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Properties

Label 30T159
30T159 1 2 1->2 19 1->19 3 2->3 20 2->20 3->1 21 3->21 4 14 4->14 22 4->22 5 15 5->15 23 5->23 6 13 6->13 24 6->24 7 11 7->11 25 7->25 8 12 8->12 26 8->26 9 10 9->10 27 9->27 10->8 28 10->28 11->9 29 11->29 12->7 30 12->30 13->5 16 13->16 14->6 17 14->17 15->4 18 15->18 16->7 16->30 17->8 17->18 17->28 18->9 18->29 19->10 19->27 20->11 20->21 20->25 21->12 21->26 22->13 22->24 23->14 23->22 23->24 24->15 24->23 25->1 25->21 26->2 26->19 26->27 27->3 27->20 28->4 28->18 29->5 29->16 29->30 30->6 30->17
Degree $30$
Order $720$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3^2:D_{10}$

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Copy content magma:G := TransitiveGroup(30, 159);
 

Group invariants

Abstract group:  $S_3^2:D_{10}$
Copy content magma:IdentifyGroup(G);
 
Order:  $720=2^{4} \cdot 3^{2} \cdot 5$
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:   not nilpotent
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $30$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $159$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(17,18)(20,21)(23,24)(26,27)(29,30)$, $(1,2,3)(4,14,6,13,5,15)(7,11,9,10,8,12)(16,30,17,28,18,29)(19,27,20,25,21,26)(22,24,23)$, $(1,19,10,28,4,22,13,16,7,25)(2,20,11,29,5,23,14,17,8,26)(3,21,12,30,6,24,15,18,9,27)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$8$:  $D_{4}$ x 2, $C_2^3$
$10$:  $D_{5}$
$16$:  $D_4\times C_2$
$20$:  $D_{10}$ x 3
$40$:  20T8
$72$:  $C_3^2:D_4$
$80$:  20T21
$144$:  12T77

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 5: $D_{5}$

Degree 6: $C_3^2:D_4$

Degree 10: $D_{10}$

Degree 15: None

Low degree siblings

30T159, 45T102

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{30}$ $1$ $1$ $0$ $()$
2A $2^{12},1^{6}$ $5$ $2$ $12$ $( 4,13)( 5,14)( 6,15)( 7,10)( 8,11)( 9,12)(16,28)(17,29)(18,30)(19,25)(20,26)(21,27)$
2B $2^{15}$ $6$ $2$ $15$ $( 1,23)( 2,24)( 3,22)( 4,26)( 5,27)( 6,25)( 7,29)( 8,30)( 9,28)(10,17)(11,18)(12,16)(13,20)(14,21)(15,19)$
2C $2^{5},1^{20}$ $6$ $2$ $5$ $(17,18)(20,21)(23,24)(26,27)(29,30)$
2D $2^{10},1^{10}$ $9$ $2$ $10$ $( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(16,18)(19,21)(22,24)(25,27)(28,30)$
2E $2^{13},1^{4}$ $30$ $2$ $13$ $( 1, 5)( 2, 4)( 3, 6)( 7,14)( 8,13)( 9,15)(10,11)(19,28)(20,29)(21,30)(22,25)(23,26)(24,27)$
2F $2^{15}$ $30$ $2$ $15$ $( 1,30)( 2,29)( 3,28)( 4,27)( 5,26)( 6,25)( 7,24)( 8,23)( 9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)$
2G $2^{14},1^{2}$ $45$ $2$ $14$ $( 1,12)( 2,11)( 3,10)( 4, 9)( 5, 8)( 6, 7)(13,15)(16,22)(17,24)(18,23)(20,21)(25,28)(26,30)(27,29)$
3A $3^{10}$ $4$ $3$ $20$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)(22,24,23)(25,27,26)(28,30,29)$
3B $3^{5},1^{15}$ $4$ $3$ $10$ $(16,17,18)(19,20,21)(22,23,24)(25,26,27)(28,29,30)$
4A $4^{5},2^{5}$ $18$ $4$ $20$ $( 1,23)( 2,22, 3,24)( 4,26)( 5,25, 6,27)( 7,29)( 8,28, 9,30)(10,17)(11,16,12,18)(13,20)(14,19,15,21)$
4B $4^{5},2^{5}$ $90$ $4$ $20$ $( 1,17)( 2,18, 3,16)( 4,29)( 5,30, 6,28)( 7,26)( 8,27, 9,25)(10,23)(11,24,12,22)(13,20)(14,21,15,19)$
5A1 $5^{6}$ $2$ $5$ $24$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3, 6, 9,12,15)(16,19,22,25,28)(17,20,23,26,29)(18,21,24,27,30)$
5A2 $5^{6}$ $2$ $5$ $24$ $( 1, 7,13, 4,10)( 2, 8,14, 5,11)( 3, 9,15, 6,12)(16,22,28,19,25)(17,23,29,20,26)(18,24,30,21,27)$
6A $6^{5}$ $12$ $6$ $25$ $( 1,24, 3,23, 2,22)( 4,27, 6,26, 5,25)( 7,30, 9,29, 8,28)(10,18,12,17,11,16)(13,21,15,20,14,19)$
6B $3^{5},2^{5},1^{5}$ $12$ $6$ $15$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(17,18)(20,21)(23,24)(26,27)(29,30)$
6C $6^{4},3^{2}$ $20$ $6$ $24$ $( 1, 2, 3)( 4,14, 6,13, 5,15)( 7,11, 9,10, 8,12)(16,29,18,28,17,30)(19,26,21,25,20,27)(22,23,24)$
6D $6^{2},3,2^{6},1^{3}$ $20$ $6$ $18$ $( 1, 7)( 2, 8)( 3, 9)(10,13)(11,14)(12,15)(16,21,17,19,18,20)(22,30,23,28,24,29)(25,27,26)$
6E $6^{2},3,2^{7},1$ $60$ $6$ $19$ $( 1, 5)( 2, 4)( 3, 6)( 7,14)( 8,13)( 9,15)(10,11)(16,18,17)(19,30,20,28,21,29)(22,27,23,25,24,26)$
6F $6^{5}$ $60$ $6$ $25$ $( 1,28, 2,30, 3,29)( 4,25, 5,27, 6,26)( 7,22, 8,24, 9,23)(10,19,11,21,12,20)(13,16,14,18,15,17)$
10A1 $10^{3}$ $12$ $10$ $27$ $( 1,17, 4,20, 7,23,10,26,13,29)( 2,18, 5,21, 8,24,11,27,14,30)( 3,16, 6,19, 9,22,12,25,15,28)$
10A3 $10^{3}$ $12$ $10$ $27$ $( 1,20,10,29, 4,23,13,17, 7,26)( 2,21,11,30, 5,24,14,18, 8,27)( 3,19,12,28, 6,22,15,16, 9,25)$
10B1 $10,5^{4}$ $12$ $10$ $25$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3, 6, 9,12,15)(16,19,22,25,28)(17,21,23,27,29,18,20,24,26,30)$
10B3 $10,5^{4}$ $12$ $10$ $25$ $( 1,10, 4,13, 7)( 2,11, 5,14, 8)( 3,12, 6,15, 9)(16,25,19,28,22)(17,27,20,30,23,18,26,21,29,24)$
10C1 $10^{2},5^{2}$ $18$ $10$ $26$ $( 1,13,10, 7, 4)( 2,15,11, 9, 5, 3,14,12, 8, 6)(16,30,25,24,19,18,28,27,22,21)(17,29,26,23,20)$
10C3 $10^{2},5^{2}$ $18$ $10$ $26$ $( 1, 7,13, 4,10)( 2, 9,14, 6,11, 3, 8,15, 5,12)(16,24,28,21,25,18,22,30,19,27)(17,23,29,20,26)$
15A1 $15^{2}$ $8$ $15$ $28$ $( 1, 8,15, 4,11, 3, 7,14, 6,10, 2, 9,13, 5,12)(16,23,30,19,26,18,22,29,21,25,17,24,28,20,27)$
15A2 $15^{2}$ $8$ $15$ $28$ $( 1,15,11, 7, 6, 2,13,12, 8, 4, 3,14,10, 9, 5)(16,30,26,22,21,17,28,27,23,19,18,29,25,24,20)$
15B1 $15,5^{3}$ $8$ $15$ $26$ $( 1,14,12, 7, 5, 3,13,11, 9, 4, 2,15,10, 8, 6)(16,28,25,22,19)(17,29,26,23,20)(18,30,27,24,21)$
15B2 $15,5^{3}$ $8$ $15$ $26$ $( 1,12, 5,13, 9, 2,10, 6,14, 7, 3,11, 4,15, 8)(16,25,19,28,22)(17,26,20,29,23)(18,27,21,30,24)$
20A1 $20,10$ $36$ $20$ $28$ $( 1,29,13,26,10,23, 7,20, 4,17)( 2,28,15,27,11,22, 9,21, 5,16, 3,30,14,25,12,24, 8,19, 6,18)$
20A3 $20,10$ $36$ $20$ $28$ $( 1,26, 7,17,13,23, 4,29,10,20)( 2,27, 9,16,14,24, 6,28,11,21, 3,25, 8,18,15,22, 5,30,12,19)$
30A1 $30$ $24$ $30$ $29$ $( 1,25, 8,17,15,24, 4,28,11,20, 3,27, 7,16,14,23, 6,30,10,19, 2,26, 9,18,13,22, 5,29,12,21)$
30A7 $30$ $24$ $30$ $29$ $( 1,29,14,27,12,22, 7,20, 5,18, 3,28,13,26,11,24, 9,19, 4,17, 2,30,15,25,10,23, 8,21, 6,16)$
30B1 $15,10,5$ $24$ $30$ $27$ $( 1, 9,14, 4,12, 2, 7,15, 5,10, 3, 8,13, 6,11)(16,22,28,19,25)(17,24,29,21,26,18,23,30,20,27)$
30B7 $15,10,5$ $24$ $30$ $27$ $( 1,14,12, 7, 5, 3,13,11, 9, 4, 2,15,10, 8, 6)(16,29,25,23,19,17,28,26,22,20)(18,30,27,24,21)$

Malle's constant $a(G)$:     $1/5$

Copy content magma:ConjugacyClasses(G);
 

Character table

36 x 36 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed