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Properties

Label 32T519
Degree $32$
Order $128$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $(C_2^2\times D_4):C_4$

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Show commands: Magma

magma: G := TransitiveGroup(32, 519);
 

Group action invariants

Degree $n$:  $32$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $519$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $(C_2^2\times D_4):C_4$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $8$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,32,4,30)(2,31,3,29)(5,26,8,28)(6,25,7,27)(9,22,11,24)(10,21,12,23)(13,17,15,19)(14,18,16,20), (1,23,8,20,4,21,5,18)(2,24,7,19,3,22,6,17)(9,32,13,28,11,30,15,26)(10,31,14,27,12,29,16,25), (1,21,2,22)(3,24,4,23)(5,18,6,17)(7,19,8,20)(9,25,12,28)(10,26,11,27)(13,29,16,32)(14,30,15,31)
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 12, $C_4\times C_2$ x 6, $C_2^3$
$16$:  $D_4\times C_2$ x 6, $C_2^2:C_4$ x 12, $C_4\times C_2^2$
$32$:  $C_2^2 \wr C_2$ x 4, $C_2 \times (C_2^2:C_4)$ x 3
$64$:  16T79

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: $C_2 \times (C_2^2:C_4)$, 16T232, 16T290

Low degree siblings

16T232 x 2, 16T290 x 4, 32T519 x 3, 32T520 x 4, 32T521, 32T668 x 8, 32T669 x 2, 32T670 x 2, 32T1918 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $128=2^{7}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $3$
Label:  128.527
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 4A 4B 4C 4D 4E 4F 4G 4H 4I1 4I-1 4J1 4J-1 8A1 8A-1 8B1 8B-1
Size 1 1 2 2 2 4 4 4 4 4 4 4 4 2 2 2 2 4 4 4 4 8 8 8 8 8 8 8 8
2 P 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 2A 2A 2A 2A 2A 2A 2A 2A 2C 2C 2B 2B 4A 4A 4B 4B
Type
128.527.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
128.527.1b R 1 1 1 1 1 −1 −1 1 1 −1 −1 −1 −1 1 1 1 1 1 −1 1 −1 −1 −1 1 1 −1 −1 1 1
128.527.1c R 1 1 1 1 1 −1 −1 1 1 −1 −1 −1 −1 1 1 1 1 1 −1 1 −1 1 1 −1 −1 1 1 −1 −1
128.527.1d R 1 1 1 1 1 −1 1 −1 −1 −1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 1 1 1 1 −1 −1
128.527.1e R 1 1 1 1 1 −1 1 −1 −1 −1 1 1 1 1 1 1 1 −1 −1 −1 −1 1 1 −1 −1 −1 −1 1 1
128.527.1f R 1 1 1 1 1 1 −1 −1 −1 1 −1 −1 −1 1 1 1 1 −1 1 −1 1 −1 −1 −1 −1 1 1 1 1
128.527.1g R 1 1 1 1 1 1 −1 −1 −1 1 −1 −1 −1 1 1 1 1 −1 1 −1 1 1 1 1 1 −1 −1 −1 −1
128.527.1h R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1
128.527.1i1 C 1 1 −1 −1 1 −1 −1 1 1 1 1 −1 1 −1 −1 1 1 −1 1 −1 −1 −i i i −i i −i −i i
128.527.1i2 C 1 1 −1 −1 1 −1 −1 1 1 1 1 −1 1 −1 −1 1 1 −1 1 −1 −1 i −i −i i −i i i −i
128.527.1j1 C 1 1 −1 −1 1 −1 1 −1 −1 1 −1 1 −1 −1 −1 1 1 1 1 1 −1 −i i i −i −i i i −i
128.527.1j2 C 1 1 −1 −1 1 −1 1 −1 −1 1 −1 1 −1 −1 −1 1 1 1 1 1 −1 i −i −i i i −i −i i
128.527.1k1 C 1 1 −1 −1 1 1 −1 −1 −1 −1 1 −1 1 −1 −1 1 1 1 −1 1 1 −i i −i i −i i −i i
128.527.1k2 C 1 1 −1 −1 1 1 −1 −1 −1 −1 1 −1 1 −1 −1 1 1 1 −1 1 1 i −i i −i i −i i −i
128.527.1l1 C 1 1 −1 −1 1 1 1 1 1 −1 −1 1 −1 −1 −1 1 1 −1 −1 −1 1 −i i −i i i −i i −i
128.527.1l2 C 1 1 −1 −1 1 1 1 1 1 −1 −1 1 −1 −1 −1 1 1 −1 −1 −1 1 i −i i −i −i i −i i
128.527.2a R 2 2 −2 −2 2 −2 0 0 0 −2 0 0 0 2 2 −2 −2 0 2 0 2 0 0 0 0 0 0 0 0
128.527.2b R 2 2 −2 −2 2 2 0 0 0 2 0 0 0 2 2 −2 −2 0 −2 0 −2 0 0 0 0 0 0 0 0
128.527.2c R 2 2 −2 2 −2 0 −2 0 0 0 2 2 −2 2 −2 −2 2 0 0 0 0 0 0 0 0 0 0 0 0
128.527.2d R 2 2 −2 2 −2 0 0 −2 2 0 0 0 0 −2 2 2 −2 −2 0 2 0 0 0 0 0 0 0 0 0
128.527.2e R 2 2 −2 2 −2 0 0 2 −2 0 0 0 0 −2 2 2 −2 2 0 −2 0 0 0 0 0 0 0 0 0
128.527.2f R 2 2 −2 2 −2 0 2 0 0 0 −2 −2 2 2 −2 −2 2 0 0 0 0 0 0 0 0 0 0 0 0
128.527.2g R 2 2 2 −2 −2 0 −2 0 0 0 −2 2 2 −2 2 −2 2 0 0 0 0 0 0 0 0 0 0 0 0
128.527.2h R 2 2 2 −2 −2 0 0 −2 2 0 0 0 0 2 −2 2 −2 2 0 −2 0 0 0 0 0 0 0 0 0
128.527.2i R 2 2 2 −2 −2 0 0 2 −2 0 0 0 0 2 −2 2 −2 −2 0 2 0 0 0 0 0 0 0 0 0
128.527.2j R 2 2 2 −2 −2 0 2 0 0 0 2 −2 −2 −2 2 −2 2 0 0 0 0 0 0 0 0 0 0 0 0
128.527.2k R 2 2 2 2 2 −2 0 0 0 2 0 0 0 −2 −2 −2 −2 0 −2 0 2 0 0 0 0 0 0 0 0
128.527.2l R 2 2 2 2 2 2 0 0 0 −2 0 0 0 −2 −2 −2 −2 0 2 0 −2 0 0 0 0 0 0 0 0
128.527.8a R 8 −8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

magma: CharacterTable(G);