28 ζ28−3+ζ283 ζ28−5+ζ285 −ζ28−5−ζ285 −ζ28−3−ζ283 −ζ28−1−ζ28

magma: CharacterTable(G);