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Properties

Label 32T599
32T599 1 19 1->19 20 1->20 23 1->23 2 2->19 2->20 24 2->24 3 17 3->17 18 3->18 21 3->21 4 4->17 4->18 22 4->22 5 5->22 28 5->28 6 6->21 27 6->27 7 7->24 25 7->25 8 8->23 26 8->26 9 9->26 31 9->31 10 10->25 32 10->32 11 11->28 29 11->29 12 12->27 30 12->30 13 13->2 14 13->14 16 13->16 14->1 15 14->15 15->4 15->16 16->3 17->5 18->6 19->7 20->8 21->10 22->9 23->11 24->12 25->15 26->16 27->14 28->13 29->18 29->30 29->31 30->17 30->32 31->19 31->32 32->20
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, 599);
 

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 $t$:  $599$
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:  $(1,19)(2,20)(3,17)(4,18)(13,16)(14,15)(29,31)(30,32)$, $(1,20)(2,19)(3,18)(4,17)(5,22)(6,21)(7,24)(8,23)(9,26)(10,25)(11,28)(12,27)(13,14)(15,16)(29,30)(31,32)$, $(1,23,11,29,18,6,27,14)(2,24,12,30,17,5,28,13)(3,21,10,32,20,8,26,16)(4,22,9,31,19,7,25,15)$
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, $C_2^2$

Degree 8: $C_4\times C_2$, $C_2^3: C_4$ x 2, $((C_8 : C_2):C_2):C_2$ x 2

Degree 16: 16T94, 16T157 x 2, 16T259 x 2, 16T261

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 3, 32T601, 32T602 x 2, 32T633, 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 $2^{16}$ $1$ $2$ $16$ $( 1,20)( 2,19)( 3,18)( 4,17)( 5,22)( 6,21)( 7,24)( 8,23)( 9,28)(10,27)(11,26)(12,25)(13,31)(14,32)(15,30)(16,29)$
2B $2^{16}$ $1$ $2$ $16$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,12)(10,11)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,28)(26,27)(29,32)(30,31)$
2C $2^{16}$ $1$ $2$ $16$ $( 1,18)( 2,17)( 3,20)( 4,19)( 5,24)( 6,23)( 7,22)( 8,21)( 9,25)(10,26)(11,27)(12,28)(13,30)(14,29)(15,31)(16,32)$
2D $2^{16}$ $2$ $2$ $16$ $( 1,17)( 2,18)( 3,19)( 4,20)( 5, 6)( 7, 8)( 9,26)(10,25)(11,28)(12,27)(13,14)(15,16)(21,22)(23,24)(29,30)(31,32)$
2E $2^{16}$ $2$ $2$ $16$ $( 1, 4)( 2, 3)( 5,21)( 6,22)( 7,23)( 8,24)( 9,11)(10,12)(13,32)(14,31)(15,29)(16,30)(17,20)(18,19)(25,27)(26,28)$
2F $2^{8},1^{16}$ $4$ $2$ $8$ $( 1, 4)( 2, 3)( 5,21)( 6,22)( 7,23)( 8,24)(17,20)(18,19)$
2G $2^{16}$ $4$ $2$ $16$ $( 1,17)( 2,18)( 3,19)( 4,20)( 5, 6)( 7, 8)( 9,28)(10,27)(11,26)(12,25)(13,31)(14,32)(15,30)(16,29)(21,22)(23,24)$
2H $2^{12},1^{8}$ $4$ $2$ $12$ $( 1,18)( 2,17)( 3,20)( 4,19)( 5,21)( 6,22)( 7,23)( 8,24)(13,16)(14,15)(29,31)(30,32)$
2I $2^{16}$ $4$ $2$ $16$ $( 1, 3)( 2, 4)( 5, 6)( 7, 8)( 9,28)(10,27)(11,26)(12,25)(13,29)(14,30)(15,32)(16,31)(17,19)(18,20)(21,22)(23,24)$
2J $2^{16}$ $4$ $2$ $16$ $( 1,18)( 2,17)( 3,20)( 4,19)( 5,21)( 6,22)( 7,23)( 8,24)( 9,11)(10,12)(13,30)(14,29)(15,31)(16,32)(25,27)(26,28)$
2K $2^{16}$ $4$ $2$ $16$ $( 1, 3)( 2, 4)( 5, 6)( 7, 8)( 9,26)(10,25)(11,28)(12,27)(13,15)(14,16)(17,19)(18,20)(21,22)(23,24)(29,32)(30,31)$
2L $2^{16}$ $4$ $2$ $16$ $( 1,28)( 2,27)( 3,25)( 4,26)( 5,15)( 6,16)( 7,13)( 8,14)( 9,20)(10,19)(11,17)(12,18)(21,29)(22,30)(23,32)(24,31)$
2M $2^{16}$ $4$ $2$ $16$ $( 1, 9)( 2,10)( 3,12)( 4,11)( 5,30)( 6,29)( 7,31)( 8,32)(13,24)(14,23)(15,22)(16,21)(17,26)(18,25)(19,27)(20,28)$
4A $4^{8}$ $4$ $4$ $24$ $( 1,12,18,28)( 2,11,17,27)( 3, 9,20,25)( 4,10,19,26)( 5,29,24,14)( 6,30,23,13)( 7,32,22,16)( 8,31,21,15)$
4B $4^{8}$ $4$ $4$ $24$ $( 1,25,18, 9)( 2,26,17,10)( 3,28,20,12)( 4,27,19,11)( 5,16,24,32)( 6,15,23,31)( 7,14,22,29)( 8,13,21,30)$
4C $4^{8}$ $8$ $4$ $24$ $( 1,11, 4, 9)( 2,12, 3,10)( 5,16,21,30)( 6,15,22,29)( 7,14,23,31)( 8,13,24,32)(17,28,20,26)(18,27,19,25)$
4D $4^{8}$ $8$ $4$ $24$ $( 1,26, 4,28)( 2,25, 3,27)( 5,29,21,15)( 6,30,22,16)( 7,32,23,13)( 8,31,24,14)( 9,20,11,17)(10,19,12,18)$
4E1 $4^{8}$ $8$ $4$ $24$ $( 1,23, 9,14)( 2,24,10,13)( 3,21,12,16)( 4,22,11,15)( 5,26,30,17)( 6,25,29,18)( 7,27,31,19)( 8,28,32,20)$
4E-1 $4^{8}$ $8$ $4$ $24$ $( 1,14, 9,23)( 2,13,10,24)( 3,16,12,21)( 4,15,11,22)( 5,17,30,26)( 6,18,29,25)( 7,19,31,27)( 8,20,32,28)$
4F1 $4^{8}$ $8$ $4$ $24$ $( 1, 8, 9,32)( 2, 7,10,31)( 3, 6,12,29)( 4, 5,11,30)(13,19,24,27)(14,20,23,28)(15,17,22,26)(16,18,21,25)$
4F-1 $4^{8}$ $8$ $4$ $24$ $( 1,32, 9, 8)( 2,31,10, 7)( 3,29,12, 6)( 4,30,11, 5)(13,27,24,19)(14,28,23,20)(15,26,22,17)(16,25,21,18)$
8A1 $8^{4}$ $8$ $8$ $28$ $( 1,22,25,29,18, 7, 9,14)( 2,21,26,30,17, 8,10,13)( 3,24,28,32,20, 5,12,16)( 4,23,27,31,19, 6,11,15)$
8A-1 $8^{4}$ $8$ $8$ $28$ $( 1,15,11,22,18,31,27, 7)( 2,16,12,21,17,32,28, 8)( 3,13,10,24,20,30,26, 5)( 4,14, 9,23,19,29,25, 6)$
8B1 $8^{4}$ $8$ $8$ $28$ $( 1, 5,25,16,18,24, 9,32)( 2, 6,26,15,17,23,10,31)( 3, 7,28,14,20,22,12,29)( 4, 8,27,13,19,21,11,30)$
8B-1 $8^{4}$ $8$ $8$ $28$ $( 1,30,11, 5,18,13,27,24)( 2,29,12, 6,17,14,28,23)( 3,31,10, 7,20,15,26,22)( 4,32, 9, 8,19,16,25,21)$

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