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Properties

Label 28T50
28T50 1 16 1->16 1->16 2 15 2->15 2->15 3 3->1 3->2 4 4->1 4->2 5 12 5->12 20 5->20 6 11 6->11 19 6->19 7 7->6 26 7->26 8 8->5 25 8->25 9 9->8 23 9->23 10 10->7 24 10->24 11->10 22 11->22 12->9 21 12->21 13 13->4 27 13->27 14 14->3 28 14->28 15->13 18 15->18 16->14 17 16->17 17->4 17->27 18->3 18->28 19->13 19->18 20->14 20->17 21->7 21->24 22->8 22->23 23->9 23->21 24->10 24->22 25->11 25->19 26->12 26->20 27->6 27->26 28->5 28->25
Degree $28$
Order $392$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_7:D_{28}$

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Copy content magma:G := TransitiveGroup(28, 50);
 

Group invariants

Abstract group:  $C_7:D_{28}$
Copy content magma:IdentifyGroup(G);
 
Order:  $392=2^{3} \cdot 7^{2}$
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 $t$:  $50$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $2$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,16,14,28,25,11,10,24,22,8,5,20,17,4,2,15,13,27,26,12,9,23,21,7,6,19,18,3)$, $(1,16,17,27,6,11,22,23,9,8,25,19,13,4)(2,15,18,28,5,12,21,24,10,7,26,20,14,3)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$8$:  $D_{4}$
$14$:  $D_{7}$ x 2
$28$:  $D_{14}$ x 2
$56$:  28T6, $D_{28}$
$196$:  14T13

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 7: None

Degree 14: 14T13

Low degree siblings

28T50 x 5

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{28}$ $1$ $1$ $0$ $()$
2A $2^{14}$ $1$ $2$ $14$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)$
2B $2^{14}$ $14$ $2$ $14$ $( 1, 4)( 2, 3)( 5,28)( 6,27)( 7,26)( 8,25)( 9,23)(10,24)(11,22)(12,21)(13,19)(14,20)(15,18)(16,17)$
2C $2^{13},1^{2}$ $98$ $2$ $13$ $( 1, 5)( 2, 6)( 7,28)( 8,27)( 9,26)(10,25)(11,23)(12,24)(13,21)(14,22)(15,20)(16,19)(17,18)$
4A $4^{7}$ $14$ $4$ $21$ $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,12,10,11)(13,15,14,16)(17,20,18,19)(21,23,22,24)(25,28,26,27)$
7A1 $7^{4}$ $2$ $7$ $24$ $( 1,17, 6,22, 9,25,13)( 2,18, 5,21,10,26,14)( 3,15,28,12,24, 7,20)( 4,16,27,11,23, 8,19)$
7A2 $7^{4}$ $2$ $7$ $24$ $( 1, 6, 9,13,17,22,25)( 2, 5,10,14,18,21,26)( 3,28,24,20,15,12, 7)( 4,27,23,19,16,11, 8)$
7A3 $7^{4}$ $2$ $7$ $24$ $( 1,22,13, 6,25,17, 9)( 2,21,14, 5,26,18,10)( 3,12,20,28, 7,15,24)( 4,11,19,27, 8,16,23)$
7B1 $7^{4}$ $2$ $7$ $24$ $( 1,17, 6,22, 9,25,13)( 2,18, 5,21,10,26,14)( 3,20, 7,24,12,28,15)( 4,19, 8,23,11,27,16)$
7B2 $7^{4}$ $2$ $7$ $24$ $( 1, 6, 9,13,17,22,25)( 2, 5,10,14,18,21,26)( 3, 7,12,15,20,24,28)( 4, 8,11,16,19,23,27)$
7B3 $7^{4}$ $2$ $7$ $24$ $( 1,22,13, 6,25,17, 9)( 2,21,14, 5,26,18,10)( 3,24,15, 7,28,20,12)( 4,23,16, 8,27,19,11)$
7C1 $7^{2},1^{14}$ $4$ $7$ $12$ $( 1, 6, 9,13,17,22,25)( 2, 5,10,14,18,21,26)$
7C2 $7^{2},1^{14}$ $4$ $7$ $12$ $( 1, 9,17,25, 6,13,22)( 2,10,18,26, 5,14,21)$
7C3 $7^{2},1^{14}$ $4$ $7$ $12$ $( 1,13,25, 9,22, 6,17)( 2,14,26,10,21, 5,18)$
7D1 $7^{4}$ $4$ $7$ $24$ $( 1,22,13, 6,25,17, 9)( 2,21,14, 5,26,18,10)( 3,15,28,12,24, 7,20)( 4,16,27,11,23, 8,19)$
7D2 $7^{4}$ $4$ $7$ $24$ $( 1,13,25, 9,22, 6,17)( 2,14,26,10,21, 5,18)( 3,28,24,20,15,12, 7)( 4,27,23,19,16,11, 8)$
7D3 $7^{4}$ $4$ $7$ $24$ $( 1, 6, 9,13,17,22,25)( 2, 5,10,14,18,21,26)( 3,12,20,28, 7,15,24)( 4,11,19,27, 8,16,23)$
7E1 $7^{4}$ $4$ $7$ $24$ $( 1, 9,17,25, 6,13,22)( 2,10,18,26, 5,14,21)( 3,28,24,20,15,12, 7)( 4,27,23,19,16,11, 8)$
7E2 $7^{4}$ $4$ $7$ $24$ $( 1,17, 6,22, 9,25,13)( 2,18, 5,21,10,26,14)( 3,24,15, 7,28,20,12)( 4,23,16, 8,27,19,11)$
7E3 $7^{4}$ $4$ $7$ $24$ $( 1,25,22,17,13, 9, 6)( 2,26,21,18,14,10, 5)( 3,20, 7,24,12,28,15)( 4,19, 8,23,11,27,16)$
14A1 $14^{2}$ $2$ $14$ $26$ $( 1,10,17,26, 6,14,22, 2, 9,18,25, 5,13,21)( 3,23,15, 8,28,19,12, 4,24,16, 7,27,20,11)$
14A3 $14^{2}$ $2$ $14$ $26$ $( 1,26,22,18,13,10, 6, 2,25,21,17,14, 9, 5)( 3, 8,12,16,20,23,28, 4, 7,11,15,19,24,27)$
14A5 $14^{2}$ $2$ $14$ $26$ $( 1,18, 6,21, 9,26,13, 2,17, 5,22,10,25,14)( 3,16,28,11,24, 8,20, 4,15,27,12,23, 7,19)$
14B1 $14^{2}$ $2$ $14$ $26$ $( 1,10,17,26, 6,14,22, 2, 9,18,25, 5,13,21)( 3,11,20,27, 7,16,24, 4,12,19,28, 8,15,23)$
14B3 $14^{2}$ $2$ $14$ $26$ $( 1,26,22,18,13,10, 6, 2,25,21,17,14, 9, 5)( 3,27,24,19,15,11, 7, 4,28,23,20,16,12, 8)$
14B5 $14^{2}$ $2$ $14$ $26$ $( 1,14,25,10,22, 5,17, 2,13,26, 9,21, 6,18)( 3,16,28,11,24, 8,20, 4,15,27,12,23, 7,19)$
14C1 $14,2^{7}$ $4$ $14$ $20$ $( 1,18, 6,21, 9,26,13, 2,17, 5,22,10,25,14)( 3, 4)( 7, 8)(11,12)(15,16)(19,20)(23,24)(27,28)$
14C3 $14,2^{7}$ $4$ $14$ $20$ $( 1,21,13, 5,25,18, 9, 2,22,14, 6,26,17,10)( 3, 4)( 7, 8)(11,12)(15,16)(19,20)(23,24)(27,28)$
14C5 $14,2^{7}$ $4$ $14$ $20$ $( 1, 2)( 3,27,24,19,15,11, 7, 4,28,23,20,16,12, 8)( 5, 6)( 9,10)(13,14)(17,18)(21,22)(25,26)$
14D1 $14^{2}$ $4$ $14$ $26$ $( 1,26,22,18,13,10, 6, 2,25,21,17,14, 9, 5)( 3,23,15, 8,28,19,12, 4,24,16, 7,27,20,11)$
14D3 $14^{2}$ $4$ $14$ $26$ $( 1, 5, 9,14,17,21,25, 2, 6,10,13,18,22,26)( 3,19, 7,23,12,27,15, 4,20, 8,24,11,28,16)$
14D5 $14^{2}$ $4$ $14$ $26$ $( 1,10,17,26, 6,14,22, 2, 9,18,25, 5,13,21)( 3,19, 7,23,12,27,15, 4,20, 8,24,11,28,16)$
14E1 $14^{2}$ $4$ $14$ $26$ $( 1, 5, 9,14,17,21,25, 2, 6,10,13,18,22,26)( 3,16,28,11,24, 8,20, 4,15,27,12,23, 7,19)$
14E3 $14^{2}$ $4$ $14$ $26$ $( 1,14,25,10,22, 5,17, 2,13,26, 9,21, 6,18)( 3,11,20,27, 7,16,24, 4,12,19,28, 8,15,23)$
14E5 $14^{2}$ $4$ $14$ $26$ $( 1,21,13, 5,25,18, 9, 2,22,14, 6,26,17,10)( 3, 8,12,16,20,23,28, 4, 7,11,15,19,24,27)$
14F1 $14^{2}$ $14$ $14$ $26$ $( 1,23,17, 8, 6,19,22, 4, 9,16,25,27,13,11)( 2,24,18, 7, 5,20,21, 3,10,15,26,28,14,12)$
14F-1 $14^{2}$ $14$ $14$ $26$ $( 1,11,13,27,25,16, 9, 4,22,19, 6, 8,17,23)( 2,12,14,28,26,15,10, 3,21,20, 5, 7,18,24)$
14F3 $14^{2}$ $14$ $14$ $26$ $( 1, 8,22,16,13,23, 6, 4,25,11,17,19, 9,27)( 2, 7,21,15,14,24, 5, 3,26,12,18,20,10,28)$
14F-3 $14^{2}$ $14$ $14$ $26$ $( 1,27, 9,19,17,11,25, 4, 6,23,13,16,22, 8)( 2,28,10,20,18,12,26, 3, 5,24,14,15,21, 7)$
14F5 $14^{2}$ $14$ $14$ $26$ $( 1,19,25,23,22,27,17, 4,13, 8, 9,11, 6,16)( 2,20,26,24,21,28,18, 3,14, 7,10,12, 5,15)$
14F-5 $14^{2}$ $14$ $14$ $26$ $( 1,16, 6,11, 9, 8,13, 4,17,27,22,23,25,19)( 2,15, 5,12,10, 7,14, 3,18,28,21,24,26,20)$
28A1 $28$ $14$ $28$ $27$ $( 1, 8,10,15,17,23,26, 3, 6,11,14,20,22,27, 2, 7, 9,16,18,24,25, 4, 5,12,13,19,21,28)$
28A3 $28$ $14$ $28$ $27$ $( 1,15,26,11,22, 7,18, 4,13,28,10,23, 6,20, 2,16,25,12,21, 8,17, 3,14,27, 9,24, 5,19)$
28A5 $28$ $14$ $28$ $27$ $( 1,23,14, 7,25,19,10, 3,22,16, 5,28,17,11, 2,24,13, 8,26,20, 9, 4,21,15, 6,27,18,12)$
28A9 $28$ $14$ $28$ $27$ $( 1,11,18,28, 6,16,21, 3, 9,19,26, 7,13,23, 2,12,17,27, 5,15,22, 4,10,20,25, 8,14,24)$
28A11 $28$ $14$ $28$ $27$ $( 1,16,26,12,22, 8,18, 3,13,27,10,24, 6,19, 2,15,25,11,21, 7,17, 4,14,28, 9,23, 5,20)$
28A13 $28$ $14$ $28$ $27$ $( 1,27,21,20,13,11, 5, 3,25,23,18,15, 9, 8, 2,28,22,19,14,12, 6, 4,26,24,17,16,10, 7)$

Malle's constant $a(G)$:     $1/12$

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Character table

47 x 47 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed