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Properties

Label 43T7
Degree $43$
Order $903$
Cyclic no
Abelian no
Solvable yes
Primitive yes
$p$-group no
Group: $C_{43}:C_{21}$

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Show commands: Magma

magma: G := TransitiveGroup(43, 7);
 

Group action invariants

Degree $n$:  $43$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $7$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_{43}:C_{21}$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  yes
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,9,38,41,25,10,4,36,23,35,14,40,16,15,6,11,13,31,21,17,24)(2,18,33,39,7,20,8,29,3,27,28,37,32,30,12,22,26,19,42,34,5), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$3$:  $C_3$
$7$:  $C_7$
$21$:  $C_{21}$

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $903=3 \cdot 7 \cdot 43$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  903.1
magma: IdentifyGroup(G);
 
Character table:

1A 3A1 3A-1 7A1 7A-1 7A2 7A-2 7A3 7A-3 21A1 21A-1 21A2 21A-2 21A4 21A-4 21A5 21A-5 21A8 21A-8 21A10 21A-10 43A1 43A-1
Size 1 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 43 21 21
3 P 1A 3A-1 3A1 7A2 7A-1 7A3 7A-2 7A1 7A-3 21A8 21A-5 21A10 21A1 21A-8 21A-10 21A-4 21A-2 21A5 21A4 21A-1 21A2 43A-1 43A1
7 P 1A 1A 1A 7A3 7A2 7A1 7A-3 7A-2 7A-1 7A-3 7A1 7A-2 7A-3 7A3 7A2 7A-2 7A-1 7A-1 7A2 7A3 7A1 43A-1 43A1
43 P 1A 3A1 3A-1 1A 1A 1A 1A 1A 1A 3A1 3A-1 3A-1 3A-1 3A-1 3A1 3A1 3A-1 3A1 3A-1 3A1 3A1 43A-1 43A1
Type
903.1.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
903.1.1b1 C 1 ζ3−1 ζ3 1 1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 1 1
903.1.1b2 C 1 ζ3 ζ3−1 1 1 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 1 1
903.1.1c1 C 1 1 1 ζ7−3 ζ73 ζ7 ζ7−1 ζ7−2 ζ72 ζ7−1 ζ7 ζ7−2 ζ72 ζ73 ζ7−3 ζ72 ζ7−2 ζ7−1 ζ7 ζ7−3 ζ73 1 1
903.1.1c2 C 1 1 1 ζ73 ζ7−3 ζ7−1 ζ7 ζ72 ζ7−2 ζ7 ζ7−1 ζ72 ζ7−2 ζ7−3 ζ73 ζ7−2 ζ72 ζ7 ζ7−1 ζ73 ζ7−3 1 1
903.1.1c3 C 1 1 1 ζ7−2 ζ72 ζ73 ζ7−3 ζ7 ζ7−1 ζ7−3 ζ73 ζ7 ζ7−1 ζ72 ζ7−2 ζ7−1 ζ7 ζ7−3 ζ73 ζ7−2 ζ72 1 1
903.1.1c4 C 1 1 1 ζ72 ζ7−2 ζ7−3 ζ73 ζ7−1 ζ7 ζ73 ζ7−3 ζ7−1 ζ7 ζ7−2 ζ72 ζ7 ζ7−1 ζ73 ζ7−3 ζ72 ζ7−2 1 1
903.1.1c5 C 1 1 1 ζ7−1 ζ7 ζ7−2 ζ72 ζ7−3 ζ73 ζ72 ζ7−2 ζ7−3 ζ73 ζ7 ζ7−1 ζ73 ζ7−3 ζ72 ζ7−2 ζ7−1 ζ7 1 1
903.1.1c6 C 1 1 1 ζ7 ζ7−1 ζ72 ζ7−2 ζ73 ζ7−3 ζ7−2 ζ72 ζ73 ζ7−3 ζ7−1 ζ7 ζ7−3 ζ73 ζ7−2 ζ72 ζ7 ζ7−1 1 1
903.1.1d1 C 1 ζ21−7 ζ217 ζ21−9 ζ219 ζ213 ζ21−3 ζ21−6 ζ216 ζ21−10 ζ2110 ζ21 ζ21−1 ζ212 ζ21−2 ζ21−8 ζ218 ζ214 ζ21−4 ζ215 ζ21−5 1 1
903.1.1d2 C 1 ζ217 ζ21−7 ζ219 ζ21−9 ζ21−3 ζ213 ζ216 ζ21−6 ζ2110 ζ21−10 ζ21−1 ζ21 ζ21−2 ζ212 ζ218 ζ21−8 ζ21−4 ζ214 ζ21−5 ζ215 1 1
903.1.1d3 C 1 ζ21−7 ζ217 ζ219 ζ21−9 ζ21−3 ζ213 ζ216 ζ21−6 ζ21−4 ζ214 ζ21−8 ζ218 ζ215 ζ21−5 ζ21 ζ21−1 ζ2110 ζ21−10 ζ212 ζ21−2 1 1
903.1.1d4 C 1 ζ217 ζ21−7 ζ21−9 ζ219 ζ213 ζ21−3 ζ21−6 ζ216 ζ214 ζ21−4 ζ218 ζ21−8 ζ21−5 ζ215 ζ21−1 ζ21 ζ21−10 ζ2110 ζ21−2 ζ212 1 1
903.1.1d5 C 1 ζ21−7 ζ217 ζ21−6 ζ216 ζ219 ζ21−9 ζ213 ζ21−3 ζ215 ζ21−5 ζ2110 ζ21−10 ζ21−1 ζ21 ζ214 ζ21−4 ζ21−2 ζ212 ζ218 ζ21−8 1 1
903.1.1d6 C 1 ζ217 ζ21−7 ζ216 ζ21−6 ζ21−9 ζ219 ζ21−3 ζ213 ζ21−5 ζ215 ζ21−10 ζ2110 ζ21 ζ21−1 ζ21−4 ζ214 ζ212 ζ21−2 ζ21−8 ζ218 1 1
903.1.1d7 C 1 ζ21−7 ζ217 ζ216 ζ21−6 ζ21−9 ζ219 ζ21−3 ζ213 ζ212 ζ21−2 ζ214 ζ21−4 ζ218 ζ21−8 ζ2110 ζ21−10 ζ21−5 ζ215 ζ21−1 ζ21 1 1
903.1.1d8 C 1 ζ217 ζ21−7 ζ21−6 ζ216 ζ219 ζ21−9 ζ213 ζ21−3 ζ21−2 ζ212 ζ21−4 ζ214 ζ21−8 ζ218 ζ21−10 ζ2110 ζ215 ζ21−5 ζ21 ζ21−1 1 1
903.1.1d9 C 1 ζ21−7 ζ217 ζ21−3 ζ213 ζ21−6 ζ216 ζ21−9 ζ219 ζ21−1 ζ21 ζ21−2 ζ212 ζ21−4 ζ214 ζ21−5 ζ215 ζ21−8 ζ218 ζ21−10 ζ2110 1 1
903.1.1d10 C 1 ζ217 ζ21−7 ζ213 ζ21−3 ζ216 ζ21−6 ζ219 ζ21−9 ζ21 ζ21−1 ζ212 ζ21−2 ζ214 ζ21−4 ζ215 ζ21−5 ζ218 ζ21−8 ζ2110 ζ21−10 1 1
903.1.1d11 C 1 ζ21−7 ζ217 ζ213 ζ21−3 ζ216 ζ21−6 ζ219 ζ21−9 ζ218 ζ21−8 ζ21−5 ζ215 ζ21−10 ζ2110 ζ21−2 ζ212 ζ21 ζ21−1 ζ21−4 ζ214 1 1
903.1.1d12 C 1 ζ217 ζ21−7 ζ21−3 ζ213 ζ21−6 ζ216 ζ21−9 ζ219 ζ21−8 ζ218 ζ215 ζ21−5 ζ2110 ζ21−10 ζ212 ζ21−2 ζ21−1 ζ21 ζ214 ζ21−4 1 1
903.1.21a1 C 21 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −ζ43−20−ζ43−19−ζ43−18−ζ43−12−ζ43−8−ζ43−7−ζ43−5−ζ43−3−ζ43−2−1−ζ43−ζ434−ζ436−ζ439−ζ4310−ζ4311−ζ4313−ζ4314−ζ4315−ζ4316−ζ4317−ζ4321 ζ43−20+ζ43−19+ζ43−18+ζ43−12+ζ43−8+ζ43−7+ζ43−5+ζ43−3+ζ43−2+ζ43+ζ434+ζ436+ζ439+ζ4310+ζ4311+ζ4313+ζ4314+ζ4315+ζ4316+ζ4317+ζ4321
903.1.21a2 C 21 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ζ43−20+ζ43−19+ζ43−18+ζ43−12+ζ43−8+ζ43−7+ζ43−5+ζ43−3+ζ43−2+ζ43+ζ434+ζ436+ζ439+ζ4310+ζ4311+ζ4313+ζ4314+ζ4315+ζ4316+ζ4317+ζ4321 −ζ43−20−ζ43−19−ζ43−18−ζ43−12−ζ43−8−ζ43−7−ζ43−5−ζ43−3−ζ43−2−1−ζ43−ζ434−ζ436−ζ439−ζ4310−ζ4311−ζ4313−ζ4314−ζ4315−ζ4316−ζ4317−ζ4321

magma: CharacterTable(G);