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Properties

Label 35T44
35T44 1 2 1->2 3 1->3 4 2->4 6 3->6 8 4->8 5 5->6 10 5->10 12 6->12 7 14 7->14 8->3 9 9->7 9->12 15 10->15 11 11->14 16 11->16 12->1 13 17 13->17 18 14->18 19 15->19 20 16->20 17->11 23 17->23 21 18->21 24 18->24 19->24 25 19->25 27 20->27 26 21->26 22 22->5 23->13 28 24->28 29 25->29 31 25->31 32 26->32 27->23 27->29 34 28->34 29->22 30 30->31 33 31->33 35 32->35 32->35 33->21 34->9 35->30
Degree $35$
Order $40320$
Cyclic no
Abelian no
Solvable no
Primitive yes
$p$-group no
Group: $S_8$

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Copy content magma:G := TransitiveGroup(35, 44);
 

Group invariants

Abstract group:  $S_8$
Copy content magma:IdentifyGroup(G);
 
Order:  $40320=2^{7} \cdot 3^{2} \cdot 5 \cdot 7$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  no
Copy content magma:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $35$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $44$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  yes
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,2,4,8,3,6,12)(5,10,15,19,25,29,22)(7,14,18,24,28,34,9)(11,16,20,27,23,13,17)(21,26,32,35,30,31,33)$, $(1,3)(5,6)(9,12)(11,14)(17,23)(18,21)(19,24)(25,31)(27,29)(32,35)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: None

Degree 7: None

Low degree siblings

8T50, 16T1838, 28T502, 30T1153

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{35}$ $1$ $1$ $0$ $()$
2A $2^{10},1^{15}$ $28$ $2$ $10$ $( 4,10)( 5,11)( 6,14)( 7,15)(17,18)(20,26)(21,23)(25,32)(31,35)(33,34)$
2B $2^{12},1^{11}$ $105$ $2$ $12$ $( 1,27)( 2,25)( 5,18)( 7,19)( 9,15)(10,20)(12,34)(13,35)(14,23)(16,31)(22,32)(24,33)$
2C $2^{14},1^{7}$ $210$ $2$ $14$ $( 1, 5)( 3, 6)( 4,22)( 8,26)( 9,18)(10,33)(12,21)(13,15)(16,30)(17,31)(23,25)(27,35)(28,34)(29,32)$
2D $2^{16},1^{3}$ $420$ $2$ $16$ $( 1,25)( 2,33)( 3, 7)( 4,29)( 5,27)( 6, 8)( 9,13)(10,14)(15,23)(16,21)(17,32)(18,24)(19,26)(20,35)(22,30)(28,31)$
3A $3^{10},1^{5}$ $112$ $3$ $20$ $( 2, 3,22)( 4,25,20)( 5, 7,17)( 6,23,34)( 8,12,24)( 9,30,19)(10,32,26)(11,15,18)(13,16,27)(14,21,33)$
3B $3^{11},1^{2}$ $1120$ $3$ $22$ $( 1,16,33)( 2,18, 7)( 3, 4,26)( 5,19,25)( 8,28,21)( 9,20,32)(10,22,15)(11,17,30)(12,13,14)(23,34,35)(24,27,31)$
4A $4^{7},2^{3},1$ $420$ $4$ $24$ $( 1,31, 5,17)( 2,20)( 3,25, 6,23)( 4,34,22,28)( 8,33,26,10)( 9,27,18,35)(11,19)(12,29,21,32)(13,16,15,30)(14,24)$
4B $4^{6},2^{4},1^{3}$ $1260$ $4$ $22$ $( 2, 7,12,35)( 3,17,29,11)( 4,26)( 5, 8,18,28)( 6,10)( 9,19,16,13)(14,23)(15,22,31,24)(20,21)(25,32,34,33)$
4C $4^{7},2,1^{5}$ $1260$ $4$ $22$ $( 1, 7,16,18)( 2,33,29, 6)( 3,30,21,35)( 4,11, 8,13)( 5,10,27,12)(14,34)(19,22,31,23)(24,26,28,25)$
4D $4^{7},2^{3},1$ $2520$ $4$ $24$ $( 1, 3, 9,12)( 2,27,30,29)( 4,18,15,21)( 5,25,35,14)( 6,11,32,31)( 7,23,10,17)( 8,22)(13,24,28,19)(20,26)(33,34)$
5A $5^{7}$ $1344$ $5$ $28$ $( 1,35,30,13, 5)( 2,28,32,12, 6)( 3,10,25,14, 4)( 7,11,31,15, 9)( 8,23,33,20,21)(16,27,17,18,19)(22,26,24,29,34)$
6A $6^{3},3^{4},2,1^{3}$ $1120$ $6$ $24$ $( 2,22, 3)( 4,26,25,10,20,32)( 5,18, 7,11,17,15)( 6,33,23,14,34,21)( 8,24,12)( 9,19,30)(13,27,16)(31,35)$
6B $6^{3},3^{5},2$ $1120$ $6$ $26$ $( 1, 7,13,28,25,12)( 2,33,14)( 3,30,29)( 4,18,17, 5,26,20)( 6,22,31)( 8,15,16,27,32,24)( 9,35,23)(10,11)(19,34,21)$
6C $6^{4},3^{2},2^{2},1$ $1680$ $6$ $26$ $( 1, 6,15, 5, 3,13)( 2,14,11)( 4,21, 9,22,12,18)( 8,26)(10,33)(16,31,23,30,17,25)(19,20,24)(27,28,29,35,34,32)$
6D $6^{4},3^{3},1^{2}$ $3360$ $6$ $26$ $( 1,24,16,27,33,31)( 2,19,18,25, 7, 5)( 3,26, 4)( 8,21,28)( 9,22,20,15,32,10)(11,30,17)(12,23,13,34,14,35)$
6E $6^{5},3,2$ $3360$ $6$ $28$ $( 1,15, 4,25,23,29)( 2,32, 9,33,17,13)( 3,21,31, 7,16,28)( 5,30,10,27,22,14)( 6,24,35, 8,18,20)(11,34,12)(19,26)$
7A $7^{5}$ $5760$ $7$ $30$ $( 1,35,31,19,20, 8,15)( 2,23,33,16,22,10,21)( 3,11,29,30,17,26, 4)( 5, 6,25,27,13,12,32)( 7,18,28, 9,14,24,34)$
8A $8^{3},4^{2},2,1$ $5040$ $8$ $28$ $( 2,29, 7,11,12, 3,35,17)( 4,21,26,20)( 5,24, 8,15,18,22,28,31)( 6,23,10,14)( 9,32,19,34,16,33,13,25)(27,30)$
10A $10^{2},5^{3}$ $4032$ $10$ $30$ $( 1,28,35,26, 6, 5, 4, 9, 8, 3)( 2,31,10,19, 7)(11,20,23,24,25)(12,27,15,22,29,32,18,13,34,21)(14,17,16,30,33)$
12A $12^{2},6,4,1$ $3360$ $12$ $30$ $( 1,16, 6,31,15,23, 5,30, 3,17,13,25)( 2,19,14,20,11,24)( 4,35,21,34, 9,32,22,27,12,28,18,29)( 8,10,26,33)$
15A $15^{2},5$ $2688$ $15$ $32$ $( 1, 2,34,35,28,22,30,32,26,13,12,24, 5, 6,29)( 3,33,15,10,20, 9,25,21, 7,14, 8,11, 4,23,31)(16,17,19,27,18)$

Malle's constant $a(G)$:     $1/10$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 3A 3B 4A 4B 4C 4D 5A 6A 6B 6C 6D 6E 7A 8A 10A 12A 15A
Size 1 28 105 210 420 112 1120 420 1260 1260 2520 1344 1120 1120 1680 3360 3360 5760 5040 4032 3360 2688
2 P 1A 1A 1A 1A 1A 3A 3B 2C 2B 2C 2C 5A 3A 3B 3A 3B 3B 7A 4B 5A 6C 15A
3 P 1A 2A 2B 2C 2D 1A 1A 4A 4B 4C 4D 5A 2A 2A 2C 2B 2D 7A 8A 10A 4A 5A
5 P 1A 2A 2B 2C 2D 3A 3B 4A 4B 4C 4D 1A 6A 6B 6C 6D 6E 7A 8A 2A 12A 3A
7 P 1A 2A 2B 2C 2D 3A 3B 4A 4B 4C 4D 5A 6A 6B 6C 6D 6E 1A 8A 10A 12A 15A
Type
40320.a.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
40320.a.1b R 1 −1 1 1 −1 1 1 −1 1 −1 1 1 −1 −1 1 1 −1 1 −1 −1 −1 1
40320.a.7a R 7 5 −1 3 1 4 1 3 −1 −1 1 2 2 −1 0 −1 1 0 −1 0 0 −1
40320.a.7b R 7 −5 −1 3 −1 4 1 −3 −1 1 1 2 −2 1 0 −1 −1 0 1 0 0 −1
40320.a.14a R 14 −4 6 2 0 −1 2 2 2 −2 0 −1 −1 2 −1 0 0 0 0 1 −1 −1
40320.a.14b R 14 4 6 2 0 −1 2 −2 2 2 0 −1 1 −2 −1 0 0 0 0 −1 1 −1
40320.a.20a R 20 10 4 4 2 5 −1 2 0 2 0 0 1 1 1 1 −1 −1 0 0 −1 0
40320.a.20b R 20 −10 4 4 −2 5 −1 −2 0 −2 0 0 −1 −1 1 1 1 −1 0 0 1 0
40320.a.21a R 21 −9 −3 1 3 6 0 −3 1 1 −1 1 0 0 −2 0 0 0 −1 1 0 1
40320.a.21b R 21 9 −3 1 −3 6 0 3 1 −1 −1 1 0 0 −2 0 0 0 1 −1 0 1
40320.a.28a R 28 10 −4 4 2 1 1 −2 0 −2 0 −2 1 1 1 −1 −1 0 0 0 1 1
40320.a.28b R 28 −10 −4 4 −2 1 1 2 0 2 0 −2 −1 −1 1 −1 1 0 0 0 −1 1
40320.a.35a R 35 −5 3 −5 3 5 2 −1 −1 −1 −1 0 1 −2 1 0 0 0 1 0 −1 0
40320.a.35b R 35 5 3 −5 −3 5 2 1 −1 1 −1 0 −1 2 1 0 0 0 −1 0 1 0
40320.a.42a R 42 0 −6 2 0 −6 0 0 2 0 −2 2 0 0 2 0 0 0 0 0 0 −1
40320.a.56a R 56 4 8 0 4 −4 −1 0 0 0 0 1 −2 1 0 −1 1 0 0 −1 0 1
40320.a.56b R 56 −4 8 0 −4 −4 −1 0 0 0 0 1 2 −1 0 −1 −1 0 0 1 0 1
40320.a.64a R 64 16 0 0 0 4 −2 0 0 0 0 −1 −2 −2 0 0 0 1 0 1 0 −1
40320.a.64b R 64 −16 0 0 0 4 −2 0 0 0 0 −1 2 2 0 0 0 1 0 −1 0 −1
40320.a.70a R 70 −10 −2 2 2 −5 1 4 −2 0 0 0 −1 −1 −1 1 −1 0 0 0 1 0
40320.a.70b R 70 10 −2 2 −2 −5 1 −4 −2 0 0 0 1 1 −1 1 1 0 0 0 −1 0
40320.a.90a R 90 0 −6 −6 0 0 0 0 2 0 2 0 0 0 0 0 0 −1 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed