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Properties

Label 45T70
45T70 1 16 1->16 26 1->26 2 18 2->18 25 2->25 3 17 3->17 27 3->27 4 4->1 6 4->6 5 5->3 5->4 6->2 6->5 7 33 7->33 38 7->38 8 31 8->31 37 8->37 9 32 9->32 39 9->39 10 10->27 36 10->36 11 11->26 35 11->35 12 12->25 34 12->34 13 13->10 15 13->15 14 14->12 14->13 15->11 15->14 16->3 40 16->40 17->2 41 17->41 18->1 42 18->42 19 19->34 44 19->44 20 20->36 43 20->43 21 21->35 45 21->45 22 22->19 24 22->24 23 23->21 23->22 24->20 24->23 25->6 25->11 26->4 26->10 27->5 27->12 28 28->9 28->45 29 29->8 29->44 30 30->7 30->43 31->29 31->33 32->28 32->31 33->30 33->32 34->14 34->21 35->15 35->20 36->13 36->19 37->9 37->17 38->8 38->16 39->7 39->18 40->38 40->42 41->37 41->40 42->39 42->41 43->24 43->29 44->22 44->28 45->23 45->30
Degree $45$
Order $450$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3\times C_5^2:S_3$

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Copy content magma:G := TransitiveGroup(45, 70);
 

Group invariants

Abstract group:  $C_3\times C_5^2:S_3$
Copy content magma:IdentifyGroup(G);
 
Order:  $450=2 \cdot 3^{2} \cdot 5^{2}$
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 $n$:  $45$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $70$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $15$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,16,3,17,2,18)(4,6,5)(7,38,8,37,9,39)(10,27,12,25,11,26)(13,15,14)(19,34,21,35,20,36)(22,24,23)(28,45,30,43,29,44)(31,33,32)(40,42,41)$, $(1,26,4)(2,25,6)(3,27,5)(7,33,30)(8,31,29)(9,32,28)(10,36,13)(11,35,15)(12,34,14)(16,40,38)(17,41,37)(18,42,39)(19,44,22)(20,43,24)(21,45,23)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $S_3$, $C_6$
$18$:  $S_3\times C_3$
$150$:  $(C_5^2 : C_3):C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$, $S_3$

Degree 5: None

Degree 9: $S_3\times C_3$

Degree 15: $(C_5^2 : C_3):C_2$

Low degree siblings

45T71

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{45}$ $1$ $1$ $0$ $()$
2A $2^{15},1^{15}$ $15$ $2$ $15$ $( 4,27)( 5,25)( 6,26)( 7,32)( 8,33)( 9,31)(13,34)(14,35)(15,36)(16,42)(17,40)(18,41)(22,45)(23,43)(24,44)$
3A1 $3^{15}$ $1$ $3$ $30$ $( 1, 2, 3)( 4, 6, 5)( 7, 9, 8)(10,11,12)(13,15,14)(16,18,17)(19,20,21)(22,24,23)(25,27,26)(28,29,30)(31,33,32)(34,36,35)(37,38,39)(40,42,41)(43,45,44)$
3A-1 $3^{15}$ $1$ $3$ $30$ $( 1, 3, 2)( 4, 5, 6)( 7, 8, 9)(10,12,11)(13,14,15)(16,17,18)(19,21,20)(22,23,24)(25,26,27)(28,30,29)(31,32,33)(34,35,36)(37,39,38)(40,41,42)(43,44,45)$
3B $3^{15}$ $50$ $3$ $30$ $( 1,33,17)( 2,32,16)( 3,31,18)( 4,34,20)( 5,35,19)( 6,36,21)( 7,37,22)( 8,39,23)( 9,38,24)(10,40,25)(11,42,27)(12,41,26)(13,45,29)(14,43,28)(15,44,30)$
3C1 $3^{15}$ $50$ $3$ $30$ $( 1,45,24)( 2,44,23)( 3,43,22)( 4,30,25)( 5,29,26)( 6,28,27)( 7,31,10)( 8,32,12)( 9,33,11)(13,38,35)(14,37,36)(15,39,34)(16,41,19)(17,42,21)(18,40,20)$
3C-1 $3^{15}$ $50$ $3$ $30$ $( 1,24,45)( 2,23,44)( 3,22,43)( 4,25,30)( 5,26,29)( 6,27,28)( 7,10,31)( 8,12,32)( 9,11,33)(13,35,38)(14,36,37)(15,34,39)(16,19,41)(17,21,42)(18,20,40)$
5A1 $5^{9}$ $3$ $5$ $36$ $( 1,19,39,10,28)( 2,20,37,11,29)( 3,21,38,12,30)( 4,42,32,22,13)( 5,40,33,23,14)( 6,41,31,24,15)( 7,45,34,27,16)( 8,43,35,25,17)( 9,44,36,26,18)$
5A-1 $5^{9}$ $3$ $5$ $36$ $( 1,28,10,39,19)( 2,29,11,37,20)( 3,30,12,38,21)( 4,13,22,32,42)( 5,14,23,33,40)( 6,15,24,31,41)( 7,16,27,34,45)( 8,17,25,35,43)( 9,18,26,36,44)$
5A2 $5^{9}$ $3$ $5$ $36$ $( 1,39,28,19,10)( 2,37,29,20,11)( 3,38,30,21,12)( 4,32,13,42,22)( 5,33,14,40,23)( 6,31,15,41,24)( 7,34,16,45,27)( 8,35,17,43,25)( 9,36,18,44,26)$
5A-2 $5^{9}$ $3$ $5$ $36$ $( 1,10,19,28,39)( 2,11,20,29,37)( 3,12,21,30,38)( 4,22,42,13,32)( 5,23,40,14,33)( 6,24,41,15,31)( 7,27,45,16,34)( 8,25,43,17,35)( 9,26,44,18,36)$
5B1 $5^{6},1^{15}$ $6$ $5$ $24$ $( 1,10,19,28,39)( 2,11,20,29,37)( 3,12,21,30,38)( 7,45,34,27,16)( 8,43,35,25,17)( 9,44,36,26,18)$
5B2 $5^{6},1^{15}$ $6$ $5$ $24$ $( 1,19,39,10,28)( 2,20,37,11,29)( 3,21,38,12,30)( 7,34,16,45,27)( 8,35,17,43,25)( 9,36,18,44,26)$
6A1 $6^{5},3^{5}$ $15$ $6$ $35$ $( 1, 2, 3)( 4,26, 5,27, 6,25)( 7,31, 8,32, 9,33)(10,11,12)(13,36,14,34,15,35)(16,41,17,42,18,40)(19,20,21)(22,44,23,45,24,43)(28,29,30)(37,38,39)$
6A-1 $6^{5},3^{5}$ $15$ $6$ $35$ $( 1, 3, 2)( 4,25, 6,27, 5,26)( 7,33, 9,32, 8,31)(10,12,11)(13,35,15,34,14,36)(16,40,18,42,17,41)(19,21,20)(22,43,24,45,23,44)(28,30,29)(37,39,38)$
10A1 $10^{3},5^{3}$ $15$ $10$ $39$ $( 1,10,19,28,39)( 2,11,20,29,37)( 3,12,21,30,38)( 4,45,42,34,32,27,22,16,13, 7)( 5,43,40,35,33,25,23,17,14, 8)( 6,44,41,36,31,26,24,18,15, 9)$
10A-1 $10^{3},5^{3}$ $15$ $10$ $39$ $( 1,39,28,19,10)( 2,37,29,20,11)( 3,38,30,21,12)( 4, 7,13,16,22,27,32,34,42,45)( 5, 8,14,17,23,25,33,35,40,43)( 6, 9,15,18,24,26,31,36,41,44)$
10A3 $10^{3},5^{3}$ $15$ $10$ $39$ $( 1,28,10,39,19)( 2,29,11,37,20)( 3,30,12,38,21)( 4,34,22, 7,42,27,13,45,32,16)( 5,35,23, 8,40,25,14,43,33,17)( 6,36,24, 9,41,26,15,44,31,18)$
10A-3 $10^{3},5^{3}$ $15$ $10$ $39$ $( 1,19,39,10,28)( 2,20,37,11,29)( 3,21,38,12,30)( 4,16,32,45,13,27,42, 7,22,34)( 5,17,33,43,14,25,40, 8,23,35)( 6,18,31,44,15,26,41, 9,24,36)$
15A1 $15^{3}$ $3$ $15$ $42$ $( 1,37,30,19,11, 3,39,29,21,10, 2,38,28,20,12)( 4,31,14,42,24, 5,32,15,40,22, 6,33,13,41,23)( 7,36,17,45,26, 8,34,18,43,27, 9,35,16,44,25)$
15A-1 $15^{3}$ $3$ $15$ $42$ $( 1,12,20,28,38, 2,10,21,29,39, 3,11,19,30,37)( 4,23,41,13,33, 6,22,40,15,32, 5,24,42,14,31)( 7,25,44,16,35, 9,27,43,18,34, 8,26,45,17,36)$
15A2 $15^{3}$ $3$ $15$ $42$ $( 1,30,11,39,21, 2,28,12,37,19, 3,29,10,38,20)( 4,14,24,32,40, 6,13,23,31,42, 5,15,22,33,41)( 7,17,26,34,43, 9,16,25,36,45, 8,18,27,35,44)$
15A-2 $15^{3}$ $3$ $15$ $42$ $( 1,20,38,10,29, 3,19,37,12,28, 2,21,39,11,30)( 4,41,33,22,15, 5,42,31,23,13, 6,40,32,24,14)( 7,44,35,27,18, 8,45,36,25,16, 9,43,34,26,17)$
15A4 $15^{3}$ $3$ $15$ $42$ $( 1,11,21,28,37, 3,10,20,30,39, 2,12,19,29,38)( 4,24,40,13,31, 5,22,41,14,32, 6,23,42,15,33)( 7,26,43,16,36, 8,27,44,17,34, 9,25,45,18,35)$
15A-4 $15^{3}$ $3$ $15$ $42$ $( 1,38,29,19,12, 2,39,30,20,10, 3,37,28,21,11)( 4,33,15,42,23, 6,32,14,41,22, 5,31,13,40,24)( 7,35,18,45,25, 9,34,17,44,27, 8,36,16,43,26)$
15A7 $15^{3}$ $3$ $15$ $42$ $( 1,29,12,39,20, 3,28,11,38,19, 2,30,10,37,21)( 4,15,23,32,41, 5,13,24,33,42, 6,14,22,31,40)( 7,18,25,34,44, 8,16,26,35,45, 9,17,27,36,43)$
15A-7 $15^{3}$ $3$ $15$ $42$ $( 1,21,37,10,30, 2,19,38,11,28, 3,20,39,12,29)( 4,40,31,22,14, 6,42,33,24,13, 5,41,32,23,15)( 7,43,36,27,17, 9,45,35,26,16, 8,44,34,25,18)$
15B1 $15^{2},3^{5}$ $6$ $15$ $38$ $( 1,21,37,10,30, 2,19,38,11,28, 3,20,39,12,29)( 4, 5, 6)( 7,35,18,45,25, 9,34,17,44,27, 8,36,16,43,26)(13,14,15)(22,23,24)(31,32,33)(40,41,42)$
15B-1 $15^{2},3^{5}$ $6$ $15$ $38$ $( 1, 2, 3)( 4,24,40,13,31, 5,22,41,14,32, 6,23,42,15,33)( 7,36,17,45,26, 8,34,18,43,27, 9,35,16,44,25)(10,11,12)(19,20,21)(28,29,30)(37,38,39)$
15B2 $15^{2},3^{5}$ $6$ $15$ $38$ $( 1,37,30,19,11, 3,39,29,21,10, 2,38,28,20,12)( 4, 6, 5)( 7,18,25,34,44, 8,16,26,35,45, 9,17,27,36,43)(13,15,14)(22,24,23)(31,33,32)(40,42,41)$
15B-2 $15^{2},3^{5}$ $6$ $15$ $38$ $( 1,38,29,19,12, 2,39,30,20,10, 3,37,28,21,11)( 4, 5, 6)( 7,17,26,34,43, 9,16,25,36,45, 8,18,27,35,44)(13,14,15)(22,23,24)(31,32,33)(40,41,42)$
30A1 $30,15$ $15$ $30$ $43$ $( 1,21,37,10,30, 2,19,38,11,28, 3,20,39,12,29)( 4,17,31,45,14,26,42, 8,24,34, 5,18,32,43,15,27,40, 9,22,35, 6,16,33,44,13,25,41, 7,23,36)$
30A-1 $30,15$ $15$ $30$ $43$ $( 1,29,12,39,20, 3,28,11,38,19, 2,30,10,37,21)( 4,26,23,45,41,17,13,36,33, 7, 6,25,22,44,40,16,15,35,32, 9, 5,27,24,43,42,18,14,34,31, 8)$
30A7 $30,15$ $15$ $30$ $43$ $( 1,38,29,19,12, 2,39,30,20,10, 3,37,28,21,11)( 4, 8,15,16,23,26,32,35,41,45, 5, 9,13,17,24,27,33,36,42,43, 6, 7,14,18,22,25,31,34,40,44)$
30A-7 $30,15$ $15$ $30$ $43$ $( 1,11,21,28,37, 3,10,20,30,39, 2,12,19,29,38)( 4,36,40,27,31,17,22, 9,14,45, 6,35,42,26,33,16,24, 8,13,44, 5,34,41,25,32,18,23, 7,15,43)$
30A11 $30,15$ $15$ $30$ $43$ $( 1,20,38,10,29, 3,19,37,12,28, 2,21,39,11,30)( 4, 9,33,34,15,17,42,44,23,27, 6, 8,32,36,14,16,41,43,22,26, 5, 7,31,35,13,18,40,45,24,25)$
30A-11 $30,15$ $15$ $30$ $43$ $( 1,30,11,39,21, 2,28,12,37,19, 3,29,10,38,20)( 4,35,24, 7,40,26,13,43,31,16, 5,36,22, 8,41,27,14,44,32,17, 6,34,23, 9,42,25,15,45,33,18)$
30A13 $30,15$ $15$ $30$ $43$ $( 1,12,20,28,38, 2,10,21,29,39, 3,11,19,30,37)( 4,43,41,34,33,26,22,17,15, 7, 5,44,42,35,31,27,23,18,13, 8, 6,45,40,36,32,25,24,16,14, 9)$
30A-13 $30,15$ $15$ $30$ $43$ $( 1,37,30,19,11, 3,39,29,21,10, 2,38,28,20,12)( 4,44,14, 7,24,17,32,26,40,34, 6,43,13, 9,23,16,31,25,42,36, 5,45,15, 8,22,18,33,27,41,35)$

Malle's constant $a(G)$:     $1/15$

Copy content magma:ConjugacyClasses(G);
 

Character table

39 x 39 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed