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Properties

Label 45T50
45T50 1 39 1->39 41 1->41 2 38 2->38 45 2->45 3 37 3->37 44 3->44 4 36 4->36 43 4->43 5 40 5->40 42 5->42 6 19 6->19 7 10 7->10 18 7->18 8 9 8->9 17 8->17 16 9->16 20 10->20 11 24 11->24 31 11->31 12 23 12->23 35 12->35 13 22 13->22 34 13->34 14 21 14->21 33 14->33 15 25 15->25 32 15->32 16->4 26 16->26 17->3 30 17->30 18->2 29 18->29 19->1 28 19->28 20->5 27 20->27 21->29 21->36 22->28 22->40 23->27 23->39 24->26 24->38 25->30 25->37 26->34 27->33 28->32 29->31 30->35 31->14 32->13 33->12 34->11 35->15 36->9 37->8 38->7 39->6 40->10 41->44 42->43
Degree $45$
Order $360$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^2:C_4\times D_5$

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Copy content magma:G := TransitiveGroup(45, 50);
 

Group invariants

Abstract group:  $C_3^2:C_4\times D_5$
Copy content magma:IdentifyGroup(G);
 
Order:  $360=2^{3} \cdot 3^{2} \cdot 5$
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 $n$:  $45$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $50$
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:  $(1,39,6,19)(2,38,7,18)(3,37,8,17)(4,36,9,16)(5,40,10,20)(11,24,26,34)(12,23,27,33)(13,22,28,32)(14,21,29,31)(15,25,30,35)(41,44)(42,43)$, $(1,41)(2,45)(3,44)(4,43)(5,42)(7,10)(8,9)(11,31)(12,35)(13,34)(14,33)(15,32)(16,26)(17,30)(18,29)(19,28)(20,27)(21,36)(22,40)(23,39)(24,38)(25,37)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $C_4\times C_2$
$10$:  $D_{5}$
$20$:  $D_{10}$
$36$:  $C_3^2:C_4$
$40$:  20T6
$72$:  12T40

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 5: $D_{5}$

Degree 9: $C_3^2:C_4$

Degree 15: None

Low degree siblings

30T99 x 2

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

Malle's constant $a(G)$:     $1/18$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 3A 3B 4A1 4A-1 4B1 4B-1 5A1 5A2 6A 6B 10A1 10A3 15A1 15A2 15B1 15B2 20A1 20A-1 20A3 20A-3
Size 1 5 9 45 4 4 9 9 45 45 2 2 20 20 18 18 8 8 8 8 18 18 18 18
2 P 1A 1A 1A 1A 3A 3B 2B 2B 2B 2B 5A2 5A1 3B 3A 5A1 5A2 15A2 15A1 15B2 15B1 10A1 10A1 10A3 10A3
3 P 1A 2A 2B 2C 1A 1A 4A-1 4A1 4B-1 4B1 5A2 5A1 2A 2A 10A3 10A1 5A1 5A2 5A2 5A1 20A3 20A-3 20A1 20A-1
5 P 1A 2A 2B 2C 3A 3B 4A1 4A-1 4B1 4B-1 1A 1A 6A 6B 2B 2B 3A 3A 3B 3B 4A1 4A-1 4A-1 4A1
Type
360.130.1a 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
360.130.1b 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
360.130.1c 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
360.130.1d 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
360.130.1e1 C 1 −1 −1 1 1 1 −i i i −i 1 1 −1 −1 −1 −1 1 1 1 1 −i i i −i
360.130.1e2 C 1 −1 −1 1 1 1 i −i −i i 1 1 −1 −1 −1 −1 1 1 1 1 i −i −i i
360.130.1f1 C 1 1 −1 −1 1 1 −i i −i i 1 1 1 1 −1 −1 1 1 1 1 −i i i −i
360.130.1f2 C 1 1 −1 −1 1 1 i −i i −i 1 1 1 1 −1 −1 1 1 1 1 i −i −i i
360.130.2a1 R 2 0 2 0 2 2 2 2 0 0 ζ5−2+ζ52 ζ5−1+ζ5 0 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5
360.130.2a2 R 2 0 2 0 2 2 2 2 0 0 ζ5−1+ζ5 ζ5−2+ζ52 0 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52
360.130.2b1 R 2 0 2 0 2 2 −2 −2 0 0 ζ5−2+ζ52 ζ5−1+ζ5 0 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5
360.130.2b2 R 2 0 2 0 2 2 −2 −2 0 0 ζ5−1+ζ5 ζ5−2+ζ52 0 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52
360.130.2c1 C 2 0 −2 0 2 2 −2ζ205 2ζ205 0 0 −ζ20−2−ζ202 ζ20−4+ζ204 0 0 −ζ20−4−ζ204 ζ20−2+ζ202 ζ20−4+ζ204 −ζ20−2−ζ202 −ζ20−2−ζ202 ζ20−4+ζ204 ζ203+ζ207 −ζ203−ζ207 ζ203−ζ205+ζ207 −ζ203+ζ205−ζ207
360.130.2c2 C 2 0 −2 0 2 2 2ζ205 −2ζ205 0 0 −ζ20−2−ζ202 ζ20−4+ζ204 0 0 −ζ20−4−ζ204 ζ20−2+ζ202 ζ20−4+ζ204 −ζ20−2−ζ202 −ζ20−2−ζ202 ζ20−4+ζ204 −ζ203−ζ207 ζ203+ζ207 −ζ203+ζ205−ζ207 ζ203−ζ205+ζ207
360.130.2c3 C 2 0 −2 0 2 2 −2ζ205 2ζ205 0 0 ζ20−4+ζ204 −ζ20−2−ζ202 0 0 ζ20−2+ζ202 −ζ20−4−ζ204 −ζ20−2−ζ202 ζ20−4+ζ204 ζ20−4+ζ204 −ζ20−2−ζ202 −ζ203+ζ205−ζ207 ζ203−ζ205+ζ207 −ζ203−ζ207 ζ203+ζ207
360.130.2c4 C 2 0 −2 0 2 2 2ζ205 −2ζ205 0 0 ζ20−4+ζ204 −ζ20−2−ζ202 0 0 ζ20−2+ζ202 −ζ20−4−ζ204 −ζ20−2−ζ202 ζ20−4+ζ204 ζ20−4+ζ204 −ζ20−2−ζ202 ζ203−ζ205+ζ207 −ζ203+ζ205−ζ207 ζ203+ζ207 −ζ203−ζ207
360.130.4a R 4 4 0 0 −2 1 0 0 0 0 4 4 1 −2 0 0 −2 −2 1 1 0 0 0 0
360.130.4b R 4 4 0 0 1 −2 0 0 0 0 4 4 −2 1 0 0 1 1 −2 −2 0 0 0 0
360.130.4c R 4 −4 0 0 −2 1 0 0 0 0 4 4 −1 2 0 0 −2 −2 1 1 0 0 0 0
360.130.4d R 4 −4 0 0 1 −2 0 0 0 0 4 4 2 −1 0 0 1 1 −2 −2 0 0 0 0
360.130.8a1 R 8 0 0 0 −4 2 0 0 0 0 4ζ5−2+4ζ52 4ζ5−1+4ζ5 0 0 0 0 −2ζ5−1−2ζ5 −2ζ5−2−2ζ52 ζ5−2+ζ52 ζ5−1+ζ5 0 0 0 0
360.130.8a2 R 8 0 0 0 −4 2 0 0 0 0 4ζ5−1+4ζ5 4ζ5−2+4ζ52 0 0 0 0 −2ζ5−2−2ζ52 −2ζ5−1−2ζ5 ζ5−1+ζ5 ζ5−2+ζ52 0 0 0 0
360.130.8b1 R 8 0 0 0 2 −4 0 0 0 0 4ζ5−2+4ζ52 4ζ5−1+4ζ5 0 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52 −2ζ5−2−2ζ52 −2ζ5−1−2ζ5 0 0 0 0
360.130.8b2 R 8 0 0 0 2 −4 0 0 0 0 4ζ5−1+4ζ5 4ζ5−2+4ζ52 0 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5 −2ζ5−1−2ζ5 −2ζ5−2−2ζ52 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed