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Properties

Label 28T101
28T101 1 22 1->22 25 1->25 2 21 2->21 26 2->26 3 23 3->23 27 3->27 4 24 4->24 28 4->28 5 20 5->20 5->26 6 19 6->19 6->25 7 18 7->18 7->28 8 17 8->17 8->27 9 9->2 9->8 10 10->1 10->7 11 11->3 11->5 12 12->4 12->6 13 13->1 13->20 14 14->2 14->19 15 15->3 15->17 16 16->4 16->18 17->12 17->22 18->11 18->21 19->9 19->24 20->10 20->23 21->7 21->15 22->8 22->16 23->5 23->14 24->6 24->13 25->11 25->15 26->12 26->16 27->10 27->13 28->9 28->14
Degree $28$
Order $896$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^4:F_8$

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Copy content magma:G := TransitiveGroup(28, 101);
 

Group invariants

Abstract group:  $C_2^4:F_8$
Copy content magma:IdentifyGroup(G);
 
Order:  $896=2^{7} \cdot 7$
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 $t$:  $101$
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:  $(1,22,8,27,13,20,10)(2,21,7,28,14,19,9)(3,23,5,26,16,18,11)(4,24,6,25,15,17,12)$, $(1,25,11,5,20,23,14,2,26,12,6,19,24,13)(3,27,10,7,18,21,15)(4,28,9,8,17,22,16)$
Copy content magma:Generators(G);
 

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Degree 7: $C_7$

Degree 14: 14T6

Low degree siblings

16T1075 x 4, 28T97 x 4, 28T101 x 3

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 $2^{8},1^{12}$ $7$ $2$ $8$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)(13,14)(15,16)(25,26)(27,28)$
2B $2^{7},1^{14}$ $8$ $2$ $7$ $( 3, 4)( 7, 8)(11,12)(15,16)(19,20)(23,24)(27,28)$
2C $2^{12},1^{4}$ $14$ $2$ $12$ $( 1, 2)( 3, 4)( 5, 7)( 6, 8)( 9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,22)(23,24)$
2D $2^{12},1^{4}$ $14$ $2$ $12$ $( 5, 8)( 6, 7)( 9,12)(10,11)(13,15)(14,16)(17,20)(18,19)(21,22)(23,24)(25,26)(27,28)$
2E $2^{12},1^{4}$ $14$ $2$ $12$ $( 1, 4)( 2, 3)( 9,11)(10,12)(13,14)(15,16)(17,19)(18,20)(21,22)(23,24)(25,28)(26,27)$
2F $2^{8},1^{12}$ $14$ $2$ $8$ $( 1, 4)( 2, 3)( 5, 7)( 6, 8)(17,20)(18,19)(21,23)(22,24)$
4A $4^{4},2^{3},1^{6}$ $56$ $4$ $15$ $( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,10)(13,16,14,15)(17,18)(23,24)(25,27,26,28)$
7A1 $7^{4}$ $64$ $7$ $24$ $( 1,25,10, 5,17,21,14)( 2,26, 9, 6,18,22,13)( 3,28,11, 8,19,24,15)( 4,27,12, 7,20,23,16)$
7A-1 $7^{4}$ $64$ $7$ $24$ $( 1,14,21,17, 5,10,25)( 2,13,22,18, 6, 9,26)( 3,15,24,19, 8,11,28)( 4,16,23,20, 7,12,27)$
7A2 $7^{4}$ $64$ $7$ $24$ $( 1,10,17,14,25, 5,21)( 2, 9,18,13,26, 6,22)( 3,11,19,15,28, 8,24)( 4,12,20,16,27, 7,23)$
7A-2 $7^{4}$ $64$ $7$ $24$ $( 1,21, 5,25,14,17,10)( 2,22, 6,26,13,18, 9)( 3,24, 8,28,15,19,11)( 4,23, 7,27,16,20,12)$
7A3 $7^{4}$ $64$ $7$ $24$ $( 1, 5,14,10,21,25,17)( 2, 6,13, 9,22,26,18)( 3, 8,15,11,24,28,19)( 4, 7,16,12,23,27,20)$
7A-3 $7^{4}$ $64$ $7$ $24$ $( 1,17,25,21,10,14, 5)( 2,18,26,22, 9,13, 6)( 3,19,28,24,11,15, 8)( 4,20,27,23,12,16, 7)$
14A1 $14,7^{2}$ $64$ $14$ $25$ $( 1,17,25,21,10,14, 5)( 2,18,26,22, 9,13, 6)( 3,20,28,23,11,16, 8, 4,19,27,24,12,15, 7)$
14A-1 $14,7^{2}$ $64$ $14$ $25$ $( 1, 5,14,10,21,25,17)( 2, 6,13, 9,22,26,18)( 3, 7,15,12,24,27,19, 4, 8,16,11,23,28,20)$
14A3 $14,7^{2}$ $64$ $14$ $25$ $( 1,21, 5,25,14,17,10)( 2,22, 6,26,13,18, 9)( 3,23, 8,27,15,20,11, 4,24, 7,28,16,19,12)$
14A-3 $14,7^{2}$ $64$ $14$ $25$ $( 1,10,17,14,25, 5,21)( 2, 9,18,13,26, 6,22)( 3,12,19,16,28, 7,24, 4,11,20,15,27, 8,23)$
14A5 $14,7^{2}$ $64$ $14$ $25$ $( 1,14,21,17, 5,10,25)( 2,13,22,18, 6, 9,26)( 3,16,24,20, 8,12,28, 4,15,23,19, 7,11,27)$
14A-5 $14,7^{2}$ $64$ $14$ $25$ $( 1,25,10, 5,17,21,14)( 2,26, 9, 6,18,22,13)( 3,27,11, 7,19,23,15, 4,28,12, 8,20,24,16)$

Malle's constant $a(G)$:     $1/7$

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Character table

1A 2A 2B 2C 2D 2E 2F 4A 7A1 7A-1 7A2 7A-2 7A3 7A-3 14A1 14A-1 14A3 14A-3 14A5 14A-5
Size 1 7 8 14 14 14 14 56 64 64 64 64 64 64 64 64 64 64 64 64
2 P 1A 1A 1A 1A 1A 1A 1A 2A 7A2 7A-2 7A-3 7A3 7A-1 7A1 7A1 7A-1 7A3 7A-3 7A-2 7A2
7 P 1A 2A 2B 2C 2D 2E 2F 4A 7A3 7A-3 7A-1 7A1 7A2 7A-2 14A3 14A-3 14A-5 14A5 14A1 14A-1
Type
896.5502.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
896.5502.1b R 1 1 −1 1 1 1 1 −1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1
896.5502.1c1 C 1 1 1 1 1 1 1 1 ζ7−3 ζ72 ζ7 ζ7−2 ζ73 ζ7−1 ζ7−3 ζ7−2 ζ7−1 ζ73 ζ72 ζ7
896.5502.1c2 C 1 1 1 1 1 1 1 1 ζ73 ζ7−2 ζ7−1 ζ72 ζ7−3 ζ7 ζ73 ζ72 ζ7 ζ7−3 ζ7−2 ζ7−1
896.5502.1c3 C 1 1 1 1 1 1 1 1 ζ7−2 ζ7−1 ζ73 ζ7 ζ72 ζ7−3 ζ7−2 ζ7 ζ7−3 ζ72 ζ7−1 ζ73
896.5502.1c4 C 1 1 1 1 1 1 1 1 ζ72 ζ7 ζ7−3 ζ7−1 ζ7−2 ζ73 ζ72 ζ7−1 ζ73 ζ7−2 ζ7 ζ7−3
896.5502.1c5 C 1 1 1 1 1 1 1 1 ζ7−1 ζ73 ζ7−2 ζ7−3 ζ7 ζ72 ζ7−1 ζ7−3 ζ72 ζ7 ζ73 ζ7−2
896.5502.1c6 C 1 1 1 1 1 1 1 1 ζ7 ζ7−3 ζ72 ζ73 ζ7−1 ζ7−2 ζ7 ζ73 ζ7−2 ζ7−1 ζ7−3 ζ72
896.5502.1d1 C 1 1 −1 1 1 1 1 −1 ζ7−3 ζ72 ζ7 ζ7−2 ζ73 ζ7−1 −ζ7−3 −ζ7−2 −ζ7−1 −ζ73 −ζ72 −ζ7
896.5502.1d2 C 1 1 −1 1 1 1 1 −1 ζ73 ζ7−2 ζ7−1 ζ72 ζ7−3 ζ7 −ζ73 −ζ72 −ζ7 −ζ7−3 −ζ7−2 −ζ7−1
896.5502.1d3 C 1 1 −1 1 1 1 1 −1 ζ7−2 ζ7−1 ζ73 ζ7 ζ72 ζ7−3 −ζ7−2 −ζ7 −ζ7−3 −ζ72 −ζ7−1 −ζ73
896.5502.1d4 C 1 1 −1 1 1 1 1 −1 ζ72 ζ7 ζ7−3 ζ7−1 ζ7−2 ζ73 −ζ72 −ζ7−1 −ζ73 −ζ7−2 −ζ7 −ζ7−3
896.5502.1d5 C 1 1 −1 1 1 1 1 −1 ζ7−1 ζ73 ζ7−2 ζ7−3 ζ7 ζ72 −ζ7−1 −ζ7−3 −ζ72 −ζ7 −ζ73 −ζ7−2
896.5502.1d6 C 1 1 −1 1 1 1 1 −1 ζ7 ζ7−3 ζ72 ζ73 ζ7−1 ζ7−2 −ζ7 −ζ73 −ζ7−2 −ζ7−1 −ζ7−3 −ζ72
896.5502.7a R 7 7 7 −1 −1 −1 −1 −1 0 0 0 0 0 0 0 0 0 0 0 0
896.5502.7b R 7 7 −7 −1 −1 −1 −1 1 0 0 0 0 0 0 0 0 0 0 0 0
896.5502.14a R 14 −2 0 −2 −2 −2 6 0 0 0 0 0 0 0 0 0 0 0 0 0
896.5502.14b R 14 −2 0 −2 −2 6 −2 0 0 0 0 0 0 0 0 0 0 0 0 0
896.5502.14c R 14 −2 0 −2 6 −2 −2 0 0 0 0 0 0 0 0 0 0 0 0 0
896.5502.14d R 14 −2 0 6 −2 −2 −2 0 0 0 0 0 0 0 0 0 0 0 0 0

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Regular extensions

Data not computed