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Properties

Label 45T48
Degree $45$
Order $360$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3:S_3\times F_5$

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Copy content magma:G := TransitiveGroup(45, 48);
 

Group invariants

Abstract group:  $C_3:S_3\times F_5$
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 $t$:  $48$
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:  $(1,22)(2,24)(3,23)(4,20)(5,19)(6,21)(7,18)(8,17)(9,16)(10,14)(11,13)(12,15)(25,45)(26,44)(27,43)(28,40)(29,42)(30,41)(31,37)(32,39)(33,38)(34,36)$, $(1,9,41,19,16,22,10,35,31,37,27,5)(2,7,40,21,17,23,12,34,32,38,26,4)(3,8,42,20,18,24,11,36,33,39,25,6)(13,29,45)(14,30,43)(15,28,44)$, $(1,15,34,37,33,45,19,24,17,30,6,7)(2,14,36,38,31,44,21,22,18,29,5,8)(3,13,35,39,32,43,20,23,16,28,4,9)(10,42,26)(11,40,27)(12,41,25)$
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, $C_2^2$
$6$:  $S_3$ x 4
$8$:  $C_4\times C_2$
$12$:  $D_{6}$ x 4
$18$:  $C_3^2:C_2$
$20$:  $F_5$
$24$:  $S_3 \times C_4$ x 4
$36$:  18T12
$40$:  $F_{5}\times C_2$
$72$:  36T41
$120$:  $F_5 \times S_3$ x 4

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$ x 4

Degree 5: $F_5$

Degree 9: $C_3^2:C_2$

Degree 15: $F_5 \times S_3$ x 4

Low degree siblings

There are no siblings 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 $2^{18},1^{9}$ $5$ $2$ $18$ $( 1,37)( 2,38)( 3,39)( 7,17)( 8,18)( 9,16)(10,30)(11,28)(12,29)(13,40)(14,41)(15,42)(22,31)(23,32)(24,33)(25,44)(26,45)(27,43)$
2B $2^{20},1^{5}$ $9$ $2$ $20$ $( 1,33)( 2,32)( 3,31)( 4,21)( 5,20)( 6,19)( 8, 9)(10,42)(11,41)(12,40)(13,29)(14,28)(15,30)(16,18)(22,39)(23,38)(24,37)(25,27)(35,36)(43,44)$
2C $2^{22},1$ $45$ $2$ $22$ $( 1, 7)( 2, 9)( 3, 8)( 4, 5)(10,45)(11,44)(12,43)(13,41)(14,40)(15,42)(16,38)(17,37)(18,39)(19,34)(20,36)(21,35)(22,32)(23,31)(24,33)(25,28)(26,30)(27,29)$
3A $3^{15}$ $2$ $3$ $30$ $( 1,31,16)( 2,32,17)( 3,33,18)( 4,34,21)( 5,35,19)( 6,36,20)( 7,38,23)( 8,39,24)( 9,37,22)(10,41,27)(11,42,25)(12,40,26)(13,45,29)(14,43,30)(15,44,28)$
3B $3^{15}$ $2$ $3$ $30$ $( 1, 2, 3)( 4, 6, 5)( 7, 8, 9)(10,12,11)(13,15,14)(16,17,18)(19,21,20)(22,23,24)(25,27,26)(28,30,29)(31,32,33)(34,36,35)(37,38,39)(40,42,41)(43,45,44)$
3C $3^{15}$ $2$ $3$ $30$ $( 1,18,32)( 2,16,33)( 3,17,31)( 4,19,36)( 5,20,34)( 6,21,35)( 7,22,39)( 8,23,37)( 9,24,38)(10,25,40)(11,26,41)(12,27,42)(13,30,44)(14,28,45)(15,29,43)$
3D $3^{15}$ $2$ $3$ $30$ $( 1,17,33)( 2,18,31)( 3,16,32)( 4,20,35)( 5,21,36)( 6,19,34)( 7,24,37)( 8,22,38)( 9,23,39)(10,26,42)(11,27,40)(12,25,41)(13,28,43)(14,29,44)(15,30,45)$
4A1 $4^{9},1^{9}$ $5$ $4$ $27$ $( 1,10,37,30)( 2,12,38,29)( 3,11,39,28)( 7,45,17,26)( 8,44,18,25)( 9,43,16,27)(13,32,40,23)(14,31,41,22)(15,33,42,24)$
4A-1 $4^{9},1^{9}$ $5$ $4$ $27$ $( 1,30,37,10)( 2,29,38,12)( 3,28,39,11)( 7,26,17,45)( 8,25,18,44)( 9,27,16,43)(13,23,40,32)(14,22,41,31)(15,24,42,33)$
4B1 $4^{9},2^{4},1$ $45$ $4$ $31$ $( 1,29,37,12)( 2,30,38,10)( 3,28,39,11)( 4,35)( 5,34)( 6,36)( 7,41,17,14)( 8,42,18,15)( 9,40,16,13)(19,21)(22,26,31,45)(23,27,32,43)(24,25,33,44)$
4B-1 $4^{9},2^{4},1$ $45$ $4$ $31$ $( 1,12,37,29)( 2,10,38,30)( 3,11,39,28)( 4,35)( 5,34)( 6,36)( 7,14,17,41)( 8,15,18,42)( 9,13,16,40)(19,21)(22,45,31,26)(23,43,32,27)(24,44,33,25)$
5A $5^{9}$ $4$ $5$ $36$ $( 1,30,10,37,19)( 2,29,12,38,21)( 3,28,11,39,20)( 4,32,13,40,23)( 5,31,14,41,22)( 6,33,15,42,24)( 7,34,17,45,26)( 8,36,18,44,25)( 9,35,16,43,27)$
6A $6^{6},3^{3}$ $10$ $6$ $36$ $( 1, 9,31,37,16,22)( 2, 7,32,38,17,23)( 3, 8,33,39,18,24)( 4,21,34)( 5,19,35)( 6,20,36)(10,43,41,30,27,14)(11,44,42,28,25,15)(12,45,40,29,26,13)$
6B $6^{6},3^{3}$ $10$ $6$ $36$ $( 1,20, 2,19, 3,21)( 4,31, 6,32, 5,33)( 7,43, 8,45, 9,44)(10,11,12)(13,22,15,23,14,24)(16,36,17,35,18,34)(25,26,27)(28,38,30,39,29,37)(40,41,42)$
6C $6^{6},3^{3}$ $10$ $6$ $36$ $( 1,32,18)( 2,33,16)( 3,31,17)( 4,44,19,13,36,30)( 5,45,20,14,34,28)( 6,43,21,15,35,29)( 7,11,22,26,39,41)( 8,10,23,25,37,40)( 9,12,24,27,38,42)$
6D $6^{6},3^{3}$ $10$ $6$ $36$ $( 1,15,17,30,33,45)( 2,14,18,29,31,44)( 3,13,16,28,32,43)( 4,27,20,40,35,11)( 5,25,21,41,36,12)( 6,26,19,42,34,10)( 7,37,24)( 8,38,22)( 9,39,23)$
10A $10^{4},5$ $36$ $10$ $40$ $( 1, 6,37,42,30,33,19,24,10,15)( 2, 4,38,40,29,32,21,23,12,13)( 3, 5,39,41,28,31,20,22,11,14)( 7,26,45,17,34)( 8,27,44,16,36, 9,25,43,18,35)$
12A1 $12^{3},3^{3}$ $10$ $12$ $39$ $( 1,14, 9,10,31,43,37,41,16,30,22,27)( 2,13, 7,12,32,45,38,40,17,29,23,26)( 3,15, 8,11,33,44,39,42,18,28,24,25)( 4,34,21)( 5,35,19)( 6,36,20)$
12A-1 $12^{3},3^{3}$ $10$ $12$ $39$ $( 1,41, 9,30,31,27,37,14,16,10,22,43)( 2,40, 7,29,32,26,38,13,17,12,23,45)( 3,42, 8,28,33,25,39,15,18,11,24,44)( 4,34,21)( 5,35,19)( 6,36,20)$
12B1 $12^{3},3^{3}$ $10$ $12$ $39$ $( 1,38,20,30, 2,39,19,29, 3,37,21,28)( 4,15,31,23, 6,14,32,24, 5,13,33,22)( 7,36,43,17, 8,35,45,18, 9,34,44,16)(10,12,11)(25,27,26)(40,42,41)$
12B-1 $12^{3},3^{3}$ $10$ $12$ $39$ $( 1,21,28,10, 2,20,30,12, 3,19,29,11)( 4,15,41,32, 6,14,40,33, 5,13,42,31)( 7, 8, 9)(16,34,44,27,17,36,43,26,18,35,45,25)(22,23,24)(37,38,39)$
12C1 $12^{3},3^{3}$ $10$ $12$ $39$ $( 1,18,32)( 2,16,33)( 3,17,31)( 4,37,44,40,19, 8,13,10,36,23,30,25)( 5,39,45,41,20, 7,14,11,34,22,28,26)( 6,38,43,42,21, 9,15,12,35,24,29,27)$
12C-1 $12^{3},3^{3}$ $10$ $12$ $39$ $( 1,44, 4,37,18,13,19, 8,32,30,36,23)( 2,43, 6,38,16,15,21, 9,33,29,35,24)( 3,45, 5,39,17,14,20, 7,31,28,34,22)(10,25,40)(11,26,41)(12,27,42)$
12D1 $12^{3},3^{3}$ $10$ $12$ $39$ $( 1,34,15,10,17, 6,30,26,33,19,45,42)( 2,36,14,12,18, 5,29,25,31,21,44,41)( 3,35,13,11,16, 4,28,27,32,20,43,40)( 7,24,37)( 8,22,38)( 9,23,39)$
12D-1 $12^{3},3^{3}$ $10$ $12$ $39$ $( 1, 7, 6,30,17,24,19,45,33,37,34,15)( 2, 8, 5,29,18,22,21,44,31,38,36,14)( 3, 9, 4,28,16,23,20,43,32,39,35,13)(10,26,42)(11,27,40)(12,25,41)$
15A $15^{3}$ $8$ $15$ $42$ $( 1,26, 6,30, 7,33,10,34,15,37,17,42,19,45,24)( 2,25, 5,29, 8,31,12,36,14,38,18,41,21,44,22)( 3,27, 4,28, 9,32,11,35,13,39,16,40,20,43,23)$
15B $15^{3}$ $8$ $15$ $42$ $( 1,14,27,37, 5,16,30,41, 9,19,31,43,10,22,35)( 2,13,26,38, 4,17,29,40, 7,21,32,45,12,23,34)( 3,15,25,39, 6,18,28,42, 8,20,33,44,11,24,36)$
15C $15^{3}$ $8$ $15$ $42$ $( 1,12,20,30,38, 3,10,21,28,37, 2,11,19,29,39)( 4,15,22,32,42, 5,13,24,31,40, 6,14,23,33,41)( 7,18,27,34,44, 9,17,25,35,45, 8,16,26,36,43)$
15D $15^{3}$ $8$ $15$ $42$ $( 1,36,23,10,44,32,19, 8,40,30,18, 4,37,25,13)( 2,35,24,12,43,33,21, 9,42,29,16, 6,38,27,15)( 3,34,22,11,45,31,20, 7,41,28,17, 5,39,26,14)$

Malle's constant $a(G)$:     $1/18$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 3A 3B 3C 3D 4A1 4A-1 4B1 4B-1 5A 6A 6B 6C 6D 10A 12A1 12A-1 12B1 12B-1 12C1 12C-1 12D1 12D-1 15A 15B 15C 15D
Size 1 5 9 45 2 2 2 2 5 5 45 45 4 10 10 10 10 36 10 10 10 10 10 10 10 10 8 8 8 8
2 P 1A 1A 1A 1A 3A 3B 3C 3D 2A 2A 2A 2A 5A 3A 3B 3C 3D 5A 6A 6A 6B 6B 6C 6C 6D 6D 15A 15B 15C 15D
3 P 1A 2A 2B 2C 1A 1A 1A 1A 4A-1 4A1 4B-1 4B1 5A 2A 2A 2A 2A 10A 4A1 4A-1 4A1 4A-1 4A1 4A-1 4A-1 4A1 5A 5A 5A 5A
5 P 1A 2A 2B 2C 3A 3B 3C 3D 4A1 4A-1 4B1 4B-1 1A 6A 6B 6C 6D 2B 12A1 12A-1 12B1 12B-1 12C1 12C-1 12D1 12D-1 3D 3A 3B 3C
Type
360.127.1a 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 1 1 1 1 1 1
360.127.1b 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 −1 −1 1 1 1 1
360.127.1c 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 1 1 1 1 1 1
360.127.1d 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 −1 −1 1 1 1 1
360.127.1e1 C 1 −1 −1 1 1 1 1 1 −i i −i i 1 −1 −1 −1 −1 −1 −i i −i −i i i i −i 1 1 1 1
360.127.1e2 C 1 −1 −1 1 1 1 1 1 i −i i −i 1 −1 −1 −1 −1 −1 i −i i i −i −i −i i 1 1 1 1
360.127.1f1 C 1 −1 1 −1 1 1 1 1 −i i i −i 1 −1 −1 −1 −1 1 −i i −i −i i i i −i 1 1 1 1
360.127.1f2 C 1 −1 1 −1 1 1 1 1 i −i −i i 1 −1 −1 −1 −1 1 i −i i i −i −i −i i 1 1 1 1
360.127.2a R 2 2 0 0 −1 −1 −1 2 2 2 0 0 2 −1 −1 2 −1 0 −1 −1 −1 2 −1 2 −1 −1 −1 2 −1 −1
360.127.2b R 2 2 0 0 −1 −1 2 −1 2 2 0 0 2 2 −1 −1 −1 0 −1 −1 2 −1 2 −1 −1 −1 −1 −1 2 −1
360.127.2c R 2 2 0 0 −1 2 −1 −1 2 2 0 0 2 −1 2 −1 −1 0 2 −1 −1 −1 −1 −1 2 −1 −1 −1 −1 2
360.127.2d R 2 2 0 0 2 −1 −1 −1 2 2 0 0 2 −1 −1 −1 2 0 −1 2 −1 −1 −1 −1 −1 2 2 −1 −1 −1
360.127.2e R 2 2 0 0 −1 −1 −1 2 −2 −2 0 0 2 −1 −1 2 −1 0 1 1 1 −2 1 −2 1 1 −1 2 −1 −1
360.127.2f R 2 2 0 0 −1 −1 2 −1 −2 −2 0 0 2 2 −1 −1 −1 0 1 1 −2 1 −2 1 1 1 −1 −1 2 −1
360.127.2g R 2 2 0 0 −1 2 −1 −1 −2 −2 0 0 2 −1 2 −1 −1 0 −2 1 1 1 1 1 −2 1 −1 −1 −1 2
360.127.2h R 2 2 0 0 2 −1 −1 −1 −2 −2 0 0 2 −1 −1 −1 2 0 1 −2 1 1 1 1 1 −2 2 −1 −1 −1
360.127.2i1 C 2 −2 0 0 −1 −1 −1 2 −2i 2i 0 0 2 1 1 −2 1 0 i −i i −2i −i 2i −i i −1 2 −1 −1
360.127.2i2 C 2 −2 0 0 −1 −1 −1 2 2i −2i 0 0 2 1 1 −2 1 0 −i i −i 2i i −2i i −i −1 2 −1 −1
360.127.2j1 C 2 −2 0 0 −1 −1 2 −1 −2i 2i 0 0 2 −2 1 1 1 0 i −i −2i i 2i −i −i i −1 −1 2 −1
360.127.2j2 C 2 −2 0 0 −1 −1 2 −1 2i −2i 0 0 2 −2 1 1 1 0 −i i 2i −i −2i i i −i −1 −1 2 −1
360.127.2k1 C 2 −2 0 0 −1 2 −1 −1 −2i 2i 0 0 2 1 −2 1 1 0 −2i −i i i −i −i 2i i −1 −1 −1 2
360.127.2k2 C 2 −2 0 0 −1 2 −1 −1 2i −2i 0 0 2 1 −2 1 1 0 2i i −i −i i i −2i −i −1 −1 −1 2
360.127.2l1 C 2 −2 0 0 2 −1 −1 −1 −2i 2i 0 0 2 1 1 1 −2 0 i 2i i i −i −i −i −2i 2 −1 −1 −1
360.127.2l2 C 2 −2 0 0 2 −1 −1 −1 2i −2i 0 0 2 1 1 1 −2 0 −i −2i −i −i i i i 2i 2 −1 −1 −1
360.127.4a R 4 0 4 0 4 4 4 4 0 0 0 0 −1 0 0 0 0 −1 0 0 0 0 0 0 0 0 −1 −1 −1 −1
360.127.4b R 4 0 −4 0 4 4 4 4 0 0 0 0 −1 0 0 0 0 1 0 0 0 0 0 0 0 0 −1 −1 −1 −1
360.127.8a R 8 0 0 0 −4 −4 −4 8 0 0 0 0 −2 0 0 0 0 0 0 0 0 0 0 0 0 0 1 −2 1 1
360.127.8b R 8 0 0 0 −4 −4 8 −4 0 0 0 0 −2 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 −2 1
360.127.8c R 8 0 0 0 −4 8 −4 −4 0 0 0 0 −2 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 −2
360.127.8d R 8 0 0 0 8 −4 −4 −4 0 0 0 0 −2 0 0 0 0 0 0 0 0 0 0 0 0 0 −2 1 1 1

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed