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Properties

Label 32T7
Degree $32$
Order $32$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_2^3:C_4$

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Show commands: Magma

magma: G := TransitiveGroup(32, 7);
 

Group action invariants

Degree $n$:  $32$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $7$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2^3:C_4$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $32$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,20,32,15)(2,19,31,16)(3,17,30,13)(4,18,29,14)(5,23,10,28)(6,24,9,27)(7,22,12,25)(8,21,11,26), (1,10)(2,9)(3,12)(4,11)(5,32)(6,31)(7,30)(8,29)(13,27)(14,28)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21), (1,19,4,17)(2,20,3,18)(5,24,8,22)(6,23,7,21)(9,28,12,26)(10,27,11,25)(13,32,16,29)(14,31,15,30)
magma: Generators(G);
 

Low degree resolvents

|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, $C_4\times C_2^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_4$ x 4, $C_2^2$ x 7, $D_{4}$ x 8

Degree 8: $C_4\times C_2$ x 6, $C_2^3$, $D_4$ x 4, $D_4\times C_2$ x 8, $C_2^2:C_4$ x 8

Degree 16: $C_4\times C_2^2$, $D_4\times C_2$ x 2, $C_2^2 : C_4$ x 4, $C_2 \times (C_2^2:C_4)$ x 4

Low degree siblings

16T21 x 4

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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,20)(14,19)(15,17)(16,18)(21,27) (22,28)(23,25)(24,26)(29,30)(31,32)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,18)(14,17)(15,19)(16,20)(21,25) (22,26)(23,27)(24,28)(29,31)(30,32)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)(17,19)(18,20)(21,23) (22,24)(25,27)(26,28)(29,32)(30,31)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,31)(10,32)(11,29)(12,30)(13,24)(14,23)(15,21) (16,22)(17,27)(18,28)(19,25)(20,26)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 6)( 2, 5)( 3, 8)( 4, 7)( 9,32)(10,31)(11,30)(12,29)(13,26)(14,25)(15,27) (16,28)(17,21)(18,22)(19,23)(20,24)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 7)( 2, 8)( 3, 5)( 4, 6)( 9,29)(10,30)(11,31)(12,32)(13,28)(14,27)(15,25) (16,26)(17,23)(18,24)(19,21)(20,22)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 8)( 2, 7)( 3, 6)( 4, 5)( 9,30)(10,29)(11,32)(12,31)(13,22)(14,21)(15,23) (16,24)(17,25)(18,26)(19,27)(20,28)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 9)( 2,10)( 3,11)( 4,12)( 5,31)( 6,32)( 7,29)( 8,30)(13,21)(14,22)(15,24) (16,23)(17,26)(18,25)(19,28)(20,27)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,12)( 2,11)( 3,10)( 4, 9)( 5,30)( 6,29)( 7,32)( 8,31)(13,23)(14,24)(15,22) (16,21)(17,28)(18,27)(19,26)(20,25)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,13, 4,16)( 2,14, 3,15)( 5,25, 8,27)( 6,26, 7,28)( 9,21,12,23)(10,22,11,24) (17,29,19,32)(18,30,20,31)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,14,32,18)( 2,13,31,17)( 3,16,30,19)( 4,15,29,20)( 5,26,10,21)( 6,25, 9,22) ( 7,27,12,24)( 8,28,11,23)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,15,32,20)( 2,16,31,19)( 3,13,30,17)( 4,14,29,18)( 5,28,10,23)( 6,27, 9,24) ( 7,25,12,22)( 8,26,11,21)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,16, 4,13)( 2,15, 3,14)( 5,27, 8,25)( 6,28, 7,26)( 9,23,12,21)(10,24,11,22) (17,32,19,29)(18,31,20,30)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,21, 4,23)( 2,22, 3,24)( 5,18, 8,20)( 6,17, 7,19)( 9,13,12,16)(10,14,11,15) (25,30,27,31)(26,29,28,32)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,22,32,25)( 2,21,31,26)( 3,23,30,28)( 4,24,29,27)( 5,17,10,13)( 6,18, 9,14) ( 7,20,12,15)( 8,19,11,16)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,23, 4,21)( 2,24, 3,22)( 5,20, 8,18)( 6,19, 7,17)( 9,16,12,13)(10,15,11,14) (25,31,27,30)(26,32,28,29)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,24,32,27)( 2,23,31,28)( 3,21,30,26)( 4,22,29,25)( 5,19,10,16)( 6,20, 9,15) ( 7,18,12,14)( 8,17,11,13)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,29)( 2,30)( 3,31)( 4,32)( 5,11)( 6,12)( 7, 9)( 8,10)(13,19)(14,20)(15,18) (16,17)(21,28)(22,27)(23,26)(24,25)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,32)( 2,31)( 3,30)( 4,29)( 5,10)( 6, 9)( 7,12)( 8,11)(13,17)(14,18)(15,20) (16,19)(21,26)(22,25)(23,28)(24,27)$

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:  32.22
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 4A1 4A-1 4B1 4B-1 4C1 4C-1 4D1 4D-1
Size 1 1 1 1 1 1 1 1 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 1A 2A 2A 2C 2C 2A 2A 2C 2C
Type
32.22.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
32.22.1b R 1 1 −1 1 −1 −1 −1 1 −1 1 1 −1 −1 −1 −1 −1 1 1 1 1
32.22.1c R 1 1 −1 1 −1 −1 −1 1 −1 1 1 −1 1 1 1 1 −1 −1 −1 −1
32.22.1d R 1 1 −1 1 −1 −1 −1 1 1 −1 −1 1 −1 −1 1 1 −1 −1 1 1
32.22.1e R 1 1 −1 1 −1 −1 −1 1 1 −1 −1 1 1 1 −1 −1 1 1 −1 −1
32.22.1f R 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 1 1 1 1 −1 −1
32.22.1g R 1 1 1 1 1 1 1 1 −1 −1 −1 −1 1 1 −1 −1 −1 −1 1 1
32.22.1h R 1 1 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1
32.22.1i1 C 1 −1 −1 −1 1 −1 1 1 −1 −1 1 1 −i i i −i −i i i −i
32.22.1i2 C 1 −1 −1 −1 1 −1 1 1 −1 −1 1 1 i −i −i i i −i −i i
32.22.1j1 C 1 −1 −1 −1 1 −1 1 1 1 1 −1 −1 −i i −i i i −i i −i
32.22.1j2 C 1 −1 −1 −1 1 −1 1 1 1 1 −1 −1 i −i i −i −i i −i i
32.22.1k1 C 1 −1 1 −1 −1 1 −1 1 −1 1 −1 1 −i i −i i −i i −i i
32.22.1k2 C 1 −1 1 −1 −1 1 −1 1 −1 1 −1 1 i −i i −i i −i i −i
32.22.1l1 C 1 −1 1 −1 −1 1 −1 1 1 −1 1 −1 −i i i −i i −i −i i
32.22.1l2 C 1 −1 1 −1 −1 1 −1 1 1 −1 1 −1 i −i −i i −i i i −i
32.22.2a R 2 −2 −2 2 −2 2 2 −2 0 0 0 0 0 0 0 0 0 0 0 0
32.22.2b R 2 −2 2 2 2 −2 −2 −2 0 0 0 0 0 0 0 0 0 0 0 0
32.22.2c R 2 2 −2 −2 2 2 −2 −2 0 0 0 0 0 0 0 0 0 0 0 0
32.22.2d R 2 2 2 −2 −2 −2 2 −2 0 0 0 0 0 0 0 0 0 0 0 0

magma: CharacterTable(G);