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Properties

Label 28T36
28T36 1 2 1->2 12 1->12 11 2->11 3 7 3->7 18 3->18 4 8 4->8 17 4->17 5 15 5->15 25 5->25 6 16 6->16 26 6->26 21 7->21 22 8->22 9 19 9->19 9->22 10 20 10->20 10->21 11->26 27 11->27 12->25 28 12->28 13 13->18 23 13->23 14 14->17 24 14->24 15->2 15->24 16->1 16->23 17->28 18->27 19->5 19->20 20->6 21->4 22->3 23->9 24->10 25->8 26->7 27->14 28->13
Degree $28$
Order $196$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_7^2$

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

Group invariants

Abstract group:  $D_7^2$
Copy content magma:IdentifyGroup(G);
 
Order:  $196=2^{2} \cdot 7^{2}$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  yes
Copy content magma:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $28$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $36$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $14$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,2)(3,7)(4,8)(5,25)(6,26)(9,22)(10,21)(11,27)(12,28)(13,18)(14,17)(15,24)(16,23)(19,20)$, $(1,12,25,8,22,3,18,27,14,24,10,20,6,16)(2,11,26,7,21,4,17,28,13,23,9,19,5,15)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$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 TypeSizeOrderIndexRepresentative
1A $1^{28}$ $1$ $1$ $0$ $()$
2A $2^{14}$ $7$ $2$ $14$ $( 1,20)( 2,19)( 3,14)( 4,13)( 5,23)( 6,24)( 7,17)( 8,18)( 9,28)(10,27)(11,21)(12,22)(15,26)(16,25)$
2B $2^{14}$ $7$ $2$ $14$ $( 1,19)( 2,20)( 3,17)( 4,18)( 5,16)( 6,15)( 7,14)( 8,13)( 9,12)(10,11)(21,27)(22,28)(23,25)(24,26)$
2C $2^{14}$ $49$ $2$ $14$ $( 1,13)( 2,14)( 3,11)( 4,12)( 5,10)( 6, 9)( 7, 8)(15,27)(16,28)(17,25)(18,26)(19,24)(20,23)(21,22)$
7A1 $7^{4}$ $2$ $7$ $24$ $( 1,14,25,10,22, 6,18)( 2,13,26, 9,21, 5,17)( 3,16,27,12,24, 8,20)( 4,15,28,11,23, 7,19)$
7A2 $7^{4}$ $2$ $7$ $24$ $( 1,25,22,18,14,10, 6)( 2,26,21,17,13, 9, 5)( 3,27,24,20,16,12, 8)( 4,28,23,19,15,11, 7)$
7A3 $7^{4}$ $2$ $7$ $24$ $( 1,10,18,25, 6,14,22)( 2, 9,17,26, 5,13,21)( 3,12,20,27, 8,16,24)( 4,11,19,28, 7,15,23)$
7B1 $7^{4}$ $2$ $7$ $24$ $( 1,10,18,25, 6,14,22)( 2, 9,17,26, 5,13,21)( 3,24,16, 8,27,20,12)( 4,23,15, 7,28,19,11)$
7B2 $7^{4}$ $2$ $7$ $24$ $( 1,18, 6,22,10,25,14)( 2,17, 5,21, 9,26,13)( 3,16,27,12,24, 8,20)( 4,15,28,11,23, 7,19)$
7B3 $7^{4}$ $2$ $7$ $24$ $( 1,25,22,18,14,10, 6)( 2,26,21,17,13, 9, 5)( 3, 8,12,16,20,24,27)( 4, 7,11,15,19,23,28)$
7C1 $7^{4}$ $4$ $7$ $24$ $( 1,18, 6,22,10,25,14)( 2,17, 5,21, 9,26,13)( 3,12,20,27, 8,16,24)( 4,11,19,28, 7,15,23)$
7C2 $7^{4}$ $4$ $7$ $24$ $( 1, 6,10,14,18,22,25)( 2, 5, 9,13,17,21,26)( 3,20, 8,24,12,27,16)( 4,19, 7,23,11,28,15)$
7C3 $7^{4}$ $4$ $7$ $24$ $( 1,22,14, 6,25,18,10)( 2,21,13, 5,26,17, 9)( 3,27,24,20,16,12, 8)( 4,28,23,19,15,11, 7)$
7D1 $7^{4}$ $4$ $7$ $24$ $( 1,22,14, 6,25,18,10)( 2,21,13, 5,26,17, 9)( 3, 8,12,16,20,24,27)( 4, 7,11,15,19,23,28)$
7D2 $7^{4}$ $4$ $7$ $24$ $( 1,10,18,25, 6,14,22)( 2, 9,17,26, 5,13,21)( 3,16,27,12,24, 8,20)( 4,15,28,11,23, 7,19)$
7D3 $7^{4}$ $4$ $7$ $24$ $( 1,14,25,10,22, 6,18)( 2,13,26, 9,21, 5,17)( 3, 8,12,16,20,24,27)( 4, 7,11,15,19,23,28)$
7E1 $7^{2},1^{14}$ $4$ $7$ $12$ $( 3,24,16, 8,27,20,12)( 4,23,15, 7,28,19,11)$
7E2 $7^{2},1^{14}$ $4$ $7$ $12$ $( 1,18, 6,22,10,25,14)( 2,17, 5,21, 9,26,13)$
7E3 $7^{2},1^{14}$ $4$ $7$ $12$ $( 1,25,22,18,14,10, 6)( 2,26,21,17,13, 9, 5)$
14A1 $14^{2}$ $14$ $14$ $26$ $( 1,12,14,24,25, 8,10,20,22, 3, 6,16,18,27)( 2,11,13,23,26, 7, 9,19,21, 4, 5,15,17,28)$
14A3 $14^{2}$ $14$ $14$ $26$ $( 1,24,10, 3,18,12,25,20, 6,27,14, 8,22,16)( 2,23, 9, 4,17,11,26,19, 5,28,13, 7,21,15)$
14A5 $14^{2}$ $14$ $14$ $26$ $( 1, 8, 6,12,10,16,14,20,18,24,22,27,25, 3)( 2, 7, 5,11, 9,15,13,19,17,23,21,28,26, 4)$
14B1 $14^{2}$ $14$ $14$ $26$ $( 1,15,10, 7,18,28,25,19, 6,11,14, 4,22,23)( 2,16, 9, 8,17,27,26,20, 5,12,13, 3,21,24)$
14B3 $14^{2}$ $14$ $14$ $26$ $( 1, 7,25,11,22,15,18,19,14,23,10,28, 6, 4)( 2, 8,26,12,21,16,17,20,13,24, 9,27, 5, 3)$
14B5 $14^{2}$ $14$ $14$ $26$ $( 1,11,18,23, 6, 7,22,19,10, 4,25,15,14,28)( 2,12,17,24, 5, 8,21,20, 9, 3,26,16,13,27)$

Malle's constant $a(G)$:     $1/12$

Copy content magma:ConjugacyClasses(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 7C2 7C3 7C1 7D2 7D3 7D1 7E2 7E3 7E1 7A1 7A3 7A2 7B1 7B3 7B2
7 P 1A 2A 2B 2C 7A3 7A1 7A2 7B3 7B1 7B2 7C3 7C1 7C2 7D3 7D1 7D2 7E3 7E1 7E2 14A3 14A5 14A1 14B3 14B5 14B1
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 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 0 0 ζ7−2+ζ72 0 ζ7−3+ζ73 ζ7−1+ζ7
196.9.2a2 R 2 0 2 0 2 2 2 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 0 0 ζ7−1+ζ7 0 ζ7−2+ζ72 ζ7−3+ζ73
196.9.2a3 R 2 0 2 0 2 2 2 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 0 0 ζ7−3+ζ73 0 ζ7−1+ζ7 ζ7−2+ζ72
196.9.2b1 R 2 2 0 0 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 2 2 2 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 0 ζ7−3+ζ73 0 0
196.9.2b2 R 2 2 0 0 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 2 2 2 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 0 ζ7−2+ζ72 0 0
196.9.2b3 R 2 2 0 0 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 2 2 2 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 0 ζ7−1+ζ7 0 0
196.9.2c1 R 2 −2 0 0 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 2 2 2 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−3+ζ73 −ζ7−2−ζ72 −ζ7−1−ζ7 0 −ζ7−3−ζ73 0 0
196.9.2c2 R 2 −2 0 0 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 2 2 2 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−2+ζ72 −ζ7−1−ζ7 −ζ7−3−ζ73 0 −ζ7−2−ζ72 0 0
196.9.2c3 R 2 −2 0 0 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 2 2 2 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−1+ζ7 −ζ7−3−ζ73 −ζ7−2−ζ72 0 −ζ7−1−ζ7 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 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 0 0 −ζ7−2−ζ72 0 −ζ7−3−ζ73 −ζ7−1−ζ7
196.9.2d2 R 2 0 −2 0 2 2 2 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 0 0 −ζ7−1−ζ7 0 −ζ7−2−ζ72 −ζ7−3−ζ73
196.9.2d3 R 2 0 −2 0 2 2 2 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 0 0 −ζ7−3−ζ73 0 −ζ7−1−ζ7 −ζ7−2−ζ72
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 −ζ7−2−1−ζ72 −ζ7−2−1−ζ72 −ζ7−3−1−ζ73 ζ7−3+2+ζ73 ζ7−1+2+ζ7 −ζ7−1−1−ζ7 −ζ7−1−1−ζ7 ζ7−2+2+ζ72 −ζ7−3−1−ζ73 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 −ζ7−1−1−ζ7 −ζ7−1−1−ζ7 −ζ7−2−1−ζ72 ζ7−2+2+ζ72 ζ7−3+2+ζ73 −ζ7−3−1−ζ73 −ζ7−3−1−ζ73 ζ7−1+2+ζ7 −ζ7−2−1−ζ72 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 −ζ7−3−1−ζ73 −ζ7−3−1−ζ73 −ζ7−1−1−ζ7 ζ7−1+2+ζ7 ζ7−2+2+ζ72 −ζ7−2−1−ζ72 −ζ7−2−1−ζ72 ζ7−3+2+ζ73 −ζ7−1−1−ζ7 0 0 0 0 0 0
196.9.4b1 R 4 0 0 0 2ζ7−3+2ζ73 2ζ7−1+2ζ7 2ζ7−2+2ζ72 2ζ7−3+2ζ73 2ζ7−1+2ζ7 2ζ7−2+2ζ72 −ζ7−1−1−ζ7 ζ7−3+2+ζ73 −ζ7−2−1−ζ72 −ζ7−3−1−ζ73 −ζ7−1−1−ζ7 −ζ7−3−1−ζ73 ζ7−2+2+ζ72 −ζ7−2−1−ζ72 ζ7−1+2+ζ7 0 0 0 0 0 0
196.9.4b2 R 4 0 0 0 2ζ7−2+2ζ72 2ζ7−3+2ζ73 2ζ7−1+2ζ7 2ζ7−2+2ζ72 2ζ7−3+2ζ73 2ζ7−1+2ζ7 −ζ7−3−1−ζ73 ζ7−2+2+ζ72 −ζ7−1−1−ζ7 −ζ7−2−1−ζ72 −ζ7−3−1−ζ73 −ζ7−2−1−ζ72 ζ7−1+2+ζ7 −ζ7−1−1−ζ7 ζ7−3+2+ζ73 0 0 0 0 0 0
196.9.4b3 R 4 0 0 0 2ζ7−1+2ζ7 2ζ7−2+2ζ72 2ζ7−3+2ζ73 2ζ7−1+2ζ7 2ζ7−2+2ζ72 2ζ7−3+2ζ73 −ζ7−2−1−ζ72 ζ7−1+2+ζ7 −ζ7−3−1−ζ73 −ζ7−1−1−ζ7 −ζ7−2−1−ζ72 −ζ7−1−1−ζ7 ζ7−3+2+ζ73 −ζ7−3−1−ζ73 ζ7−2+2+ζ72 0 0 0 0 0 0
196.9.4c1 R 4 0 0 0 2ζ7−3+2ζ73 2ζ7−1+2ζ7 2ζ7−2+2ζ72 2ζ7−1+2ζ7 2ζ7−2+2ζ72 2ζ7−3+2ζ73 ζ7−2+2+ζ72 −ζ7−3−1−ζ73 ζ7−3+2+ζ73 −ζ7−2−1−ζ72 −ζ7−3−1−ζ73 ζ7−1+2+ζ7 −ζ7−2−1−ζ72 −ζ7−1−1−ζ7 −ζ7−1−1−ζ7 0 0 0 0 0 0
196.9.4c2 R 4 0 0 0 2ζ7−2+2ζ72 2ζ7−3+2ζ73 2ζ7−1+2ζ7 2ζ7−3+2ζ73 2ζ7−1+2ζ7 2ζ7−2+2ζ72 ζ7−1+2+ζ7 −ζ7−2−1−ζ72 ζ7−2+2+ζ72 −ζ7−1−1−ζ7 −ζ7−2−1−ζ72 ζ7−3+2+ζ73 −ζ7−1−1−ζ7 −ζ7−3−1−ζ73 −ζ7−3−1−ζ73 0 0 0 0 0 0
196.9.4c3 R 4 0 0 0 2ζ7−1+2ζ7 2ζ7−2+2ζ72 2ζ7−3+2ζ73 2ζ7−2+2ζ72 2ζ7−3+2ζ73 2ζ7−1+2ζ7 ζ7−3+2+ζ73 −ζ7−1−1−ζ7 ζ7−1+2+ζ7 −ζ7−3−1−ζ73 −ζ7−1−1−ζ7 ζ7−2+2+ζ72 −ζ7−3−1−ζ73 −ζ7−2−1−ζ72 −ζ7−2−1−ζ72 0 0 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed