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Properties

Label 32T1130
Degree $32$
Order $128$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_2^5:C_4$

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Copy content magma:G := TransitiveGroup(32, 1130);
 

Group invariants

Abstract group:  $C_2^5:C_4$
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 $t$:  $1130$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $4$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,20,31,23,2,19,32,24)(3,17,30,22,4,18,29,21)(5,14,9,27,6,13,10,28)(7,15,11,25,8,16,12,26)$, $(1,17,11,27)(2,18,12,28)(3,20,10,26)(4,19,9,25)(5,24,30,16)(6,23,29,15)(7,22,32,13)(8,21,31,14)$, $(1,16)(2,15)(3,14)(4,13)(5,28)(6,27)(7,26)(8,25)(9,22)(10,21)(11,24)(12,23)(17,29)(18,30)(19,31)(20,32)$
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, $C_2 \times (C_2^2:C_4)$
$64$:  $((C_8 : C_2):C_2):C_2$ x 2, 16T76

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

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

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

Degree 16: 16T76, 16T166 x 2

Low degree siblings

16T227 x 4, 16T259 x 8, 16T261 x 4, 16T273 x 4, 16T283 x 4, 32T506, 32T507 x 2, 32T508 x 4, 32T595 x 4, 32T596 x 8, 32T597 x 2, 32T598 x 4, 32T599 x 4, 32T601, 32T602 x 2, 32T633, 32T657, 32T1130, 32T1796

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 $2^{16}$ $1$ $2$ $16$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,12)(10,11)(13,15)(14,16)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31)$
2B $2^{16}$ $1$ $2$ $16$ $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,11)(10,12)(13,16)(14,15)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)$
2C $2^{16}$ $1$ $2$ $16$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)$
2D $2^{16}$ $2$ $2$ $16$ $( 1,10)( 2, 9)( 3,11)( 4,12)( 5,31)( 6,32)( 7,29)( 8,30)(13,23)(14,24)(15,22)(16,21)(17,26)(18,25)(19,28)(20,27)$
2E $2^{16}$ $2$ $2$ $16$ $( 1,11)( 2,12)( 3,10)( 4, 9)( 5,30)( 6,29)( 7,32)( 8,31)(13,22)(14,21)(15,23)(16,24)(17,27)(18,28)(19,25)(20,26)$
2F $2^{16}$ $4$ $2$ $16$ $( 1,17)( 2,18)( 3,20)( 4,19)( 5,23)( 6,24)( 7,21)( 8,22)( 9,25)(10,26)(11,27)(12,28)(13,31)(14,32)(15,30)(16,29)$
2G $2^{16}$ $4$ $2$ $16$ $( 1,20)( 2,19)( 3,17)( 4,18)( 5,22)( 6,21)( 7,24)( 8,23)( 9,28)(10,27)(11,26)(12,25)(13,30)(14,29)(15,31)(16,32)$
2H $2^{16}$ $4$ $2$ $16$ $( 1,31)( 2,32)( 3,30)( 4,29)( 5,10)( 6, 9)( 7,12)( 8,11)(13,17)(14,18)(15,20)(16,19)(21,28)(22,27)(23,26)(24,25)$
2I $2^{16}$ $4$ $2$ $16$ $( 1,30)( 2,29)( 3,31)( 4,32)( 5,11)( 6,12)( 7, 9)( 8,10)(13,20)(14,19)(15,17)(16,18)(21,25)(22,26)(23,27)(24,28)$
2J $2^{16}$ $4$ $2$ $16$ $( 1,22)( 2,21)( 3,23)( 4,24)( 5,20)( 6,19)( 7,18)( 8,17)( 9,16)(10,15)(11,13)(12,14)(25,29)(26,30)(27,31)(28,32)$
2K $2^{16}$ $4$ $2$ $16$ $( 1,23)( 2,24)( 3,22)( 4,21)( 5,17)( 6,18)( 7,19)( 8,20)( 9,14)(10,13)(11,15)(12,16)(25,32)(26,31)(27,30)(28,29)$
2L $2^{16}$ $4$ $2$ $16$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,12)(10,11)(13,24)(14,23)(15,21)(16,22)(17,26)(18,25)(19,28)(20,27)(29,31)(30,32)$
2M $2^{12},1^{8}$ $4$ $2$ $12$ $( 5, 6)( 7, 8)(13,21)(14,22)(15,24)(16,23)(17,27)(18,28)(19,25)(20,26)(29,30)(31,32)$
4A $4^{8}$ $4$ $4$ $24$ $( 1,29, 2,30)( 3,32, 4,31)( 5,11, 6,12)( 7, 9, 8,10)(13,26,14,25)(15,27,16,28)(17,24,18,23)(19,22,20,21)$
4B $4^{8}$ $4$ $4$ $24$ $( 1,32, 2,31)( 3,29, 4,30)( 5,10, 6, 9)( 7,12, 8,11)(13,27,14,28)(15,26,16,25)(17,21,18,22)(19,23,20,24)$
4C $4^{8}$ $8$ $4$ $24$ $( 1,27,11,17)( 2,28,12,18)( 3,26,10,20)( 4,25, 9,19)( 5,16,30,24)( 6,15,29,23)( 7,13,32,22)( 8,14,31,21)$
4D $4^{8}$ $8$ $4$ $24$ $( 1,26,11,20)( 2,25,12,19)( 3,27,10,17)( 4,28, 9,18)( 5,14,30,21)( 6,13,29,22)( 7,15,32,23)( 8,16,31,24)$
4E1 $4^{6},2^{4}$ $8$ $4$ $22$ $( 1, 3)( 2, 4)( 5,31, 6,32)( 7,29, 8,30)( 9,11)(10,12)(13,20,21,26)(14,19,22,25)(15,17,24,27)(16,18,23,28)$
4E-1 $4^{6},2^{4}$ $8$ $4$ $22$ $( 1, 3)( 2, 4)( 5,32, 6,31)( 7,30, 8,29)( 9,11)(10,12)(13,26,21,20)(14,25,22,19)(15,27,24,17)(16,28,23,18)$
4F1 $4^{6},2^{2},1^{4}$ $8$ $4$ $20$ $( 5,30, 6,29)( 7,32, 8,31)( 9,10)(11,12)(13,17,21,27)(14,18,22,28)(15,20,24,26)(16,19,23,25)$
4F-1 $4^{6},2^{2},1^{4}$ $8$ $4$ $20$ $( 5,29, 6,30)( 7,31, 8,32)( 9,10)(11,12)(13,27,21,17)(14,28,22,18)(15,26,24,20)(16,25,23,19)$
8A1 $8^{4}$ $8$ $8$ $28$ $( 1,24,32,19, 2,23,31,20)( 3,21,29,18, 4,22,30,17)( 5,28,10,13, 6,27, 9,14)( 7,26,12,16, 8,25,11,15)$
8A-1 $8^{4}$ $8$ $8$ $28$ $( 1,15, 8,19, 2,16, 7,20)( 3,13, 5,18, 4,14, 6,17)( 9,22,29,28,10,21,30,27)(11,24,31,26,12,23,32,25)$
8B1 $8^{4}$ $8$ $8$ $28$ $( 1,21,32,18, 2,22,31,17)( 3,24,29,19, 4,23,30,20)( 5,25,10,15, 6,26, 9,16)( 7,27,12,14, 8,28,11,13)$
8B-1 $8^{4}$ $8$ $8$ $28$ $( 1,13, 8,18, 2,14, 7,17)( 3,15, 5,19, 4,16, 6,20)( 9,23,29,25,10,24,30,26)(11,21,31,27,12,22,32,28)$

Malle's constant $a(G)$:     $1/12$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 4A 4B 4C 4D 4E1 4E-1 4F1 4F-1 8A1 8A-1 8B1 8B-1
Size 1 1 1 1 2 2 4 4 4 4 4 4 4 4 4 4 8 8 8 8 8 8 8 8 8 8
2 P 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 2C 2C 2E 2E 2M 2M 2M 2M 4B 4B 4B 4B
Type
128.850.1a 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
128.850.1b 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
128.850.1c 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
128.850.1d 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
128.850.1e 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
128.850.1f 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
128.850.1g 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
128.850.1h 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
128.850.1i1 C 1 −1 −1 1 −1 1 −1 1 1 −1 −1 1 1 −1 1 −1 −i i −1 1 i −i i −i −i i
128.850.1i2 C 1 −1 −1 1 −1 1 −1 1 1 −1 −1 1 1 −1 1 −1 i −i −1 1 −i i −i i i −i
128.850.1j1 C 1 −1 −1 1 −1 1 1 −1 1 −1 −1 1 −1 1 1 −1 −i i 1 −1 i −i −i i i −i
128.850.1j2 C 1 −1 −1 1 −1 1 1 −1 1 −1 −1 1 −1 1 1 −1 i −i 1 −1 −i i i −i −i i
128.850.1k1 C 1 1 1 1 1 1 −1 −1 1 1 −1 −1 −1 −1 −1 −1 −i i 1 1 −i i i −i i −i
128.850.1k2 C 1 1 1 1 1 1 −1 −1 1 1 −1 −1 −1 −1 −1 −1 i −i 1 1 i −i −i i −i i
128.850.1l1 C 1 1 1 1 1 1 1 1 1 1 −1 −1 1 1 −1 −1 −i i −1 −1 −i i −i i −i i
128.850.1l2 C 1 1 1 1 1 1 1 1 1 1 −1 −1 1 1 −1 −1 i −i −1 −1 i −i i −i i −i
128.850.2a R 2 −2 −2 2 −2 2 0 0 −2 2 −2 2 0 0 −2 2 0 0 0 0 0 0 0 0 0 0
128.850.2b R 2 −2 −2 2 −2 2 0 0 −2 2 2 −2 0 0 2 −2 0 0 0 0 0 0 0 0 0 0
128.850.2c R 2 2 2 2 2 2 0 0 −2 −2 −2 −2 0 0 2 2 0 0 0 0 0 0 0 0 0 0
128.850.2d R 2 2 2 2 2 2 0 0 −2 −2 2 2 0 0 −2 −2 0 0 0 0 0 0 0 0 0 0
128.850.4a R 4 −4 −4 4 4 −4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
128.850.4b R 4 4 4 4 −4 −4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
128.850.4c R 4 −4 4 −4 0 0 −2 2 0 0 0 0 −2 2 0 0 0 0 0 0 0 0 0 0 0 0
128.850.4d R 4 −4 4 −4 0 0 2 −2 0 0 0 0 2 −2 0 0 0 0 0 0 0 0 0 0 0 0
128.850.4e R 4 4 −4 −4 0 0 −2 −2 0 0 0 0 2 2 0 0 0 0 0 0 0 0 0 0 0 0
128.850.4f R 4 4 −4 −4 0 0 2 2 0 0 0 0 −2 −2 0 0 0 0 0 0 0 0 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed