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Properties

Label 32T570
Degree $32$
Order $128$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_2^4.D_4$

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Copy content magma:G := TransitiveGroup(32, 570);
 

Group invariants

Abstract group:  $C_2^4.D_4$
Copy content magma:IdentifyGroup(G);
 
Order:  $128=2^{7}$
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:  $4$
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $32$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $570$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $8$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,3)(2,4)(9,20,11,17)(10,19,12,18)(13,22,16,24)(14,21,15,23)(29,31)(30,32)$, $(1,24,2,23)(3,22,4,21)(5,20,6,19)(7,17,8,18)(9,28,12,25)(10,27,11,26)(13,31,15,30)(14,32,16,29)$, $(1,8,3,6)(2,7,4,5)(9,23,11,21)(10,24,12,22)(13,18,16,19)(14,17,15,20)(25,31,27,29)(26,32,28,30)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_4$ x 4, $C_2^2$ x 7
$8$:  $D_{4}$ x 4, $C_4\times C_2$ x 6, $C_2^3$
$16$:  $D_4\times C_2$ x 2, $C_2^2:C_4$ x 4, $C_4\times C_2^2$
$32$:  $C_2^3 : C_4 $ x 2, $C_2 \times (C_2^2:C_4)$
$64$:  16T76

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_4$ x 2, $C_2^2$, $D_{4}$ x 4

Degree 8: $C_4\times C_2$, $D_4\times C_2$ x 2, $C_2^2:C_4$ x 4

Degree 16: $C_2 \times (C_2^2:C_4)$, 16T250 x 2

Low degree siblings

16T234 x 2, 16T250 x 2, 16T297 x 4, 16T310 x 2, 32T524 x 2, 32T525, 32T526 x 2, 32T569 x 2, 32T689 x 4, 32T690 x 4, 32T718 x 2, 32T719

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{32}$ $1$ $1$ $0$ $()$
2A $2^{16}$ $1$ $2$ $16$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,16)(14,15)(17,20)(18,19)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)$
2B $2^{16}$ $2$ $2$ $16$ $( 1, 7)( 2, 8)( 3, 5)( 4, 6)( 9,13)(10,14)(11,16)(12,15)(17,22)(18,21)(19,23)(20,24)(25,32)(26,31)(27,30)(28,29)$
2C $2^{16}$ $2$ $2$ $16$ $( 1, 7)( 2, 8)( 3, 5)( 4, 6)( 9,16)(10,15)(11,13)(12,14)(17,24)(18,23)(19,21)(20,22)(25,32)(26,31)(27,30)(28,29)$
2D $2^{8},1^{16}$ $2$ $2$ $8$ $( 9,11)(10,12)(13,16)(14,15)(17,20)(18,19)(21,23)(22,24)$
2E $2^{16}$ $4$ $2$ $16$ $( 1,26)( 2,25)( 3,28)( 4,27)( 5,29)( 6,30)( 7,31)( 8,32)( 9,23)(10,24)(11,21)(12,22)(13,19)(14,20)(15,17)(16,18)$
2F $2^{16}$ $4$ $2$ $16$ $( 1, 6)( 2, 5)( 3, 8)( 4, 7)( 9,15)(10,16)(11,14)(12,13)(17,21)(18,22)(19,24)(20,23)(25,31)(26,32)(27,29)(28,30)$
2G $2^{16}$ $4$ $2$ $16$ $( 1,31)( 2,32)( 3,29)( 4,30)( 5,28)( 6,27)( 7,26)( 8,25)( 9,19)(10,20)(11,18)(12,17)(13,23)(14,24)(15,22)(16,21)$
2H $2^{16}$ $4$ $2$ $16$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,12)(10,11)(13,15)(14,16)(17,18)(19,20)(21,22)(23,24)(25,28)(26,27)(29,32)(30,31)$
2I $2^{16}$ $8$ $2$ $16$ $( 1,29)( 2,30)( 3,31)( 4,32)( 5,28)( 6,27)( 7,26)( 8,25)( 9,12)(10,11)(13,14)(15,16)(17,18)(19,20)(21,24)(22,23)$
4A $4^{8}$ $4$ $4$ $24$ $( 1,32, 3,30)( 2,31, 4,29)( 5,27, 7,25)( 6,28, 8,26)( 9,20,11,17)(10,19,12,18)(13,24,16,22)(14,23,15,21)$
4B $4^{8}$ $4$ $4$ $24$ $( 1,25, 3,27)( 2,26, 4,28)( 5,30, 7,32)( 6,29, 8,31)( 9,24,11,22)(10,23,12,21)(13,20,16,17)(14,19,15,18)$
4C $4^{4},2^{4},1^{8}$ $8$ $4$ $16$ $( 1, 3)( 2, 4)( 9,20,11,17)(10,19,12,18)(13,22,16,24)(14,21,15,23)(29,31)(30,32)$
4D $4^{4},2^{8}$ $8$ $4$ $20$ $( 1, 7, 3, 5)( 2, 8, 4, 6)( 9,22)(10,21)(11,24)(12,23)(13,20)(14,19)(15,18)(16,17)(25,30,27,32)(26,29,28,31)$
4E $4^{8}$ $8$ $4$ $24$ $( 1,26, 3,28)( 2,25, 4,27)( 5,31, 7,29)( 6,32, 8,30)( 9,14,11,15)(10,13,12,16)(17,23,20,21)(18,24,19,22)$
4F1 $4^{8}$ $8$ $4$ $24$ $( 1,16,31,21)( 2,15,32,22)( 3,13,29,23)( 4,14,30,24)( 5, 9,28,19)( 6,10,27,20)( 7,11,26,18)( 8,12,25,17)$
4F-1 $4^{8}$ $8$ $4$ $24$ $( 1,21,31,16)( 2,22,32,15)( 3,23,29,13)( 4,24,30,14)( 5,19,28, 9)( 6,20,27,10)( 7,18,26,11)( 8,17,25,12)$
4G1 $4^{8}$ $8$ $4$ $24$ $( 1,24, 2,23)( 3,22, 4,21)( 5,20, 6,19)( 7,17, 8,18)( 9,28,12,25)(10,27,11,26)(13,31,15,30)(14,32,16,29)$
4G-1 $4^{8}$ $8$ $4$ $24$ $( 1,15, 4,13)( 2,16, 3,14)( 5,12, 8, 9)( 6,11, 7,10)(17,25,18,26)(19,28,20,27)(21,29,22,30)(23,31,24,32)$
4H1 $4^{8}$ $8$ $4$ $24$ $( 1,11,31,18)( 2,12,32,17)( 3, 9,29,19)( 4,10,30,20)( 5,13,28,23)( 6,14,27,24)( 7,16,26,21)( 8,15,25,22)$
4H-1 $4^{8}$ $8$ $4$ $24$ $( 1,18,31,11)( 2,17,32,12)( 3,19,29, 9)( 4,20,30,10)( 5,23,28,13)( 6,24,27,14)( 7,21,26,16)( 8,22,25,15)$
4I1 $4^{8}$ $8$ $4$ $24$ $( 1,17, 4,19)( 2,18, 3,20)( 5,22, 8,23)( 6,21, 7,24)( 9,31,10,32)(11,29,12,30)(13,28,14,27)(15,25,16,26)$
4I-1 $4^{8}$ $8$ $4$ $24$ $( 1,10, 2, 9)( 3,12, 4,11)( 5,14, 6,13)( 7,15, 8,16)(17,30,19,31)(18,29,20,32)(21,26,24,27)(22,25,23,28)$

Malle's constant $a(G)$:     $1/8$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F1 4F-1 4G1 4G-1 4H1 4H-1 4I1 4I-1
Size 1 1 2 2 2 4 4 4 4 8 4 4 8 8 8 8 8 8 8 8 8 8 8
2 P 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 2A 2A 2D 2D 2A 2G 2G 2H 2H 2G 2G 2H 2H
Type
128.859.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
128.859.1b R 1 1 −1 −1 1 −1 −1 1 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1
128.859.1c R 1 1 −1 −1 1 −1 −1 1 1 −1 1 −1 1 1 1 1 −1 1 −1 −1 −1 1 −1
128.859.1d R 1 1 −1 −1 1 −1 −1 1 1 1 1 −1 −1 −1 −1 1 −1 1 −1 1 1 −1 1
128.859.1e R 1 1 −1 −1 1 −1 −1 1 1 1 1 −1 −1 1 −1 −1 1 −1 1 1 −1 1 −1
128.859.1f R 1 1 1 1 1 1 1 1 1 −1 1 1 −1 −1 −1 1 1 1 1 −1 −1 −1 −1
128.859.1g R 1 1 1 1 1 1 1 1 1 −1 1 1 −1 1 −1 −1 −1 −1 −1 −1 1 1 1
128.859.1h R 1 1 1 1 1 1 1 1 1 1 1 1 1 −1 1 −1 −1 −1 −1 1 −1 −1 −1
128.859.1i1 C 1 1 −1 −1 1 1 1 −1 −1 −1 1 −1 −1 −i 1 i −i −i i 1 i i −i
128.859.1i2 C 1 1 −1 −1 1 1 1 −1 −1 −1 1 −1 −1 i 1 −i i i −i 1 −i −i i
128.859.1j1 C 1 1 −1 −1 1 1 1 −1 −1 1 1 −1 1 −i −1 −i i i −i −1 i i −i
128.859.1j2 C 1 1 −1 −1 1 1 1 −1 −1 1 1 −1 1 i −1 i −i −i i −1 −i −i i
128.859.1k1 C 1 1 1 1 1 −1 −1 −1 −1 −1 1 1 1 −i −1 −i −i i i 1 −i i i
128.859.1k2 C 1 1 1 1 1 −1 −1 −1 −1 −1 1 1 1 i −1 i i −i −i 1 i −i −i
128.859.1l1 C 1 1 1 1 1 −1 −1 −1 −1 1 1 1 −1 −i 1 i i −i −i −1 −i i i
128.859.1l2 C 1 1 1 1 1 −1 −1 −1 −1 1 1 1 −1 i 1 −i −i i i −1 i −i −i
128.859.2a R 2 2 −2 −2 2 −2 2 2 −2 0 −2 2 0 0 0 0 0 0 0 0 0 0 0
128.859.2b R 2 2 −2 −2 2 2 −2 −2 2 0 −2 2 0 0 0 0 0 0 0 0 0 0 0
128.859.2c R 2 2 2 2 2 −2 2 −2 2 0 −2 −2 0 0 0 0 0 0 0 0 0 0 0
128.859.2d R 2 2 2 2 2 2 −2 2 −2 0 −2 −2 0 0 0 0 0 0 0 0 0 0 0
128.859.4a R 4 4 −4 4 −4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
128.859.4b R 4 4 4 −4 −4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
128.859.8a R 8 −8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed