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Properties

Label 32T633
32T633 1 4 1->4 13 1->13 17 1->17 2 3 2->3 14 2->14 18 2->18 15 3->15 19 3->19 16 4->16 20 4->20 5 6 5->6 8 5->8 5->18 7 6->7 6->17 7->8 7->19 8->20 9 9->7 25 9->25 26 9->26 10 10->8 10->25 10->26 11 11->6 27 11->27 28 11->28 12 12->5 12->27 12->28 13->14 13->15 13->27 14->16 14->28 15->16 15->26 16->25 17->20 30 17->30 18->19 29 18->29 32 19->32 31 20->31 21 21->4 22 21->22 24 21->24 22->3 23 22->23 23->1 23->24 24->2 25->21 26->22 27->23 28->24 29->12 29->30 29->31 30->11 30->32 31->10 31->32 32->9
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, 633);
 

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 $t$:  $633$
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:  $(1,4)(2,3)(5,6)(7,8)(9,25)(10,26)(11,28)(12,27)(13,14)(15,16)(17,20)(18,19)(21,22)(23,24)(29,30)(31,32)$, $(1,17)(2,18)(3,19)(4,20)(5,8)(6,7)(9,26)(10,25)(11,27)(12,28)(13,15)(14,16)(21,24)(22,23)(29,31)(30,32)$, $(1,13,27,23)(2,14,28,24)(3,15,26,22)(4,16,25,21)(5,18,29,12)(6,17,30,11)(7,19,32,9)(8,20,31,10)$
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: $C_4$ x 2, $C_2^2$, $D_{4}$ x 4

Degree 8: $C_4\times C_2$, $D_4$ x 2, $C_2^2:C_4$ x 2, $((C_8 : C_2):C_2):C_2$ x 4

Degree 16: $C_2^2 : C_4$, 16T170 x 2, 16T273 x 4

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, 32T657, 32T1130 x 2, 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 $2^{16}$ $1$ $2$ $16$ $( 1,17)( 2,18)( 3,19)( 4,20)( 5,24)( 6,23)( 7,22)( 8,21)( 9,26)(10,25)(11,27)(12,28)(13,30)(14,29)(15,32)(16,31)$
2B $2^{16}$ $1$ $2$ $16$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,24)(22,23)(25,28)(26,27)(29,31)(30,32)$
2C $2^{16}$ $1$ $2$ $16$ $( 1,19)( 2,20)( 3,17)( 4,18)( 5,21)( 6,22)( 7,23)( 8,24)( 9,27)(10,28)(11,26)(12,25)(13,32)(14,31)(15,30)(16,29)$
2D $2^{16}$ $2$ $2$ $16$ $( 1, 3)( 2, 4)( 5,24)( 6,23)( 7,22)( 8,21)( 9,11)(10,12)(13,30)(14,29)(15,32)(16,31)(17,19)(18,20)(25,28)(26,27)$
2E $2^{8},1^{16}$ $2$ $2$ $8$ $( 1,19)( 2,20)( 3,17)( 4,18)( 9,27)(10,28)(11,26)(12,25)$
2F $2^{16}$ $4$ $2$ $16$ $( 1,18)( 2,17)( 3,20)( 4,19)( 5, 6)( 7, 8)( 9,12)(10,11)(13,14)(15,16)(21,22)(23,24)(25,27)(26,28)(29,30)(31,32)$
2G $2^{16}$ $4$ $2$ $16$ $( 1, 2)( 3, 4)( 5,23)( 6,24)( 7,21)( 8,22)( 9,28)(10,27)(11,25)(12,26)(13,29)(14,30)(15,31)(16,32)(17,18)(19,20)$
2H $2^{16}$ $4$ $2$ $16$ $( 1,17)( 2,18)( 3,19)( 4,20)( 5,24)( 6,23)( 7,22)( 8,21)( 9,11)(10,12)(13,15)(14,16)(25,28)(26,27)(29,31)(30,32)$
2I $2^{8},1^{16}$ $4$ $2$ $8$ $( 9,27)(10,28)(11,26)(12,25)(13,32)(14,31)(15,30)(16,29)$
2J $2^{16}$ $4$ $2$ $16$ $( 1,20)( 2,19)( 3,18)( 4,17)( 5,23)( 6,24)( 7,21)( 8,22)( 9,28)(10,27)(11,25)(12,26)(13,16)(14,15)(29,32)(30,31)$
2K $2^{16}$ $4$ $2$ $16$ $( 1, 4)( 2, 3)( 5, 6)( 7, 8)( 9,12)(10,11)(13,31)(14,32)(15,29)(16,30)(17,20)(18,19)(21,22)(23,24)(25,27)(26,28)$
2L $2^{16}$ $4$ $2$ $16$ $( 1,11)( 2,12)( 3, 9)( 4,10)( 5,14)( 6,13)( 7,15)( 8,16)(17,27)(18,28)(19,26)(20,25)(21,31)(22,32)(23,30)(24,29)$
2M $2^{16}$ $4$ $2$ $16$ $( 1,27)( 2,28)( 3,26)( 4,25)( 5,29)( 6,30)( 7,32)( 8,31)( 9,19)(10,20)(11,17)(12,18)(13,23)(14,24)(15,22)(16,21)$
4A $4^{8}$ $4$ $4$ $24$ $( 1,27,19, 9)( 2,28,20,10)( 3,26,17,11)( 4,25,18,12)( 5,29,21,16)( 6,30,22,15)( 7,32,23,13)( 8,31,24,14)$
4B $4^{8}$ $4$ $4$ $24$ $( 1,11,19,26)( 2,12,20,25)( 3, 9,17,27)( 4,10,18,28)( 5,14,21,31)( 6,13,22,32)( 7,15,23,30)( 8,16,24,29)$
4C $4^{4},2^{8}$ $8$ $4$ $20$ $( 1,12,19,25)( 2,11,20,26)( 3,10,17,28)( 4, 9,18,27)( 5,30)( 6,29)( 7,31)( 8,32)(13,24)(14,23)(15,21)(16,22)$
4D $4^{4},2^{8}$ $8$ $4$ $20$ $( 1,28,19,10)( 2,27,20, 9)( 3,25,17,12)( 4,26,18,11)( 5,13)( 6,14)( 7,16)( 8,15)(21,32)(22,31)(23,29)(24,30)$
4E1 $4^{8}$ $8$ $4$ $24$ $( 1,13,27,23)( 2,14,28,24)( 3,15,26,22)( 4,16,25,21)( 5,18,29,12)( 6,17,30,11)( 7,19,32, 9)( 8,20,31,10)$
4E-1 $4^{8}$ $8$ $4$ $24$ $( 1,23,27,13)( 2,24,28,14)( 3,22,26,15)( 4,21,25,16)( 5,12,29,18)( 6,11,30,17)( 7, 9,32,19)( 8,10,31,20)$
4F1 $4^{8}$ $8$ $4$ $24$ $( 1,30,27, 6)( 2,29,28, 5)( 3,32,26, 7)( 4,31,25, 8)( 9,22,19,15)(10,21,20,16)(11,23,17,13)(12,24,18,14)$
4F-1 $4^{8}$ $8$ $4$ $24$ $( 1, 6,27,30)( 2, 5,28,29)( 3, 7,26,32)( 4, 8,25,31)( 9,15,19,22)(10,16,20,21)(11,13,17,23)(12,14,18,24)$
8A1 $8^{4}$ $8$ $8$ $28$ $( 1,29,11, 8,19,16,26,24)( 2,30,12, 7,20,15,25,23)( 3,31, 9, 5,17,14,27,21)( 4,32,10, 6,18,13,28,22)$
8A-1 $8^{4}$ $8$ $8$ $28$ $( 1, 5,11,31,19,21,26,14)( 2, 6,12,32,20,22,25,13)( 3, 8, 9,29,17,24,27,16)( 4, 7,10,30,18,23,28,15)$
8B1 $8^{4}$ $8$ $8$ $28$ $( 1,14,11,21,19,31,26, 5)( 2,13,12,22,20,32,25, 6)( 3,16, 9,24,17,29,27, 8)( 4,15,10,23,18,30,28, 7)$
8B-1 $8^{4}$ $8$ $8$ $28$ $( 1,24,11,16,19, 8,26,29)( 2,23,12,15,20, 7,25,30)( 3,21, 9,14,17, 5,27,31)( 4,22,10,13,18, 6,28,32)$

Malle's constant $a(G)$:     $1/8$

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