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Properties

Label 8T25
8T25 1 2 1->2 3 1->3 5 1->5 8 1->8 2->3 6 2->6 2->6 2->8 4 3->4 7 3->7 4->5 4->5 4->6 4->8 5->7 5->7 6->3 6->7 7->1
Degree $8$
Order $56$
Cyclic no
Abelian no
Solvable yes
Primitive yes
$p$-group no
Group: $C_2^3:C_7$

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Copy content magma:G := TransitiveGroup(8, 25);
 

Group invariants

Abstract group:  $C_2^3:C_7$
Copy content magma:IdentifyGroup(G);
 
Order:  $56=2^{3} \cdot 7$
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 $n$:  $8$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $25$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
CHM label:   $E(8):7=F_{56}(8)$
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  yes
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,3)(2,8)(4,6)(5,7)$, $(1,2,6,3,4,5,7)$, $(1,8)(2,3)(4,5)(6,7)$, $(1,5)(2,6)(3,7)(4,8)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$7$:  $C_7$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Low degree siblings

14T6, 28T11

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{8}$ $1$ $1$ $0$ $()$
2A $2^{4}$ $7$ $2$ $4$ $(1,2)(3,8)(4,7)(5,6)$
7A1 $7,1$ $8$ $7$ $6$ $(1,8,4,6,3,5,2)$
7A-1 $7,1$ $8$ $7$ $6$ $(1,2,5,3,6,4,8)$
7A2 $7,1$ $8$ $7$ $6$ $(1,4,3,2,8,6,5)$
7A-2 $7,1$ $8$ $7$ $6$ $(1,5,6,8,2,3,4)$
7A3 $7,1$ $8$ $7$ $6$ $(1,6,2,4,5,8,3)$
7A-3 $7,1$ $8$ $7$ $6$ $(1,3,8,5,4,2,6)$

Malle's constant $a(G)$:     $1/4$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 7A1 7A-1 7A2 7A-2 7A3 7A-3
Size 1 7 8 8 8 8 8 8
2 P 1A 1A 7A2 7A-2 7A-3 7A3 7A-1 7A1
7 P 1A 2A 7A3 7A-3 7A-1 7A1 7A2 7A-2
Type
56.11.1a R 1 1 1 1 1 1 1 1
56.11.1b1 C 1 1 ζ7−3 ζ7−2 ζ7−1 ζ73 ζ72 ζ7
56.11.1b2 C 1 1 ζ73 ζ72 ζ7 ζ7−3 ζ7−2 ζ7−1
56.11.1b3 C 1 1 ζ7−2 ζ7 ζ7−3 ζ72 ζ7−1 ζ73
56.11.1b4 C 1 1 ζ72 ζ7−1 ζ73 ζ7−2 ζ7 ζ7−3
56.11.1b5 C 1 1 ζ7−1 ζ7−3 ζ72 ζ7 ζ73 ζ7−2
56.11.1b6 C 1 1 ζ7 ζ73 ζ7−2 ζ7−1 ζ7−3 ζ72
56.11.7a R 7 −1 0 0 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

$f_{ 1 } =$ $x^{8} + \left(8 t^{7} - 8 t^{6} + 4 t^{5} + 44 t^{4} + 8 t^{3} - 72 t^{2} + 32 t + 24\right) x^{6} + \left(16 t^{10} + 16 t^{9} + 32 t^{8} + 112 t^{7} + 240 t^{6} + 128 t^{5} + 208 t^{4} + 128 t^{3} - 496 t^{2} - 1408 t - 576\right) x^{5} + \left(16 t^{14} - 48 t^{13} + 296 t^{12} - 16 t^{11} + 254 t^{10} + 3132 t^{9} + 878 t^{8} + 4440 t^{7} + 14096 t^{6} + 11312 t^{5} + 9840 t^{4} + 28024 t^{3} + 22680 t^{2} + 5464 t + 632\right) x^{4} + \left(64 t^{17} - 384 t^{16} + 1760 t^{15} - 3264 t^{14} + 5216 t^{13} + 4704 t^{12} - 3968 t^{11} + 26080 t^{10} + 40224 t^{9} + 25312 t^{8} + 112800 t^{7} + 142944 t^{6} + 119744 t^{5} + 241408 t^{4} + 217600 t^{3} - 17600 t^{2} - 77184 t - 19456\right) x^{3} + \left(192 t^{20} - 1312 t^{19} + 6464 t^{18} - 16424 t^{17} + 31608 t^{16} - 13012 t^{15} - 13572 t^{14} + 102612 t^{13} + 103284 t^{12} - 76824 t^{11} + 487728 t^{10} + 676496 t^{9} + 334984 t^{8} + 793760 t^{7} + 1952512 t^{6} + 1668064 t^{5} + 594224 t^{4} + 175392 t^{3} + 546048 t^{2} + 493408 t + 118368\right) x^{2} + \left(256 t^{23} - 1984 t^{22} + 10176 t^{21} - 29616 t^{20} + 60720 t^{19} - 53168 t^{18} - 10240 t^{17} + 144816 t^{16} + 33424 t^{15} - 390336 t^{14} + 816416 t^{13} + 131424 t^{12} - 1535648 t^{11} + 209744 t^{10} + 933760 t^{9} - 6847808 t^{8} - 9344576 t^{7} - 5302016 t^{6} - 14400768 t^{5} - 30617344 t^{4} - 26949056 t^{3} - 10302784 t^{2} - 1531008 t - 64384\right) x + \left(256 t^{26} - 1600 t^{25} + 7504 t^{24} - 16912 t^{23} + 31496 t^{22} - 12464 t^{21} + 32737 t^{20} + 6180 t^{19} + 244678 t^{18} + 193164 t^{17} + 225065 t^{16} + 794392 t^{15} + 2715920 t^{14} + 862232 t^{13} + 593080 t^{12} + 8972960 t^{11} + 8883632 t^{10} - 7135704 t^{9} + 3026152 t^{8} + 28484512 t^{7} + 5926992 t^{6} - 23207072 t^{5} + 21056176 t^{4} + 77797696 t^{3} + 65774384 t^{2} + 23777632 t + 3256912\right)$ Copy content Toggle raw display