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Properties

Label 28T253
28T253 1 7 1->7 8 1->8 2 2->7 2->8 3 17 3->17 22 3->22 4 18 4->18 21 4->21 5 12 5->12 27 5->27 6 11 6->11 28 6->28 7->21 26 7->26 8->22 25 8->25 9 16 9->16 19 9->19 10 15 10->15 20 10->20 11->2 11->25 12->1 12->26 13 13->11 13->20 14 14->12 14->19 15->1 15->6 16->2 16->5 24 17->24 23 18->23 19->5 19->10 20->6 20->9 21->23 21->28 22->24 22->27 23->9 23->14 24->10 24->13 25->4 25->15 26->3 26->16 27->13 27->18 28->14 28->17
Degree $28$
Order $3528$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $F_7\wr C_2$

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Copy content magma:G := TransitiveGroup(28, 253);
 

Group invariants

Abstract group:  $F_7\wr C_2$
Copy content magma:IdentifyGroup(G);
 
Order:  $3528=2^{3} \cdot 3^{2} \cdot 7^{2}$
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 $t$:  $253$
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:  $(1,8,25,4,21,28,17,24,13,20,9,16,5,12)(2,7,26,3,22,27,18,23,14,19,10,15,6,11)$, $(1,7,21,23,9,19,5,27,13,11,25,15)(2,8,22,24,10,20,6,28,14,12,26,16)(3,17)(4,18)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_2^2$
$6$:  $S_3$, $C_6$ x 3
$8$:  $D_{4}$
$12$:  $D_{6}$, $C_6\times C_2$
$18$:  $S_3\times C_3$
$24$:  $(C_6\times C_2):C_2$, $D_4 \times C_3$
$36$:  $C_6\times S_3$
$72$:  12T42

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 7: None

Degree 14: 14T45

Low degree siblings

14T45, 28T251, 28T252, 42T368, 42T369, 42T370, 42T371, 42T372

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 $2^{14}$ $14$ $2$ $14$ $( 1, 2)( 3, 8)( 4, 7)( 5, 6)( 9,10)(11,28)(12,27)(13,14)(15,24)(16,23)(17,18)(19,20)(21,22)(25,26)$
2B $2^{14}$ $42$ $2$ $14$ $( 1,12)( 2,11)( 3, 6)( 4, 5)( 7,18)( 8,17)( 9,24)(10,23)(13,16)(14,15)(19,26)(20,25)(21,28)(22,27)$
2C $2^{12},1^{4}$ $49$ $2$ $12$ $( 1, 9)( 2,10)( 7,27)( 8,28)(11,23)(12,24)(13,25)(14,26)(15,19)(16,20)(17,21)(18,22)$
3A1 $3^{4},1^{16}$ $14$ $3$ $8$ $( 1,21,17)( 2,22,18)( 5, 9,25)( 6,10,26)$
3A-1 $3^{4},1^{16}$ $14$ $3$ $8$ $( 1,17,21)( 2,18,22)( 5,25, 9)( 6,26,10)$
3B1 $3^{8},1^{4}$ $49$ $3$ $16$ $( 1,21,17)( 2,22,18)( 3,11,15)( 4,12,16)( 5, 9,25)( 6,10,26)( 7,27,23)( 8,28,24)$
3B-1 $3^{8},1^{4}$ $49$ $3$ $16$ $( 1,17,21)( 2,18,22)( 3,15,11)( 4,16,12)( 5,25, 9)( 6,26,10)( 7,23,27)( 8,24,28)$
3C $3^{8},1^{4}$ $98$ $3$ $16$ $( 1, 5,21)( 2, 6,22)( 7,11,19)( 8,12,20)(13,25,17)(14,26,18)(15,27,23)(16,28,24)$
4A $4^{6},2^{2}$ $294$ $4$ $20$ $( 1,15, 9,19)( 2,16,10,20)( 3, 5)( 4, 6)( 7,25,27,13)( 8,26,28,14)(11,17,23,21)(12,18,24,22)$
6A1 $6^{2},2^{8}$ $14$ $6$ $18$ $( 1, 2)( 3,24,27,12,19,16)( 4,23,28,11,20,15)( 5, 6)( 7, 8)( 9,10)(13,14)(17,18)(21,22)(25,26)$
6A-1 $6^{2},2^{8}$ $14$ $6$ $18$ $( 1, 2)( 3,16,19,12,27,24)( 4,15,20,11,28,23)( 5, 6)( 7, 8)( 9,10)(13,14)(17,18)(21,22)(25,26)$
6B1 $6^{4},1^{4}$ $49$ $6$ $20$ $( 1,13,17, 9,25,21)( 2,14,18,10,26,22)( 7,23,19,27,11,15)( 8,24,20,28,12,16)$
6B-1 $6^{4},1^{4}$ $49$ $6$ $20$ $( 1,21,25, 9,17,13)( 2,22,26,10,18,14)( 7,15,11,27,19,23)( 8,16,12,28,20,24)$
6C $6^{4},1^{4}$ $98$ $6$ $20$ $( 1,21,25, 9,17,13)( 2,22,26,10,18,14)( 7,23,19,27,11,15)( 8,24,20,28,12,16)$
6D1 $6^{4},2^{2}$ $98$ $6$ $22$ $( 1,18,21, 2,17,22)( 3,24,11, 8,15,28)( 4,23,12, 7,16,27)( 5,26, 9, 6,25,10)(13,14)(19,20)$
6D-1 $6^{4},2^{2}$ $98$ $6$ $22$ $( 1,22,17, 2,21,18)( 3,28,15, 8,11,24)( 4,27,16, 7,12,23)( 5,10,25, 6, 9,26)(13,14)(19,20)$
6E1 $6^{4},2^{2}$ $98$ $6$ $22$ $( 1,22, 5, 2,21, 6)( 3, 4)( 7,16,11,28,19,24)( 8,15,12,27,20,23)( 9,10)(13,18,25,14,17,26)$
6E-1 $6^{4},2^{2}$ $98$ $6$ $22$ $( 1, 6,21, 2, 5,22)( 3, 4)( 7,24,19,28,11,16)( 8,23,20,27,12,15)( 9,10)(13,26,17,14,25,18)$
6F1 $6^{2},2^{8}$ $98$ $6$ $18$ $( 1,18,21, 2,17,22)( 3,12)( 4,11)( 5,26, 9, 6,25,10)( 7, 8)(13,14)(15,28)(16,27)(19,24)(20,23)$
6F-1 $6^{2},2^{8}$ $98$ $6$ $18$ $( 1,22,17, 2,21,18)( 3,12)( 4,11)( 5,10,25, 6, 9,26)( 7, 8)(13,14)(15,28)(16,27)(19,24)(20,23)$
6G1 $6^{2},2^{6},1^{4}$ $98$ $6$ $16$ $( 1, 9)( 2,10)( 3, 7,27,15,11,19)( 4, 8,28,16,12,20)(13,25)(14,26)(17,21)(18,22)$
6G-1 $6^{2},2^{6},1^{4}$ $98$ $6$ $16$ $( 1, 9)( 2,10)( 3,19,11,15,27, 7)( 4,20,12,16,28, 8)(13,25)(14,26)(17,21)(18,22)$
6H1 $6^{4},2^{2}$ $294$ $6$ $22$ $( 1,28,17,12,21, 8)( 2,27,18,11,22, 7)( 3,10,19, 6,23,26)( 4, 9,20, 5,24,25)(13,16)(14,15)$
6H-1 $6^{4},2^{2}$ $294$ $6$ $22$ $( 1, 8,21,12,17,28)( 2, 7,22,11,18,27)( 3,26,23, 6,19,10)( 4,25,24, 5,20, 9)(13,16)(14,15)$
7A $7^{2},1^{14}$ $12$ $7$ $12$ $( 1, 5, 9,13,17,21,25)( 2, 6,10,14,18,22,26)$
7B $7^{4}$ $36$ $7$ $24$ $( 1,17, 5,21, 9,25,13)( 2,18, 6,22,10,26,14)( 3,23,15, 7,27,19,11)( 4,24,16, 8,28,20,12)$
12A1 $12^{2},2^{2}$ $294$ $12$ $24$ $( 1,11,13,15,17, 7, 9,23,25,19,21,27)( 2,12,14,16,18, 8,10,24,26,20,22,28)( 3, 5)( 4, 6)$
12A-1 $12^{2},2^{2}$ $294$ $12$ $24$ $( 1,27,21,19,25,23, 9, 7,17,15,13,11)( 2,28,22,20,26,24,10, 8,18,16,14,12)( 3, 5)( 4, 6)$
14A $14,2^{7}$ $84$ $14$ $20$ $( 1,18, 5,22, 9,26,13, 2,17, 6,21,10,25,14)( 3,12)( 4,11)( 7, 8)(15,28)(16,27)(19,24)(20,23)$
14B $14^{2}$ $252$ $14$ $26$ $( 1, 8,17,28, 5,20,21,12, 9, 4,25,24,13,16)( 2, 7,18,27, 6,19,22,11,10, 3,26,23,14,15)$
21A1 $7^{2},3^{4},1^{2}$ $84$ $21$ $20$ $( 1,21,13, 5,25,17, 9)( 2,22,14, 6,26,18,10)( 3,27,19)( 4,28,20)(11,15,23)(12,16,24)$
21A-1 $7^{2},3^{4},1^{2}$ $84$ $21$ $20$ $( 1, 9,17,25, 5,13,21)( 2,10,18,26, 6,14,22)( 3,19,27)( 4,20,28)(11,23,15)(12,24,16)$
42A1 $14,6^{2},2$ $84$ $42$ $24$ $( 1,26,21,18,13,10, 5, 2,25,22,17,14, 9, 6)( 3,24,27,12,19,16)( 4,23,28,11,20,15)( 7, 8)$
42A-1 $14,6^{2},2$ $84$ $42$ $24$ $( 1, 6, 9,14,17,22,25, 2, 5,10,13,18,21,26)( 3,16,19,12,27,24)( 4,15,20,11,28,23)( 7, 8)$

Malle's constant $a(G)$:     $1/8$

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Character table

35 x 35 character table

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Regular extensions

Data not computed