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Properties

Label 28T36
Degree $28$
Order $196$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_7^2$

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Show commands: Magma

magma: G := TransitiveGroup(28, 36);
 

Group action invariants

Degree $n$:  $28$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $36$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_7^2$
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:  (1,2)(3,7)(4,8)(5,25)(6,26)(9,22)(10,21)(11,27)(12,28)(13,18)(14,17)(15,24)(16,23)(19,20), (1,12,25,8,22,3,18,27,14,24,10,20,6,16)(2,11,26,7,21,4,17,28,13,23,9,19,5,15)
magma: Generators(G);
 

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 7: None

Degree 14: 14T13

Low degree siblings

14T13 x 3, 28T36 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderRepresentative
$1^{28}$ $1$ $1$ $()$
$7^{2},1^{14}$ $4$ $7$ $( 3, 8,12,16,20,24,27)( 4, 7,11,15,19,23,28)$
$7^{2},1^{14}$ $4$ $7$ $( 3,12,20,27, 8,16,24)( 4,11,19,28, 7,15,23)$
$7^{2},1^{14}$ $4$ $7$ $( 3,16,27,12,24, 8,20)( 4,15,28,11,23, 7,19)$
$2^{14}$ $49$ $2$ $( 1, 2)( 3, 4)( 5,25)( 6,26)( 7,27)( 8,28)( 9,22)(10,21)(11,24)(12,23)(13,18) (14,17)(15,20)(16,19)$
$2^{14}$ $7$ $2$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23) (22,24)(25,27)(26,28)$
$14^{2}$ $14$ $14$ $( 1, 3, 6, 8,10,12,14,16,18,20,22,24,25,27)( 2, 4, 5, 7, 9,11,13,15,17,19,21, 23,26,28)$
$14^{2}$ $14$ $14$ $( 1, 3,10,12,18,20,25,27, 6, 8,14,16,22,24)( 2, 4, 9,11,17,19,26,28, 5, 7,13, 15,21,23)$
$14^{2}$ $14$ $14$ $( 1, 3,14,16,25,27,10,12,22,24, 6, 8,18,20)( 2, 4,13,15,26,28, 9,11,21,23, 5, 7,17,19)$
$2^{14}$ $7$ $2$ $( 1, 4)( 2, 3)( 5,27)( 6,28)( 7,25)( 8,26)( 9,24)(10,23)(11,22)(12,21)(13,20) (14,19)(15,18)(16,17)$
$14^{2}$ $14$ $14$ $( 1, 4, 6,28,10,23,14,19,18,15,22,11,25, 7)( 2, 3, 5,27, 9,24,13,20,17,16,21, 12,26, 8)$
$14^{2}$ $14$ $14$ $( 1, 4,10,23,18,15,25, 7, 6,28,14,19,22,11)( 2, 3, 9,24,17,16,26, 8, 5,27,13, 20,21,12)$
$14^{2}$ $14$ $14$ $( 1, 4,14,19,25, 7,10,23,22,11, 6,28,18,15)( 2, 3,13,20,26, 8, 9,24,21,12, 5, 27,17,16)$
$7^{4}$ $2$ $7$ $( 1, 6,10,14,18,22,25)( 2, 5, 9,13,17,21,26)( 3, 8,12,16,20,24,27) ( 4, 7,11,15,19,23,28)$
$7^{4}$ $4$ $7$ $( 1, 6,10,14,18,22,25)( 2, 5, 9,13,17,21,26)( 3,12,20,27, 8,16,24) ( 4,11,19,28, 7,15,23)$
$7^{4}$ $4$ $7$ $( 1, 6,10,14,18,22,25)( 2, 5, 9,13,17,21,26)( 3,16,27,12,24, 8,20) ( 4,15,28,11,23, 7,19)$
$7^{4}$ $4$ $7$ $( 1, 6,10,14,18,22,25)( 2, 5, 9,13,17,21,26)( 3,20, 8,24,12,27,16) ( 4,19, 7,23,11,28,15)$
$7^{4}$ $4$ $7$ $( 1, 6,10,14,18,22,25)( 2, 5, 9,13,17,21,26)( 3,24,16, 8,27,20,12) ( 4,23,15, 7,28,19,11)$
$7^{4}$ $2$ $7$ $( 1, 6,10,14,18,22,25)( 2, 5, 9,13,17,21,26)( 3,27,24,20,16,12, 8) ( 4,28,23,19,15,11, 7)$
$7^{4}$ $2$ $7$ $( 1,10,18,25, 6,14,22)( 2, 9,17,26, 5,13,21)( 3,12,20,27, 8,16,24) ( 4,11,19,28, 7,15,23)$
$7^{4}$ $4$ $7$ $( 1,10,18,25, 6,14,22)( 2, 9,17,26, 5,13,21)( 3,16,27,12,24, 8,20) ( 4,15,28,11,23, 7,19)$
$7^{4}$ $4$ $7$ $( 1,10,18,25, 6,14,22)( 2, 9,17,26, 5,13,21)( 3,20, 8,24,12,27,16) ( 4,19, 7,23,11,28,15)$
$7^{4}$ $2$ $7$ $( 1,10,18,25, 6,14,22)( 2, 9,17,26, 5,13,21)( 3,24,16, 8,27,20,12) ( 4,23,15, 7,28,19,11)$
$7^{4}$ $2$ $7$ $( 1,14,25,10,22, 6,18)( 2,13,26, 9,21, 5,17)( 3,16,27,12,24, 8,20) ( 4,15,28,11,23, 7,19)$
$7^{4}$ $2$ $7$ $( 1,14,25,10,22, 6,18)( 2,13,26, 9,21, 5,17)( 3,20, 8,24,12,27,16) ( 4,19, 7,23,11,28,15)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $196=2^{2} \cdot 7^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  196.9
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 7A1 7A2 7A3 7B1 7B2 7B3 7C1 7C2 7C3 7D1 7D2 7D3 7E1 7E2 7E3 14A1 14A3 14A5 14B1 14B3 14B5
Size 1 7 7 49 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 14 14 14 14 14 14
2 P 1A 1A 1A 1A 7A2 7A3 7A1 7B2 7B3 7B1 7D2 7C1 7D3 7C3 7D1 7E2 7E1 7C2 7E3 7B3 7A2 7B2 7A1 7B1 7A3
7 P 1A 2A 2B 2C 7A3 7A1 7A2 7B3 7B1 7B2 7D3 7C2 7D1 7C1 7D2 7E3 7E2 7C3 7E1 14B5 14A1 14B1 14A3 14B3 14A5
Type
196.9.1a 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
196.9.1b 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
196.9.1c 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
196.9.1d 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
196.9.2a1 R 2 0 2 0 2 2 2 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 0 0 0 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73
196.9.2a2 R 2 0 2 0 2 2 2 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 0 0 0 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72
196.9.2a3 R 2 0 2 0 2 2 2 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 0 0 0 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7
196.9.2b1 R 2 2 0 0 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 2 2 2 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 0 0 0
196.9.2b2 R 2 2 0 0 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 2 2 2 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 0 0 0
196.9.2b3 R 2 2 0 0 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 2 2 2 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 0 0 0
196.9.2c1 R 2 −2 0 0 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 2 2 2 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 −ζ7−2−ζ72 −ζ7−1−ζ7 −ζ7−3−ζ73 0 0 0
196.9.2c2 R 2 −2 0 0 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 2 2 2 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 −ζ7−1−ζ7 −ζ7−3−ζ73 −ζ7−2−ζ72 0 0 0
196.9.2c3 R 2 −2 0 0 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 2 2 2 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 −ζ7−3−ζ73 −ζ7−2−ζ72 −ζ7−1−ζ7 0 0 0
196.9.2d1 R 2 0 −2 0 2 2 2 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 0 0 0 −ζ7−2−ζ72 −ζ7−1−ζ7 −ζ7−3−ζ73
196.9.2d2 R 2 0 −2 0 2 2 2 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 0 0 0 −ζ7−1−ζ7 −ζ7−3−ζ73 −ζ7−2−ζ72
196.9.2d3 R 2 0 −2 0 2 2 2 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 0 0 0 −ζ7−3−ζ73 −ζ7−2−ζ72 −ζ7−1−ζ7
196.9.4a1 R 4 0 0 0 2ζ7−3+2ζ73 2ζ7−1+2ζ7 2ζ7−2+2ζ72 2ζ7−2+2ζ72 2ζ7−3+2ζ73 2ζ7−1+2ζ7 −ζ7−3−1−ζ73 −ζ7−1−1−ζ7 −ζ7−2−1−ζ72 −ζ7−2−1−ζ72 −ζ7−3−1−ζ73 −ζ7−1−1−ζ7 ζ7−3+2+ζ73 ζ7−1+2+ζ7 ζ7−2+2+ζ72 0 0 0 0 0 0
196.9.4a2 R 4 0 0 0 2ζ7−2+2ζ72 2ζ7−3+2ζ73 2ζ7−1+2ζ7 2ζ7−1+2ζ7 2ζ7−2+2ζ72 2ζ7−3+2ζ73 −ζ7−2−1−ζ72 −ζ7−3−1−ζ73 −ζ7−1−1−ζ7 −ζ7−1−1−ζ7 −ζ7−2−1−ζ72 −ζ7−3−1−ζ73 ζ7−2+2+ζ72 ζ7−3+2+ζ73 ζ7−1+2+ζ7 0 0 0 0 0 0
196.9.4a3 R 4 0 0 0 2ζ7−1+2ζ7 2ζ7−2+2ζ72 2ζ7−3+2ζ73 2ζ7−3+2ζ73 2ζ7−1+2ζ7 2ζ7−2+2ζ72 −ζ7−1−1−ζ7 −ζ7−2−1−ζ72 −ζ7−3−1−ζ73 −ζ7−3−1−ζ73 −ζ7−1−1−ζ7 −ζ7−2−1−ζ72 ζ7−1+2+ζ7 ζ7−2+2+ζ72 ζ7−3+2+ζ73 0 0 0 0 0 0
196.9.4b1 R 4 0 0 0 2ζ7−3+2ζ73 2ζ7−1+2ζ7 2ζ7−2+2ζ72 2ζ7−1+2ζ7 2ζ7−2+2ζ72 2ζ7−3+2ζ73 −ζ7−1−1−ζ7 −ζ7−2−1−ζ72 −ζ7−3−1−ζ73 ζ7−2+2+ζ72 ζ7−3+2+ζ73 ζ7−1+2+ζ7 −ζ7−2−1−ζ72 −ζ7−3−1−ζ73 −ζ7−1−1−ζ7 0 0 0 0 0 0
196.9.4b2 R 4 0 0 0 2ζ7−2+2ζ72 2ζ7−3+2ζ73 2ζ7−1+2ζ7 2ζ7−3+2ζ73 2ζ7−1+2ζ7 2ζ7−2+2ζ72 −ζ7−3−1−ζ73 −ζ7−1−1−ζ7 −ζ7−2−1−ζ72 ζ7−1+2+ζ7 ζ7−2+2+ζ72 ζ7−3+2+ζ73 −ζ7−1−1−ζ7 −ζ7−2−1−ζ72 −ζ7−3−1−ζ73 0 0 0 0 0 0
196.9.4b3 R 4 0 0 0 2ζ7−1+2ζ7 2ζ7−2+2ζ72 2ζ7−3+2ζ73 2ζ7−2+2ζ72 2ζ7−3+2ζ73 2ζ7−1+2ζ7 −ζ7−2−1−ζ72 −ζ7−3−1−ζ73 −ζ7−1−1−ζ7 ζ7−3+2+ζ73 ζ7−1+2+ζ7 ζ7−2+2+ζ72 −ζ7−3−1−ζ73 −ζ7−1−1−ζ7 −ζ7−2−1−ζ72 0 0 0 0 0 0
196.9.4c1 R 4 0 0 0 2ζ7−3+2ζ73 2ζ7−1+2ζ7 2ζ7−2+2ζ72 2ζ7−3+2ζ73 2ζ7−1+2ζ7 2ζ7−2+2ζ72 ζ7−1+2+ζ7 ζ7−2+2+ζ72 ζ7−3+2+ζ73 −ζ7−1−1−ζ7 −ζ7−2−1−ζ72 −ζ7−3−1−ζ73 −ζ7−3−1−ζ73 −ζ7−1−1−ζ7 −ζ7−2−1−ζ72 0 0 0 0 0 0
196.9.4c2 R 4 0 0 0 2ζ7−2+2ζ72 2ζ7−3+2ζ73 2ζ7−1+2ζ7 2ζ7−2+2ζ72 2ζ7−3+2ζ73 2ζ7−1+2ζ7 ζ7−3+2+ζ73 ζ7−1+2+ζ7 ζ7−2+2+ζ72 −ζ7−3−1−ζ73 −ζ7−1−1−ζ7 −ζ7−2−1−ζ72 −ζ7−2−1−ζ72 −ζ7−3−1−ζ73 −ζ7−1−1−ζ7 0 0 0 0 0 0
196.9.4c3 R 4 0 0 0 2ζ7−1+2ζ7 2ζ7−2+2ζ72 2ζ7−3+2ζ73 2ζ7−1+2ζ7 2ζ7−2+2ζ72 2ζ7−3+2ζ73 ζ7−2+2+ζ72 ζ7−3+2+ζ73 ζ7−1+2+ζ7 −ζ7−2−1−ζ72 −ζ7−3−1−ζ73 −ζ7−1−1−ζ7 −ζ7−1−1−ζ7 −ζ7−2−1−ζ72 −ζ7−3−1−ζ73 0 0 0 0 0 0

magma: CharacterTable(G);