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Properties

Label 45T41
Degree $45$
Order $360$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_5.S_3^2$

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Show commands: Magma

magma: G := TransitiveGroup(45, 41);
 

Group action invariants

Degree $n$:  $45$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $41$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_5.S_3^2$
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:  (1,36,23,11,43,33,21,8,42,28,18,6,37,26,13,3,35,24,12,44,32,19,9,41,30,16,5,38,25,14)(2,34,22,10,45,31,20,7,40,29,17,4,39,27,15), (1,11,39,30,3,10,37,28,2,12,38,29)(4,35,6,34,5,36)(7,13,16,40,9,14,17,42,8,15,18,41)(19,20,21)(22,43,33,27,23,44,31,25,24,45,32,26)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$6$:  $S_3$ x 2
$8$:  $C_4\times C_2$
$12$:  $D_{6}$ x 2, $C_3 : C_4$ x 2
$20$:  $F_5$
$24$:  $S_3 \times C_4$, 24T6
$36$:  $S_3^2$
$40$:  $F_{5}\times C_2$
$60$:  $C_{15} : C_4$
$72$:  24T60
$120$:  $F_5 \times S_3$, 30T17

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$ x 2

Degree 5: $F_5$

Degree 9: $S_3^2$

Degree 15: $C_{15} : C_4$, $F_5 \times S_3$

Low degree siblings

30T83

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

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

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:  360.128
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 3B 3C 4A1 4A-1 4B1 4B-1 5A 6A 6B 6C 6D 6E 10A 12A1 12A-1 15A1 15A-1 15B 15C1 15C-1 30A1 30A-1
Size 1 3 5 15 2 2 4 15 15 45 45 4 6 10 10 20 30 12 30 30 4 4 8 8 8 12 12
2 P 1A 1A 1A 1A 3A 3B 3C 2B 2B 2B 2B 5A 3B 3A 3B 3C 3B 5A 6B 6B 15A1 15A-1 15B 15C1 15C-1 15A1 15A-1
3 P 1A 2A 2B 2C 1A 1A 1A 4A-1 4A1 4B-1 4B1 5A 2A 2B 2B 2B 2C 10A 4A1 4A-1 5A 5A 5A 5A 5A 10A 10A
5 P 1A 2A 2B 2C 3A 3B 3C 4A1 4A-1 4B1 4B-1 1A 6A 6B 6C 6D 6E 2A 12A1 12A-1 3B 3B 3A 3C 3C 6A 6A
Type
360.128.1a 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
360.128.1b 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
360.128.1c 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
360.128.1d 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
360.128.1e1 C 1 −1 −1 1 1 1 1 −i i i −i 1 −1 −1 −1 −1 1 −1 i −i 1 1 1 1 1 −1 −1
360.128.1e2 C 1 −1 −1 1 1 1 1 i −i −i i 1 −1 −1 −1 −1 1 −1 −i i 1 1 1 1 1 −1 −1
360.128.1f1 C 1 1 −1 −1 1 1 1 −i i −i i 1 1 −1 −1 −1 −1 1 i −i 1 1 1 1 1 1 1
360.128.1f2 C 1 1 −1 −1 1 1 1 i −i i −i 1 1 −1 −1 −1 −1 1 −i i 1 1 1 1 1 1 1
360.128.2a R 2 0 2 0 −1 2 −1 2 2 0 0 2 0 −1 2 −1 0 0 −1 −1 2 2 −1 −1 −1 0 0
360.128.2b R 2 2 2 2 2 −1 −1 0 0 0 0 2 −1 2 −1 −1 −1 2 0 0 −1 −1 2 −1 −1 −1 −1
360.128.2c R 2 −2 2 −2 2 −1 −1 0 0 0 0 2 1 2 −1 −1 1 −2 0 0 −1 −1 2 −1 −1 1 1
360.128.2d R 2 0 2 0 −1 2 −1 −2 −2 0 0 2 0 −1 2 −1 0 0 1 1 2 2 −1 −1 −1 0 0
360.128.2e S 2 −2 −2 2 2 −1 −1 0 0 0 0 2 1 −2 1 1 −1 −2 0 0 −1 −1 2 −1 −1 1 1
360.128.2f S 2 2 −2 −2 2 −1 −1 0 0 0 0 2 −1 −2 1 1 1 2 0 0 −1 −1 2 −1 −1 −1 −1
360.128.2g1 C 2 0 −2 0 −1 2 −1 −2i 2i 0 0 2 0 1 −2 1 0 0 −i i 2 2 −1 −1 −1 0 0
360.128.2g2 C 2 0 −2 0 −1 2 −1 2i −2i 0 0 2 0 1 −2 1 0 0 i −i 2 2 −1 −1 −1 0 0
360.128.4a R 4 4 0 0 4 4 4 0 0 0 0 −1 4 0 0 0 0 −1 0 0 −1 −1 −1 −1 −1 −1 −1
360.128.4b R 4 0 4 0 −2 −2 1 0 0 0 0 4 0 −2 −2 1 0 0 0 0 −2 −2 −2 1 1 0 0
360.128.4c R 4 −4 0 0 4 4 4 0 0 0 0 −1 −4 0 0 0 0 1 0 0 −1 −1 −1 −1 −1 1 1
360.128.4d S 4 0 −4 0 −2 −2 1 0 0 0 0 4 0 2 2 −1 0 0 0 0 −2 −2 −2 1 1 0 0
360.128.4e1 C 4 4 0 0 4 −2 −2 0 0 0 0 −1 −2 0 0 0 0 −1 0 0 2−2ζ15−ζ152+ζ153−2ζ154+ζ155−ζ157 −1+2ζ15+ζ152−ζ153+2ζ154−ζ155+ζ157 −1 2−2ζ15−ζ152+ζ153−2ζ154+ζ155−ζ157 −1+2ζ15+ζ152−ζ153+2ζ154−ζ155+ζ157 2−2ζ15−ζ152+ζ153−2ζ154+ζ155−ζ157 −1+2ζ15+ζ152−ζ153+2ζ154−ζ155+ζ157
360.128.4e2 C 4 4 0 0 4 −2 −2 0 0 0 0 −1 −2 0 0 0 0 −1 0 0 −1+2ζ15+ζ152−ζ153+2ζ154−ζ155+ζ157 2−2ζ15−ζ152+ζ153−2ζ154+ζ155−ζ157 −1 −1+2ζ15+ζ152−ζ153+2ζ154−ζ155+ζ157 2−2ζ15−ζ152+ζ153−2ζ154+ζ155−ζ157 −1+2ζ15+ζ152−ζ153+2ζ154−ζ155+ζ157 2−2ζ15−ζ152+ζ153−2ζ154+ζ155−ζ157
360.128.4f1 C 4 −4 0 0 4 −2 −2 0 0 0 0 −1 2 0 0 0 0 1 0 0 2−2ζ15−ζ152+ζ153−2ζ154+ζ155−ζ157 −1+2ζ15+ζ152−ζ153+2ζ154−ζ155+ζ157 −1 2−2ζ15−ζ152+ζ153−2ζ154+ζ155−ζ157 −1+2ζ15+ζ152−ζ153+2ζ154−ζ155+ζ157 −2+2ζ15+ζ152−ζ153+2ζ154−ζ155+ζ157 1−2ζ15−ζ152+ζ153−2ζ154+ζ155−ζ157
360.128.4f2 C 4 −4 0 0 4 −2 −2 0 0 0 0 −1 2 0 0 0 0 1 0 0 −1+2ζ15+ζ152−ζ153+2ζ154−ζ155+ζ157 2−2ζ15−ζ152+ζ153−2ζ154+ζ155−ζ157 −1 −1+2ζ15+ζ152−ζ153+2ζ154−ζ155+ζ157 2−2ζ15−ζ152+ζ153−2ζ154+ζ155−ζ157 1−2ζ15−ζ152+ζ153−2ζ154+ζ155−ζ157 −2+2ζ15+ζ152−ζ153+2ζ154−ζ155+ζ157
360.128.8a R 8 0 0 0 −4 8 −4 0 0 0 0 −2 0 0 0 0 0 0 0 0 −2 −2 1 1 1 0 0
360.128.8b1 C 8 0 0 0 −4 −4 2 0 0 0 0 −2 0 0 0 0 0 0 0 0 4−4ζ15−2ζ152+2ζ153−4ζ154+2ζ155−2ζ157 −2+4ζ15+2ζ152−2ζ153+4ζ154−2ζ155+2ζ157 1 −2+2ζ15+ζ152−ζ153+2ζ154−ζ155+ζ157 1−2ζ15−ζ152+ζ153−2ζ154+ζ155−ζ157 0 0
360.128.8b2 C 8 0 0 0 −4 −4 2 0 0 0 0 −2 0 0 0 0 0 0 0 0 −2+4ζ15+2ζ152−2ζ153+4ζ154−2ζ155+2ζ157 4−4ζ15−2ζ152+2ζ153−4ζ154+2ζ155−2ζ157 1 1−2ζ15−ζ152+ζ153−2ζ154+ζ155−ζ157 −2+2ζ15+ζ152−ζ153+2ζ154−ζ155+ζ157 0 0

magma: CharacterTable(G);