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Properties

Label 8T22
Degree $8$
Order $32$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $Q_8:C_2^2$

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Show commands: Magma

magma: G := TransitiveGroup(8, 22);
 

Group action invariants

Degree $n$:  $8$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $22$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $Q_8:C_2^2$
CHM label:   $E(8):D_{4}=[2^{3}]2^{2}$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,3)(2,8)(4,6)(5,7), (2,3)(6,7), (2,3)(4,5), (1,8)(2,3)(4,5)(6,7), (1,5)(2,6)(3,7)(4,8)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 15
$4$:  $C_2^2$ x 35
$8$:  $C_2^3$ x 15
$16$:  $C_2^4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Low degree siblings

8T22 x 5, 16T23 x 9, 32T9

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 1, 1, 1, 1 $ $2$ $2$ $(4,5)(6,7)$
$ 2, 2, 1, 1, 1, 1 $ $2$ $2$ $(2,3)(6,7)$
$ 2, 2, 1, 1, 1, 1 $ $2$ $2$ $(2,3)(4,5)$
$ 2, 2, 2, 2 $ $2$ $2$ $(1,2)(3,8)(4,6)(5,7)$
$ 2, 2, 2, 2 $ $2$ $2$ $(1,2)(3,8)(4,7)(5,6)$
$ 4, 4 $ $2$ $4$ $(1,2,8,3)(4,6,5,7)$
$ 4, 4 $ $2$ $4$ $(1,2,8,3)(4,7,5,6)$
$ 2, 2, 2, 2 $ $2$ $2$ $(1,4)(2,6)(3,7)(5,8)$
$ 4, 4 $ $2$ $4$ $(1,4,8,5)(2,6,3,7)$
$ 2, 2, 2, 2 $ $2$ $2$ $(1,4)(2,7)(3,6)(5,8)$
$ 4, 4 $ $2$ $4$ $(1,4,8,5)(2,7,3,6)$
$ 2, 2, 2, 2 $ $2$ $2$ $(1,6)(2,4)(3,5)(7,8)$
$ 4, 4 $ $2$ $4$ $(1,6,8,7)(2,4,3,5)$
$ 4, 4 $ $2$ $4$ $(1,6,8,7)(2,5,3,4)$
$ 2, 2, 2, 2 $ $2$ $2$ $(1,6)(2,5)(3,4)(7,8)$
$ 2, 2, 2, 2 $ $1$ $2$ $(1,8)(2,3)(4,5)(6,7)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $32=2^{5}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $2$
Label:  32.49
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 4A 4B 4C 4D 4E 4F
Size 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
2 P 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 2A 2A 2A 2A 2A 2A
Type
32.49.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
32.49.1b R 1 1 −1 −1 −1 −1 −1 1 1 1 −1 −1 1 −1 1 1 1
32.49.1c R 1 1 −1 −1 −1 1 1 1 1 −1 1 −1 −1 1 −1 −1 1
32.49.1d R 1 1 −1 −1 1 −1 1 −1 1 −1 −1 1 1 1 −1 1 −1
32.49.1e R 1 1 −1 −1 1 1 −1 −1 1 1 1 1 −1 −1 1 −1 −1
32.49.1f R 1 1 −1 1 −1 −1 −1 1 −1 −1 1 1 −1 1 1 1 −1
32.49.1g R 1 1 −1 1 −1 1 1 1 −1 1 −1 1 1 −1 −1 −1 −1
32.49.1h R 1 1 −1 1 1 −1 1 −1 −1 1 1 −1 −1 −1 −1 1 1
32.49.1i R 1 1 −1 1 1 1 −1 −1 −1 −1 −1 −1 1 1 1 −1 1
32.49.1j R 1 1 1 −1 −1 −1 1 −1 −1 −1 1 1 1 −1 1 −1 1
32.49.1k R 1 1 1 −1 −1 1 −1 −1 −1 1 −1 1 −1 1 −1 1 1
32.49.1l R 1 1 1 −1 1 −1 −1 1 −1 1 1 −1 1 1 −1 −1 −1
32.49.1m R 1 1 1 −1 1 1 1 1 −1 −1 −1 −1 −1 −1 1 1 −1
32.49.1n R 1 1 1 1 −1 −1 1 −1 1 1 −1 −1 −1 1 1 −1 −1
32.49.1o R 1 1 1 1 −1 1 −1 −1 1 −1 1 −1 1 −1 −1 1 −1
32.49.1p R 1 1 1 1 1 −1 −1 1 1 −1 −1 1 −1 −1 −1 −1 1
32.49.4a R 4 −4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

magma: CharacterTable(G);