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Properties

Label 28T38
28T38 1 15 1->15 24 1->24 2 16 2->16 23 2->23 3 26 3->26 4 25 4->25 5 6 5->6 27 5->27 28 6->28 7 7->1 21 7->21 8 8->2 22 8->22 9 9->3 10 10->4 11 11->5 11->26 12 12->6 12->25 13 13->7 13->28 14 14->8 14->27 15->10 16->9 17 17->12 18 18->11 19 19->13 20 19->20 20->14 21->16 22->15 23->17 24->18 25->20 26->19 27->21 28->22
Degree $28$
Order $224$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^2\times F_8$

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Copy content magma:G := TransitiveGroup(28, 38);
 

Group invariants

Abstract group:  $C_2^2\times F_8$
Copy content magma:IdentifyGroup(G);
 
Order:  $224=2^{5} \cdot 7$
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 $t$:  $38$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $4$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,24,18,11,5,27,21,16,9,3,26,19,13,7)(2,23,17,12,6,28,22,15,10,4,25,20,14,8)$, $(1,15)(2,16)(5,6)(7,21)(8,22)(11,26)(12,25)(13,28)(14,27)(19,20)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$7$:  $C_7$
$14$:  $C_{14}$ x 3
$28$:  28T2
$56$:  $C_2^3:C_7$
$112$:  14T9 x 3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Degree 7: $C_7$

Degree 14: 14T9 x 3

Low degree siblings

28T38 x 5, 28T39 x 3, 32T2228

Siblings are shown with degree $\leq 47$

A number field with this Galois group has 5 arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{28}$ $1$ $1$ $0$ $()$
2A $2^{14}$ $1$ $2$ $14$ $( 1,16)( 2,15)( 3,18)( 4,17)( 5,19)( 6,20)( 7,21)( 8,22)( 9,24)(10,23)(11,26)(12,25)(13,27)(14,28)$
2B $2^{14}$ $1$ $2$ $14$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)$
2C $2^{14}$ $1$ $2$ $14$ $( 1,15)( 2,16)( 3,17)( 4,18)( 5,20)( 6,19)( 7,22)( 8,21)( 9,23)(10,24)(11,25)(12,26)(13,28)(14,27)$
2D $2^{12},1^{4}$ $7$ $2$ $12$ $( 1,15)( 2,16)( 3,18)( 4,17)( 5,19)( 6,20)( 7, 8)(11,25)(12,26)(13,14)(21,22)(27,28)$
2E $2^{10},1^{8}$ $7$ $2$ $10$ $( 1, 2)( 7,22)( 8,21)( 9,24)(10,23)(11,12)(13,28)(14,27)(15,16)(25,26)$
2F $2^{10},1^{8}$ $7$ $2$ $10$ $( 1,16)( 2,15)( 3,17)( 4,18)( 5,20)( 6,19)( 9,10)(11,26)(12,25)(23,24)$
2G $2^{10},1^{8}$ $7$ $2$ $10$ $( 3, 4)( 5, 6)( 7,21)( 8,22)( 9,23)(10,24)(13,27)(14,28)(17,18)(19,20)$
7A1 $7^{4}$ $8$ $7$ $24$ $( 1,18, 5,21, 9,26,13)( 2,17, 6,22,10,25,14)( 3,19, 7,24,11,27,16)( 4,20, 8,23,12,28,15)$
7A-1 $7^{4}$ $8$ $7$ $24$ $( 1,13,26, 9,21, 5,18)( 2,14,25,10,22, 6,17)( 3,16,27,11,24, 7,19)( 4,15,28,12,23, 8,20)$
7A2 $7^{4}$ $8$ $7$ $24$ $( 1, 5, 9,13,18,21,26)( 2, 6,10,14,17,22,25)( 3, 7,11,16,19,24,27)( 4, 8,12,15,20,23,28)$
7A-2 $7^{4}$ $8$ $7$ $24$ $( 1,26,21,18,13, 9, 5)( 2,25,22,17,14,10, 6)( 3,27,24,19,16,11, 7)( 4,28,23,20,15,12, 8)$
7A3 $7^{4}$ $8$ $7$ $24$ $( 1,21,13, 5,26,18, 9)( 2,22,14, 6,25,17,10)( 3,24,16, 7,27,19,11)( 4,23,15, 8,28,20,12)$
7A-3 $7^{4}$ $8$ $7$ $24$ $( 1, 9,18,26, 5,13,21)( 2,10,17,25, 6,14,22)( 3,11,19,27, 7,16,24)( 4,12,20,28, 8,15,23)$
14A1 $14^{2}$ $8$ $14$ $26$ $( 1,24,18,11, 5,27,21,16, 9, 3,26,19,13, 7)( 2,23,17,12, 6,28,22,15,10, 4,25,20,14, 8)$
14A-1 $14^{2}$ $8$ $14$ $26$ $( 1, 7,13,19,26, 3, 9,16,21,27, 5,11,18,24)( 2, 8,14,20,25, 4,10,15,22,28, 6,12,17,23)$
14A3 $14^{2}$ $8$ $14$ $26$ $( 1,11,21, 3,13,24, 5,16,26, 7,18,27, 9,19)( 2,12,22, 4,14,23, 6,15,25, 8,17,28,10,20)$
14A-3 $14^{2}$ $8$ $14$ $26$ $( 1,19, 9,27,18, 7,26,16, 5,24,13, 3,21,11)( 2,20,10,28,17, 8,25,15, 6,23,14, 4,22,12)$
14A5 $14^{2}$ $8$ $14$ $26$ $( 1,27,26,24,21,19,18,16,13,11, 9, 7, 5, 3)( 2,28,25,23,22,20,17,15,14,12,10, 8, 6, 4)$
14A-5 $14^{2}$ $8$ $14$ $26$ $( 1, 3, 5, 7, 9,11,13,16,18,19,21,24,26,27)( 2, 4, 6, 8,10,12,14,15,17,20,22,23,25,28)$
14B1 $14^{2}$ $8$ $14$ $26$ $( 1,10,18,25, 5,14,21, 2, 9,17,26, 6,13,22)( 3,12,19,28, 7,15,24, 4,11,20,27, 8,16,23)$
14B-1 $14^{2}$ $8$ $14$ $26$ $( 1,22,13, 6,26,17, 9, 2,21,14, 5,25,18,10)( 3,23,16, 8,27,20,11, 4,24,15, 7,28,19,12)$
14B3 $14^{2}$ $8$ $14$ $26$ $( 1,25,21,17,13,10, 5, 2,26,22,18,14, 9, 6)( 3,28,24,20,16,12, 7, 4,27,23,19,15,11, 8)$
14B-3 $14^{2}$ $8$ $14$ $26$ $( 1, 6, 9,14,18,22,26, 2, 5,10,13,17,21,25)( 3, 8,11,15,19,23,27, 4, 7,12,16,20,24,28)$
14B5 $14^{2}$ $8$ $14$ $26$ $( 1,14,26,10,21, 6,18, 2,13,25, 9,22, 5,17)( 3,15,27,12,24, 8,19, 4,16,28,11,23, 7,20)$
14B-5 $14^{2}$ $8$ $14$ $26$ $( 1,17, 5,22, 9,25,13, 2,18, 6,21,10,26,14)( 3,20, 7,23,11,28,16, 4,19, 8,24,12,27,15)$
14C1 $14^{2}$ $8$ $14$ $26$ $( 1,23,18,12, 5,28,21,15, 9, 4,26,20,13, 8)( 2,24,17,11, 6,27,22,16,10, 3,25,19,14, 7)$
14C-1 $14^{2}$ $8$ $14$ $26$ $( 1, 8,13,20,26, 4, 9,15,21,28, 5,12,18,23)( 2, 7,14,19,25, 3,10,16,22,27, 6,11,17,24)$
14C3 $14^{2}$ $8$ $14$ $26$ $( 1,12,21, 4,13,23, 5,15,26, 8,18,28, 9,20)( 2,11,22, 3,14,24, 6,16,25, 7,17,27,10,19)$
14C-3 $14^{2}$ $8$ $14$ $26$ $( 1,20, 9,28,18, 8,26,15, 5,23,13, 4,21,12)( 2,19,10,27,17, 7,25,16, 6,24,14, 3,22,11)$
14C5 $14^{2}$ $8$ $14$ $26$ $( 1,28,26,23,21,20,18,15,13,12, 9, 8, 5, 4)( 2,27,25,24,22,19,17,16,14,11,10, 7, 6, 3)$
14C-5 $14^{2}$ $8$ $14$ $26$ $( 1, 4, 5, 8, 9,12,13,15,18,20,21,23,26,28)( 2, 3, 6, 7,10,11,14,16,17,19,22,24,25,27)$

Malle's constant $a(G)$:     $1/10$

Copy content magma:ConjugacyClasses(G);
 

Character table

32 x 32 character table

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Regular extensions

Data not computed