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Properties

Label 45T21
Degree $45$
Order $180$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{15}:D_6$

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Copy content magma:G := TransitiveGroup(45, 21);
 

Group invariants

Abstract group:  $C_{15}:D_6$
Copy content magma:IdentifyGroup(G);
 
Order:  $180=2^{2} \cdot 3^{2} \cdot 5$
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 $t$:  $21$
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:  $(1,7,29,35,10,17,37,44,21,25)(2,9,30,36,11,18,39,45,19,26)(3,8,28,34,12,16,38,43,20,27)(4,42,31,22,15,5,41,32,23,13)(6,40,33,24,14)$, $(1,28,2,29,3,30)(4,25,6,27,5,26)(7,24,8,22,9,23)(10,20,11,21,12,19)(13,18,15,17,14,16)(31,44,33,43,32,45)(34,42,36,41,35,40)(37,38,39)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$ x 2
$10$:  $D_{5}$
$12$:  $D_{6}$ x 2
$20$:  $D_{10}$
$36$:  $S_3^2$
$60$:  $D_5\times S_3$ x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$ x 2

Degree 5: $D_{5}$

Degree 9: $S_3^2$

Degree 15: $D_5\times S_3$ x 2

Low degree siblings

30T43

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 $2^{20},1^{5}$ $9$ $2$ $20$ $( 1,16)( 2,17)( 3,18)( 5, 6)( 7,39)( 8,37)( 9,38)(10,27)(11,25)(12,26)(13,14)(19,35)(20,36)(21,34)(22,24)(28,45)(29,43)(30,44)(32,33)(40,42)$
2B $2^{21},1^{3}$ $15$ $2$ $21$ $( 1,39)( 2,37)( 3,38)( 4, 6)( 7,17)( 8,18)( 9,16)(10,30)(11,29)(12,28)(13,42)(14,41)(15,40)(19,21)(22,32)(23,33)(24,31)(25,44)(26,43)(27,45)(34,36)$
2C $2^{21},1^{3}$ $15$ $2$ $21$ $( 1,15)( 2,14)( 3,13)( 4,10)( 5,12)( 6,11)(16,43)(17,44)(18,45)(19,40)(20,42)(21,41)(22,38)(23,37)(24,39)(25,35)(26,36)(27,34)(28,32)(29,31)(30,33)$
3A $3^{15}$ $2$ $3$ $30$ $( 1,16,31)( 2,18,33)( 3,17,32)( 4,21,34)( 5,20,35)( 6,19,36)( 7,22,38)( 8,23,37)( 9,24,39)(10,27,41)(11,26,40)(12,25,42)(13,28,44)(14,30,45)(15,29,43)$
3B $3^{15}$ $2$ $3$ $30$ $( 1, 2, 3)( 4, 6, 5)( 7, 8, 9)(10,11,12)(13,15,14)(16,18,17)(19,20,21)(22,23,24)(25,27,26)(28,29,30)(31,33,32)(34,36,35)(37,39,38)(40,42,41)(43,45,44)$
3C $3^{15}$ $4$ $3$ $30$ $( 1,18,32)( 2,17,31)( 3,16,33)( 4,19,35)( 5,21,36)( 6,20,34)( 7,23,39)( 8,24,38)( 9,22,37)(10,26,42)(11,25,41)(12,27,40)(13,29,45)(14,28,43)(15,30,44)$
5A1 $5^{9}$ $2$ $5$ $36$ $( 1,10,21,29,37)( 2,11,19,30,39)( 3,12,20,28,38)( 4,15,23,31,41)( 5,13,22,32,42)( 6,14,24,33,40)( 7,17,25,35,44)( 8,16,27,34,43)( 9,18,26,36,45)$
5A2 $5^{9}$ $2$ $5$ $36$ $( 1,21,37,10,29)( 2,19,39,11,30)( 3,20,38,12,28)( 4,23,41,15,31)( 5,22,42,13,32)( 6,24,40,14,33)( 7,25,44,17,35)( 8,27,43,16,34)( 9,26,45,18,36)$
6A $6^{7},3$ $30$ $6$ $37$ $( 1,24,16,39,31, 9)( 2,23,18,37,33, 8)( 3,22,17,38,32, 7)( 4,36,21, 6,34,19)( 5,35,20)(10,14,27,30,41,45)(11,15,26,29,40,43)(12,13,25,28,42,44)$
6B $6^{7},3$ $30$ $6$ $37$ $( 1,14, 3,15, 2,13)( 4,11, 5,10, 6,12)( 7, 8, 9)(16,45,17,43,18,44)(19,42,21,40,20,41)(22,37,24,38,23,39)(25,34,26,35,27,36)(28,31,30,32,29,33)$
10A1 $10^{4},5$ $18$ $10$ $40$ $( 1,27,21,43,37,16,10,34,29, 8)( 2,25,19,44,39,17,11,35,30, 7)( 3,26,20,45,38,18,12,36,28, 9)( 4,15,23,31,41)( 5,14,22,33,42, 6,13,24,32,40)$
10A3 $10^{4},5$ $18$ $10$ $40$ $( 1,43,10, 8,21,16,29,27,37,34)( 2,44,11, 7,19,17,30,25,39,35)( 3,45,12, 9,20,18,28,26,38,36)( 4,31,15,41,23)( 5,33,13,40,22, 6,32,14,42,24)$
15A1 $15^{3}$ $4$ $15$ $42$ $( 1, 4, 8,10,15,16,21,23,27,29,31,34,37,41,43)( 2, 6, 9,11,14,18,19,24,26,30,33,36,39,40,45)( 3, 5, 7,12,13,17,20,22,25,28,32,35,38,42,44)$
15A2 $15^{3}$ $4$ $15$ $42$ $( 1, 8,15,21,27,31,37,43, 4,10,16,23,29,34,41)( 2, 9,14,19,26,33,39,45, 6,11,18,24,30,36,40)( 3, 7,13,20,25,32,38,44, 5,12,17,22,28,35,42)$
15B1 $15^{3}$ $4$ $15$ $42$ $( 1,12,19,29,38, 2,10,20,30,37, 3,11,21,28,39)( 4,13,24,31,42, 6,15,22,33,41, 5,14,23,32,40)( 7,18,27,35,45, 8,17,26,34,44, 9,16,25,36,43)$
15B2 $15^{3}$ $4$ $15$ $42$ $( 1,20,39,10,28, 2,21,38,11,29, 3,19,37,12,30)( 4,22,40,15,32, 6,23,42,14,31, 5,24,41,13,33)( 7,26,43,17,36, 8,25,45,16,35, 9,27,44,18,34)$
15C1 $15^{3}$ $4$ $15$ $42$ $( 1, 5, 9,10,13,18,21,22,26,29,32,36,37,42,45)( 2, 4, 7,11,15,17,19,23,25,30,31,35,39,41,44)( 3, 6, 8,12,14,16,20,24,27,28,33,34,38,40,43)$
15C-1 $15^{3}$ $4$ $15$ $42$ $( 1, 6, 7,10,14,17,21,24,25,29,33,35,37,40,44)( 2, 5, 8,11,13,16,19,22,27,30,32,34,39,42,43)( 3, 4, 9,12,15,18,20,23,26,28,31,36,38,41,45)$
15C2 $15^{3}$ $4$ $15$ $42$ $( 1, 9,13,21,26,32,37,45, 5,10,18,22,29,36,42)( 2, 7,15,19,25,31,39,44, 4,11,17,23,30,35,41)( 3, 8,14,20,27,33,38,43, 6,12,16,24,28,34,40)$
15C-2 $15^{3}$ $4$ $15$ $42$ $( 1, 7,14,21,25,33,37,44, 6,10,17,24,29,35,40)( 2, 8,13,19,27,32,39,43, 5,11,16,22,30,34,42)( 3, 9,15,20,26,31,38,45, 4,12,18,23,28,36,41)$

Malle's constant $a(G)$:     $1/20$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 3A 3B 3C 5A1 5A2 6A 6B 10A1 10A3 15A1 15A2 15B1 15B2 15C1 15C-1 15C2 15C-2
Size 1 9 15 15 2 2 4 2 2 30 30 18 18 4 4 4 4 4 4 4 4
2 P 1A 1A 1A 1A 3A 3B 3C 5A2 5A1 3A 3B 5A2 5A1 15A2 15A1 15B2 15B1 15C2 15C-2 15C-1 15C1
3 P 1A 2A 2B 2C 1A 1A 1A 5A2 5A1 2B 2C 10A3 10A1 5A1 5A2 5A2 5A1 5A1 5A1 5A2 5A2
5 P 1A 2A 2B 2C 3A 3B 3C 1A 1A 6A 6B 2A 2A 3A 3A 3B 3B 3C 3C 3C 3C
Type
180.30.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
180.30.1b R 1 −1 −1 1 1 1 1 1 1 −1 1 −1 −1 1 1 1 1 1 1 1 1
180.30.1c R 1 −1 1 −1 1 1 1 1 1 1 −1 −1 −1 1 1 1 1 1 1 1 1
180.30.1d R 1 1 −1 −1 1 1 1 1 1 −1 −1 1 1 1 1 1 1 1 1 1 1
180.30.2a R 2 0 0 2 2 −1 −1 2 2 0 −1 0 0 −1 2 −1 −1 −1 −1 2 −1
180.30.2b R 2 0 2 0 −1 2 −1 2 2 −1 0 0 0 −1 −1 2 2 −1 −1 −1 −1
180.30.2c R 2 0 −2 0 −1 2 −1 2 2 1 0 0 0 −1 −1 2 2 −1 −1 −1 −1
180.30.2d R 2 0 0 −2 2 −1 −1 2 2 0 1 0 0 −1 2 −1 −1 −1 −1 2 −1
180.30.2e1 R 2 2 0 0 2 2 2 ζ5−2+ζ52 ζ5−1+ζ5 0 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52
180.30.2e2 R 2 2 0 0 2 2 2 ζ5−1+ζ5 ζ5−2+ζ52 0 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5
180.30.2f1 R 2 −2 0 0 2 2 2 ζ5−2+ζ52 ζ5−1+ζ5 0 0 −ζ5−2−ζ52 −ζ5−1−ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52
180.30.2f2 R 2 −2 0 0 2 2 2 ζ5−1+ζ5 ζ5−2+ζ52 0 0 −ζ5−1−ζ5 −ζ5−2−ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5
180.30.4a R 4 0 0 0 −2 −2 1 4 4 0 0 0 0 1 −2 −2 −2 1 1 −2 1
180.30.4b1 R 4 0 0 0 −2 4 −2 2ζ5−2+2ζ52 2ζ5−1+2ζ5 0 0 0 0 −ζ5−1−ζ5 −ζ5−2−ζ52 2ζ5−2+2ζ52 2ζ5−1+2ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52
180.30.4b2 R 4 0 0 0 −2 4 −2 2ζ5−1+2ζ5 2ζ5−2+2ζ52 0 0 0 0 −ζ5−2−ζ52 −ζ5−1−ζ5 2ζ5−1+2ζ5 2ζ5−2+2ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5
180.30.4c1 R 4 0 0 0 4 −2 −2 2ζ5−2+2ζ52 2ζ5−1+2ζ5 0 0 0 0 −ζ5−1−ζ5 2ζ5−2+2ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 2ζ5−1+2ζ5 −ζ5−2−ζ52
180.30.4c2 R 4 0 0 0 4 −2 −2 2ζ5−1+2ζ5 2ζ5−2+2ζ52 0 0 0 0 −ζ5−2−ζ52 2ζ5−1+2ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 2ζ5−2+2ζ52 −ζ5−1−ζ5
180.30.4d1 C 4 0 0 0 −2 −2 1 2ζ5−2+2ζ52 2ζ5−1+2ζ5 0 0 0 0 −2ζ5−2−2−3ζ5−2ζ52 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 2ζ5−2−ζ52 ζ5−2+1+3ζ5+ζ52 −ζ5−1−ζ5 −ζ5−2+2ζ52
180.30.4d2 C 4 0 0 0 −2 −2 1 2ζ5−2+2ζ52 2ζ5−1+2ζ5 0 0 0 0 ζ5−2+1+3ζ5+ζ52 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2+2ζ52 −2ζ5−2−2−3ζ5−2ζ52 −ζ5−1−ζ5 2ζ5−2−ζ52
180.30.4d3 C 4 0 0 0 −2 −2 1 2ζ5−1+2ζ5 2ζ5−2+2ζ52 0 0 0 0 2ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 ζ5−2+1+3ζ5+ζ52 −ζ5−2+2ζ52 −ζ5−2−ζ52 −2ζ5−2−2−3ζ5−2ζ52
180.30.4d4 C 4 0 0 0 −2 −2 1 2ζ5−1+2ζ5 2ζ5−2+2ζ52 0 0 0 0 −ζ5−2+2ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −2ζ5−2−2−3ζ5−2ζ52 2ζ5−2−ζ52 −ζ5−2−ζ52 ζ5−2+1+3ζ5+ζ52

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed