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Properties

Label 45T46
Degree $45$
Order $360$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_9\times F_5$

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Show commands: Magma

magma: G := TransitiveGroup(45, 46);
 

Group action invariants

Degree $n$:  $45$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $46$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_9\times F_5$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,36,15,12,16,5,28,26,32,19,44,42,2,35,14,11,17,4,30,25,33,21,43,41,3,34,13,10,18,6,29,27,31,20,45,40)(7,23,37,8,24,38,9,22,39), (1,26,19,45,37,17,10,34,30,7)(2,27,21,43,38,16,12,35,29,9)(3,25,20,44,39,18,11,36,28,8)(4,13,23,32,40)(5,15,22,33,41,6,14,24,31,42)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$6$:  $S_3$
$8$:  $C_4\times C_2$
$12$:  $D_{6}$
$18$:  $D_{9}$
$20$:  $F_5$
$24$:  $S_3 \times C_4$
$36$:  $D_{18}$
$40$:  $F_{5}\times C_2$
$72$:  36T45
$120$:  $F_5 \times S_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 5: $F_5$

Degree 9: $D_{9}$

Degree 15: $F_5 \times S_3$

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $360=2^{3} \cdot 3^{2} \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  360.39
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 4A1 4A-1 4B1 4B-1 5A 6A 9A1 9A2 9A4 10A 12A1 12A-1 15A 18A1 18A5 18A7 36A1 36A-1 36A5 36A-5 36A7 36A-7 45A1 45A2 45A4
Size 1 5 9 45 2 5 5 45 45 4 10 2 2 2 36 10 10 8 10 10 10 10 10 10 10 10 10 8 8 8
2 P 1A 1A 1A 1A 3A 2A 2A 2A 2A 5A 3A 9A2 9A4 9A1 5A 6A 6A 15A 9A1 9A4 9A2 18A1 18A1 18A5 18A7 18A7 18A5 45A1 45A2 45A4
3 P 1A 2A 2B 2C 1A 4A-1 4A1 4B-1 4B1 5A 2A 3A 3A 3A 10A 4A1 4A-1 5A 6A 6A 6A 12A1 12A-1 12A1 12A-1 12A1 12A-1 15A 15A 15A
5 P 1A 2A 2B 2C 3A 4A1 4A-1 4B1 4B-1 1A 6A 9A4 9A1 9A2 2B 12A1 12A-1 3A 18A5 18A7 18A1 36A5 36A-5 36A-7 36A-1 36A1 36A7 9A2 9A4 9A1
Type
360.39.1a 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
360.39.1b 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
360.39.1c 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
360.39.1d 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
360.39.1e1 C 1 −1 −1 1 1 −i i −i i 1 −1 1 1 1 −1 i −i 1 −1 −1 −1 −i i −i i i −i 1 1 1
360.39.1e2 C 1 −1 −1 1 1 i −i i −i 1 −1 1 1 1 −1 −i i 1 −1 −1 −1 i −i i −i −i i 1 1 1
360.39.1f1 C 1 −1 1 −1 1 −i i i −i 1 −1 1 1 1 1 i −i 1 −1 −1 −1 −i i −i i i −i 1 1 1
360.39.1f2 C 1 −1 1 −1 1 i −i −i i 1 −1 1 1 1 1 −i i 1 −1 −1 −1 i −i i −i −i i 1 1 1
360.39.2a R 2 2 0 0 2 2 2 0 0 2 2 −1 −1 −1 0 2 2 2 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1
360.39.2b R 2 2 0 0 2 −2 −2 0 0 2 2 −1 −1 −1 0 −2 −2 2 −1 −1 −1 1 1 1 1 1 1 −1 −1 −1
360.39.2c1 C 2 −2 0 0 2 −2i 2i 0 0 2 −2 −1 −1 −1 0 2i −2i 2 1 1 1 i −i i −i −i i −1 −1 −1
360.39.2c2 C 2 −2 0 0 2 2i −2i 0 0 2 −2 −1 −1 −1 0 −2i 2i 2 1 1 1 −i i −i i i −i −1 −1 −1
360.39.2d1 R 2 2 0 0 −1 2 2 0 0 2 −1 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−2+ζ92 0 −1 −1 −1 ζ9−2+ζ92 ζ9−1+ζ9 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−1+ζ9 ζ9−4+ζ94 ζ9−4+ζ94 ζ9−2+ζ92 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−2+ζ92
360.39.2d2 R 2 2 0 0 −1 2 2 0 0 2 −1 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−1+ζ9 0 −1 −1 −1 ζ9−1+ζ9 ζ9−4+ζ94 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−4+ζ94 ζ9−2+ζ92 ζ9−2+ζ92 ζ9−1+ζ9 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−1+ζ9
360.39.2d3 R 2 2 0 0 −1 2 2 0 0 2 −1 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−4+ζ94 0 −1 −1 −1 ζ9−4+ζ94 ζ9−2+ζ92 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−2+ζ92 ζ9−1+ζ9 ζ9−1+ζ9 ζ9−4+ζ94 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−4+ζ94
360.39.2e1 R 2 2 0 0 −1 −2 −2 0 0 2 −1 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−2+ζ92 0 1 1 −1 ζ9−2+ζ92 ζ9−1+ζ9 ζ9−4+ζ94 −ζ9−1−ζ9 −ζ9−1−ζ9 −ζ9−4−ζ94 −ζ9−4−ζ94 −ζ9−2−ζ92 −ζ9−2−ζ92 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−2+ζ92
360.39.2e2 R 2 2 0 0 −1 −2 −2 0 0 2 −1 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−1+ζ9 0 1 1 −1 ζ9−1+ζ9 ζ9−4+ζ94 ζ9−2+ζ92 −ζ9−4−ζ94 −ζ9−4−ζ94 −ζ9−2−ζ92 −ζ9−2−ζ92 −ζ9−1−ζ9 −ζ9−1−ζ9 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−1+ζ9
360.39.2e3 R 2 2 0 0 −1 −2 −2 0 0 2 −1 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−4+ζ94 0 1 1 −1 ζ9−4+ζ94 ζ9−2+ζ92 ζ9−1+ζ9 −ζ9−2−ζ92 −ζ9−2−ζ92 −ζ9−1−ζ9 −ζ9−1−ζ9 −ζ9−4−ζ94 −ζ9−4−ζ94 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−4+ζ94
360.39.2f1 C 2 −2 0 0 −1 −2ζ369 2ζ369 0 0 2 1 −ζ36−2−ζ362 ζ36−4+ζ364 ζ36−8+ζ368 0 −ζ369 ζ369 −1 −ζ36−8−ζ368 −ζ36−4−ζ364 ζ36−2+ζ362 ζ36−ζ365−ζ367 −ζ36+ζ365+ζ367 ζ367+ζ3611 −ζ367−ζ3611 ζ36−ζ365+ζ3611 −ζ36+ζ365−ζ3611 −ζ36−2−ζ362 ζ36−4+ζ364 ζ36−8+ζ368
360.39.2f2 C 2 −2 0 0 −1 2ζ369 −2ζ369 0 0 2 1 −ζ36−2−ζ362 ζ36−4+ζ364 ζ36−8+ζ368 0 ζ369 −ζ369 −1 −ζ36−8−ζ368 −ζ36−4−ζ364 ζ36−2+ζ362 −ζ36+ζ365+ζ367 ζ36−ζ365−ζ367 −ζ367−ζ3611 ζ367+ζ3611 −ζ36+ζ365−ζ3611 ζ36−ζ365+ζ3611 −ζ36−2−ζ362 ζ36−4+ζ364 ζ36−8+ζ368
360.39.2f3 C 2 −2 0 0 −1 −2ζ369 2ζ369 0 0 2 1 ζ36−8+ζ368 −ζ36−2−ζ362 ζ36−4+ζ364 0 −ζ369 ζ369 −1 −ζ36−4−ζ364 ζ36−2+ζ362 −ζ36−8−ζ368 ζ367+ζ3611 −ζ367−ζ3611 −ζ36+ζ365−ζ3611 ζ36−ζ365+ζ3611 −ζ36+ζ365+ζ367 ζ36−ζ365−ζ367 ζ36−8+ζ368 −ζ36−2−ζ362 ζ36−4+ζ364
360.39.2f4 C 2 −2 0 0 −1 2ζ369 −2ζ369 0 0 2 1 ζ36−8+ζ368 −ζ36−2−ζ362 ζ36−4+ζ364 0 ζ369 −ζ369 −1 −ζ36−4−ζ364 ζ36−2+ζ362 −ζ36−8−ζ368 −ζ367−ζ3611 ζ367+ζ3611 ζ36−ζ365+ζ3611 −ζ36+ζ365−ζ3611 ζ36−ζ365−ζ367 −ζ36+ζ365+ζ367 ζ36−8+ζ368 −ζ36−2−ζ362 ζ36−4+ζ364
360.39.2f5 C 2 −2 0 0 −1 −2ζ369 2ζ369 0 0 2 1 ζ36−4+ζ364 ζ36−8+ζ368 −ζ36−2−ζ362 0 −ζ369 ζ369 −1 ζ36−2+ζ362 −ζ36−8−ζ368 −ζ36−4−ζ364 −ζ36+ζ365−ζ3611 ζ36−ζ365+ζ3611 ζ36−ζ365−ζ367 −ζ36+ζ365+ζ367 −ζ367−ζ3611 ζ367+ζ3611 ζ36−4+ζ364 ζ36−8+ζ368 −ζ36−2−ζ362
360.39.2f6 C 2 −2 0 0 −1 2ζ369 −2ζ369 0 0 2 1 ζ36−4+ζ364 ζ36−8+ζ368 −ζ36−2−ζ362 0 ζ369 −ζ369 −1 ζ36−2+ζ362 −ζ36−8−ζ368 −ζ36−4−ζ364 ζ36−ζ365+ζ3611 −ζ36+ζ365−ζ3611 −ζ36+ζ365+ζ367 ζ36−ζ365−ζ367 ζ367+ζ3611 −ζ367−ζ3611 ζ36−4+ζ364 ζ36−8+ζ368 −ζ36−2−ζ362
360.39.4a R 4 0 4 0 4 0 0 0 0 −1 0 4 4 4 −1 0 0 −1 0 0 0 0 0 0 0 0 0 −1 −1 −1
360.39.4b R 4 0 −4 0 4 0 0 0 0 −1 0 4 4 4 1 0 0 −1 0 0 0 0 0 0 0 0 0 −1 −1 −1
360.39.8a R 8 0 0 0 8 0 0 0 0 −2 0 −4 −4 −4 0 0 0 −2 0 0 0 0 0 0 0 0 0 1 1 1
360.39.8b1 R 8 0 0 0 −4 0 0 0 0 −2 0 4ζ9−4+4ζ94 4ζ9−1+4ζ9 4ζ9−2+4ζ92 0 0 0 1 0 0 0 0 0 0 0 0 0 −ζ9−4−ζ94 −ζ9−1−ζ9 −ζ9−2−ζ92
360.39.8b2 R 8 0 0 0 −4 0 0 0 0 −2 0 4ζ9−2+4ζ92 4ζ9−4+4ζ94 4ζ9−1+4ζ9 0 0 0 1 0 0 0 0 0 0 0 0 0 −ζ9−2−ζ92 −ζ9−4−ζ94 −ζ9−1−ζ9
360.39.8b3 R 8 0 0 0 −4 0 0 0 0 −2 0 4ζ9−1+4ζ9 4ζ9−2+4ζ92 4ζ9−4+4ζ94 0 0 0 1 0 0 0 0 0 0 0 0 0 −ζ9−1−ζ9 −ζ9−2−ζ92 −ζ9−4−ζ94

magma: CharacterTable(G);