↑

Properties

Label 45T266
45T266 1 18 1->18 31 1->31 2 16 2->16 32 2->32 3 17 3->17 33 3->33 4 4->16 21 4->21 5 5->18 19 5->19 6 6->17 20 6->20 7 7->19 22 7->22 8 8->20 24 8->24 9 9->21 23 9->23 10 10->24 26 10->26 11 11->22 27 11->27 12 12->23 25 12->25 13 13->8 28 13->28 14 14->7 29 14->29 15 15->9 30 15->30 16->10 16->33 17->11 17->32 18->12 18->31 19->14 35 19->35 20->15 36 20->36 21->13 34 21->34 38 22->38 43 22->43 39 23->39 45 23->45 37 24->37 44 24->44 25->3 40 25->40 26->1 42 26->42 27->2 41 27->41 28->4 28->45 29->5 29->43 30->6 30->44 31->1 31->35 32->2 32->34 33->3 33->36 34->4 34->39 35->6 35->37 36->5 36->38 37->9 37->40 38->7 38->41 39->8 39->42 40->10 40->26 41->11 41->25 42->12 42->27 43->15 43->30 44->14 44->28 45->13 45->29
Degree $45$
Order $2700$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{15}^2:(C_2\times C_6)$

Related objects

Downloads

Learn more

Show commands: Magma

Copy content magma:G := TransitiveGroup(45, 266);
 

Group invariants

Abstract group:  $C_{15}^2:(C_2\times C_6)$
Copy content magma:IdentifyGroup(G);
 
Order:  $2700=2^{2} \cdot 3^{3} \cdot 5^{2}$
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 $t$:  $266$
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:  $(1,18,31)(2,16,33,3,17,32)(4,21,34)(5,19,35,6,20,36)(7,22,38)(8,24,37,9,23,39)(10,26,42,12,25,40)(11,27,41)(13,28,45)(14,29,43,15,30,44)$, $(1,31,35,37,40,26)(2,32,34,39,42,27)(3,33,36,38,41,25)(4,16,10,24,44,28)(5,18,12,23,45,29)(6,17,11,22,43,30)(7,19,14)(8,20,15,9,21,13)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_2^2$
$6$:  $S_3$, $C_6$ x 3
$12$:  $D_{6}$, $C_6\times C_2$
$18$:  $S_3\times C_3$
$36$:  $C_6\times S_3$
$54$:  $C_3^2 : C_6$
$108$:  18T41
$150$:  $(C_5^2 : C_3):C_2$
$300$:  30T68
$900$:  45T125

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 5: None

Degree 9: $C_3^2 : S_3 $

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

Low degree siblings

45T266 x 5

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 $2^{15},1^{15}$ $9$ $2$ $15$ $( 2, 3)( 5, 6)( 8, 9)(10,12)(14,15)(16,17)(19,20)(23,24)(25,26)(29,30)(32,33)(35,36)(37,39)(40,42)(43,44)$
2B $2^{18},1^{9}$ $25$ $2$ $18$ $( 1,11)( 2,12)( 3,10)( 4,31)( 5,32)( 6,33)( 7,27)( 8,25)( 9,26)(13,22)(14,23)(15,24)(19,39)(20,37)(21,38)(34,45)(35,43)(36,44)$
2C $2^{21},1^{3}$ $225$ $2$ $21$ $( 1,39)( 2,38)( 3,37)( 4,42)( 5,41)( 6,40)( 7,34)( 8,36)( 9,35)(10,30)(11,29)(12,28)(13,32)(14,31)(15,33)(16,26)(17,25)(18,27)(19,21)(22,23)(43,44)$
3A $3^{15}$ $2$ $3$ $30$ $( 1, 2, 3)( 4, 6, 5)( 7, 9, 8)(10,11,12)(13,15,14)(16,18,17)(19,20,21)(22,24,23)(25,27,26)(28,29,30)(31,33,32)(34,36,35)(37,38,39)(40,42,41)(43,45,44)$
3B $3^{10},1^{15}$ $6$ $3$ $20$ $( 4, 6, 5)( 7, 8, 9)(13,15,14)(16,17,18)(22,24,23)(25,26,27)(31,33,32)(34,35,36)(40,42,41)(43,44,45)$
3C1 $3^{15}$ $75$ $3$ $30$ $( 1, 5,26)( 2, 4,25)( 3, 6,27)( 7,30,33)( 8,29,31)( 9,28,32)(10,15,34)(11,14,36)(12,13,35)(16,39,41)(17,38,42)(18,37,40)(19,22,43)(20,24,45)(21,23,44)$
3C-1 $3^{15}$ $75$ $3$ $30$ $( 1,26, 5)( 2,25, 4)( 3,27, 6)( 7,33,30)( 8,31,29)( 9,32,28)(10,34,15)(11,36,14)(12,35,13)(16,41,39)(17,42,38)(18,40,37)(19,43,22)(20,45,24)(21,44,23)$
3D1 $3^{15}$ $150$ $3$ $30$ $( 1,25, 4)( 2,27, 6)( 3,26, 5)( 7,33,29)( 8,31,28)( 9,32,30)(10,36,14)(11,35,13)(12,34,15)(16,41,38)(17,42,37)(18,40,39)(19,45,24)(20,44,23)(21,43,22)$
3D-1 $3^{15}$ $150$ $3$ $30$ $( 1, 4,25)( 2, 6,27)( 3, 5,26)( 7,29,33)( 8,28,31)( 9,30,32)(10,14,36)(11,13,35)(12,15,34)(16,38,41)(17,37,42)(18,39,40)(19,24,45)(20,23,44)(21,22,43)$
5A1 $5^{6},1^{15}$ $6$ $5$ $24$ $( 1,21,38,11,28)( 2,19,39,12,29)( 3,20,37,10,30)( 7,34,18,45,27)( 8,35,16,43,25)( 9,36,17,44,26)$
5A2 $5^{6},1^{15}$ $6$ $5$ $24$ $( 1,38,28,21,11)( 2,39,29,19,12)( 3,37,30,20,10)( 7,18,27,34,45)( 8,16,25,35,43)( 9,17,26,36,44)$
5B1 $5^{9}$ $6$ $5$ $36$ $( 1,21,38,11,28)( 2,19,39,12,29)( 3,20,37,10,30)( 4,41,31,22,13)( 5,42,32,23,14)( 6,40,33,24,15)( 7,45,34,27,18)( 8,43,35,25,16)( 9,44,36,26,17)$
5B2 $5^{9}$ $6$ $5$ $36$ $( 1,38,28,21,11)( 2,39,29,19,12)( 3,37,30,20,10)( 4,31,13,41,22)( 5,32,14,42,23)( 6,33,15,40,24)( 7,34,18,45,27)( 8,35,16,43,25)( 9,36,17,44,26)$
6A $6^{6},3^{3}$ $50$ $6$ $36$ $( 1,10, 2,11, 3,12)( 4,32, 6,31, 5,33)( 7,25, 9,27, 8,26)(13,23,15,22,14,24)(16,17,18)(19,38,20,39,21,37)(28,30,29)(34,43,36,45,35,44)(40,41,42)$
6B1 $6^{6},3^{3}$ $75$ $6$ $36$ $( 1,26,32,11,17,42)( 2,25,31,12,16,41)( 3,27,33,10,18,40)( 4,39,35,22,19, 8)( 5,38,36,23,21, 9)( 6,37,34,24,20, 7)(13,29,43)(14,28,44)(15,30,45)$
6B-1 $6^{6},3^{3}$ $75$ $6$ $36$ $( 1,42,17,11,32,26)( 2,41,16,12,31,25)( 3,40,18,10,33,27)( 4, 8,19,22,35,39)( 5, 9,21,23,36,38)( 6, 7,20,24,34,37)(13,43,29)(14,44,28)(15,45,30)$
6C $6^{4},3^{2},2^{6},1^{3}$ $150$ $6$ $30$ $( 4, 5, 6)( 7,17, 8,18, 9,16)(10,37)(11,38)(12,39)(13,42,15,41,14,40)(19,29)(20,30)(21,28)(22,32,24,31,23,33)(25,45,26,43,27,44)(34,36,35)$
6D1 $6^{6},3^{3}$ $150$ $6$ $36$ $( 1,41,25,38, 4,16)( 2,40,27,39, 6,18)( 3,42,26,37, 5,17)( 7,12,33,34,29,15)( 8,11,31,35,28,13)( 9,10,32,36,30,14)(19,24,45)(20,23,44)(21,22,43)$
6D-1 $6^{6},3^{3}$ $150$ $6$ $36$ $( 1,16, 4,38,25,41)( 2,18, 6,39,27,40)( 3,17, 5,37,26,42)( 7,15,29,34,33,12)( 8,13,28,35,31,11)( 9,14,30,36,32,10)(19,45,24)(20,44,23)(21,43,22)$
6E1 $6^{7},3$ $225$ $6$ $37$ $( 1,16, 5,39,26,41)( 2,17, 4,38,25,42)( 3,18, 6,37,27,40)( 7,15,30,34,33,10)( 8,14,29,36,31,11)( 9,13,28,35,32,12)(19,44,22,21,43,23)(20,45,24)$
6E-1 $6^{7},3$ $225$ $6$ $37$ $( 1,41,26,39, 5,16)( 2,42,25,38, 4,17)( 3,40,27,37, 6,18)( 7,10,33,34,30,15)( 8,11,31,36,29,14)( 9,12,32,35,28,13)(19,23,43,21,22,44)(20,24,45)$
6F1 $6^{5},3^{5}$ $225$ $6$ $35$ $( 1, 4,18)( 2, 5,17, 3, 6,16)( 7,38,41)( 8,39,42, 9,37,40)(10,15,25,12,14,26)(11,13,27)(19,23,36,20,24,35)(21,22,34)(28,31,45)(29,32,44,30,33,43)$
6F-1 $6^{5},3^{5}$ $225$ $6$ $35$ $( 1,18, 4)( 2,16, 6, 3,17, 5)( 7,41,38)( 8,40,37, 9,42,39)(10,26,14,12,25,15)(11,27,13)(19,35,24,20,36,23)(21,34,22)(28,45,31)(29,43,33,30,44,32)$
10A1 $10^{3},5^{3}$ $54$ $10$ $39$ $( 1,28,11,38,21)( 2,30,12,37,19, 3,29,10,39,20)( 4,33,13,40,22, 6,31,15,41,24)( 5,32,14,42,23)( 7,43,34,25,18, 8,45,35,27,16)( 9,44,36,26,17)$
10A3 $10^{3},5^{3}$ $54$ $10$ $39$ $( 1,38,28,21,11)( 2,37,29,20,12, 3,39,30,19,10)( 4,40,31,24,13, 6,41,33,22,15)( 5,42,32,23,14)( 7,25,45,16,34, 8,27,43,18,35)( 9,26,44,17,36)$
10B1 $10^{2},5^{2},2^{5},1^{5}$ $54$ $10$ $31$ $( 1,19,38,12,28, 2,21,39,11,29)( 3,20,37,10,30)( 4,32,13,42,22, 5,31,14,41,23)( 6,33,15,40,24)( 8, 9)(16,17)(25,26)(35,36)(43,44)$
10B3 $10^{2},5^{2},2^{5},1^{5}$ $54$ $10$ $31$ $( 1,12,21,29,38, 2,11,19,28,39)( 3,10,20,30,37)( 4,42,31,23,13, 5,41,32,22,14)( 6,40,33,24,15)( 8, 9)(16,17)(25,26)(35,36)(43,44)$
15A1 $15^{2},3^{5}$ $12$ $15$ $38$ $( 1,37,29,21,10, 2,38,30,19,11, 3,39,28,20,12)( 4, 5, 6)( 7,16,26,34,43, 9,18,25,36,45, 8,17,27,35,44)(13,14,15)(22,23,24)(31,32,33)(40,41,42)$
15A2 $15^{2},3^{5}$ $12$ $15$ $38$ $( 1,29,10,38,19, 3,28,12,37,21, 2,30,11,39,20)( 4, 6, 5)( 7,26,43,18,36, 8,27,44,16,34, 9,25,45,17,35)(13,15,14)(22,24,23)(31,33,32)(40,42,41)$
15B1 $15,5^{3},3^{5}$ $12$ $15$ $36$ $( 1,21,38,11,28)( 2,19,39,12,29)( 3,20,37,10,30)( 4,32,15,41,23, 6,31,14,40,22, 5,33,13,42,24)( 7, 9, 8)(16,18,17)(25,27,26)(34,36,35)(43,45,44)$
15B2 $15,5^{3},3^{5}$ $12$ $15$ $36$ $( 1,38,28,21,11)( 2,39,29,19,12)( 3,37,30,20,10)( 4,15,23,31,40, 5,13,24,32,41, 6,14,22,33,42)( 7, 8, 9)(16,17,18)(25,26,27)(34,35,36)(43,44,45)$
15C1 $15^{2},5^{3}$ $12$ $15$ $40$ $( 1,38,28,21,11)( 2,39,29,19,12)( 3,37,30,20,10)( 4,32,15,41,23, 6,31,14,40,22, 5,33,13,42,24)( 7,36,16,45,26, 8,34,17,43,27, 9,35,18,44,25)$
15C2 $15^{2},5^{3}$ $12$ $15$ $40$ $( 1,28,11,38,21)( 2,29,12,39,19)( 3,30,10,37,20)( 4,15,23,31,40, 5,13,24,32,41, 6,14,22,33,42)( 7,16,26,34,43, 9,18,25,36,45, 8,17,27,35,44)$
15D1 $15^{2},5^{3}$ $12$ $15$ $40$ $( 1,10,19,28,37, 2,11,20,29,38, 3,12,21,30,39)( 4,33,14,41,24, 5,31,15,42,22, 6,32,13,40,23)( 7,18,27,34,45)( 8,16,25,35,43)( 9,17,26,36,44)$
15D2 $15^{2},5^{3}$ $12$ $15$ $40$ $( 1,28,11,38,21)( 2,29,12,39,19)( 3,30,10,37,20)( 4,32,15,41,23, 6,31,14,40,22, 5,33,13,42,24)( 7,44,35,27,17, 8,45,36,25,18, 9,43,34,26,16)$
15E1 $15^{3}$ $12$ $15$ $42$ $( 1,39,30,21,12, 3,38,29,20,11, 2,37,28,19,10)( 4,40,32,22,15, 5,41,33,23,13, 6,42,31,24,14)( 7,26,43,18,36, 8,27,44,16,34, 9,25,45,17,35)$
15E2 $15^{3}$ $12$ $15$ $42$ $( 1,20,39,11,30, 2,21,37,12,28, 3,19,38,10,29)( 4,23,40,13,32, 6,22,42,15,31, 5,24,41,14,33)( 7,16,26,34,43, 9,18,25,36,45, 8,17,27,35,44)$
15F1 $15^{2},5^{3}$ $12$ $15$ $40$ $( 1,11,21,28,38)( 2,12,19,29,39)( 3,10,20,30,37)( 4,32,15,41,23, 6,31,14,40,22, 5,33,13,42,24)( 7,17,25,34,44, 8,18,26,35,45, 9,16,27,36,43)$
15F2 $15^{2},5^{3}$ $12$ $15$ $40$ $( 1,21,38,11,28)( 2,19,39,12,29)( 3,20,37,10,30)( 4,15,23,31,40, 5,13,24,32,41, 6,14,22,33,42)( 7,25,44,18,35, 9,27,43,17,34, 8,26,45,16,36)$
15G1 $15^{2},1^{15}$ $12$ $15$ $28$ $( 4,32,15,41,23, 6,31,14,40,22, 5,33,13,42,24)( 7,26,43,18,36, 8,27,44,16,34, 9,25,45,17,35)$
15G2 $15^{2},1^{15}$ $12$ $15$ $28$ $( 4,15,23,31,40, 5,13,24,32,41, 6,14,22,33,42)( 7,43,36,27,16, 9,45,35,26,18, 8,44,34,25,17)$
15H1 $15,5^{3},3^{5}$ $12$ $15$ $36$ $( 1, 3, 2)( 4,33,14,41,24, 5,31,15,42,22, 6,32,13,40,23)( 7,27,45,18,34)( 8,25,43,16,35)( 9,26,44,17,36)(10,12,11)(19,21,20)(28,30,29)(37,39,38)$
15H2 $15,5^{3},3^{5}$ $12$ $15$ $36$ $( 1,38,28,21,11)( 2,39,29,19,12)( 3,37,30,20,10)( 4, 6, 5)( 7,16,26,34,43, 9,18,25,36,45, 8,17,27,35,44)(13,15,14)(22,24,23)(31,33,32)(40,42,41)$

Malle's constant $a(G)$:     $1/15$

Copy content magma:ConjugacyClasses(G);
 

Character table

44 x 44 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed