Properties

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

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

 Label Cycle Type Size Order Index Representative 1A $1^{43}$ $1$ $1$ $0$ $()$ 3A1 $3^{14},1$ $43$ $3$ $28$ $( 2,37, 7)( 3,30,13)( 4,23,19)( 5,16,25)( 6, 9,31)( 8,38,43)(10,24,12)(11,17,18)(14,39,36)(15,32,42)(20,40,29)(21,33,35)(22,26,41)(27,34,28)$ 3A-1 $3^{14},1$ $43$ $3$ $28$ $( 2, 7,37)( 3,13,30)( 4,19,23)( 5,25,16)( 6,31, 9)( 8,43,38)(10,12,24)(11,18,17)(14,36,39)(15,42,32)(20,29,40)(21,35,33)(22,41,26)(27,28,34)$ 7A1 $7^{6},1$ $43$ $7$ $36$ $( 2,36,22, 5,12,42,17)( 3,28,43, 9,23,40,33)( 4,20,21,13,34,38, 6)( 7,39,41,25,24,32,11)( 8,31,19,29,35,30,27)(10,15,18,37,14,26,16)$ 7A-1 $7^{6},1$ $43$ $7$ $36$ $( 2, 5,17,22,42,36,12)( 3, 9,33,43,40,28,23)( 4,13, 6,21,38,20,34)( 7,25,11,41,32,39,24)( 8,29,27,19,30,31,35)(10,37,16,18,26,15,14)$ 7A2 $7^{6},1$ $43$ $7$ $36$ $( 2,42, 5,36,17,12,22)( 3,40, 9,28,33,23,43)( 4,38,13,20, 6,34,21)( 7,32,25,39,11,24,41)( 8,30,29,31,27,35,19)(10,26,37,15,16,14,18)$ 7A-2 $7^{6},1$ $43$ $7$ $36$ $( 2,17,42,12, 5,22,36)( 3,33,40,23, 9,43,28)( 4, 6,38,34,13,21,20)( 7,11,32,24,25,41,39)( 8,27,30,35,29,19,31)(10,16,26,14,37,18,15)$ 7A3 $7^{6},1$ $43$ $7$ $36$ $( 2,12,36,42,22,17, 5)( 3,23,28,40,43,33, 9)( 4,34,20,38,21, 6,13)( 7,24,39,32,41,11,25)( 8,35,31,30,19,27,29)(10,14,15,26,18,16,37)$ 7A-3 $7^{6},1$ $43$ $7$ $36$ $( 2,22,12,17,36, 5,42)( 3,43,23,33,28, 9,40)( 4,21,34, 6,20,13,38)( 7,41,24,11,39,25,32)( 8,19,35,27,31,29,30)(10,18,14,16,15,37,26)$ 21A1 $21^{2},1$ $43$ $21$ $40$ $( 2,18,32,12,16,41,36,37,11,42,10,25,22,14, 7,17,15,24, 5,26,39)( 3,35,20,23,31,38,28,30,21,40,19, 6,43,27,13,33,29, 4, 9, 8,34)$ 21A-1 $21^{2},1$ $43$ $21$ $40$ $( 2,32,16,36,11,10,22, 7,15, 5,39,18,12,41,37,42,25,14,17,24,26)( 3,20,31,28,21,19,43,13,29, 9,34,35,23,38,30,40, 6,27,33, 4, 8)$ 21A2 $21^{2},1$ $43$ $21$ $40$ $( 2,24,14,42,41,18, 5, 7,10,36,32,26,17,25,37,12,39,15,22,11,16)( 3, 4,27,40,38,35, 9,13,19,28,20, 8,33, 6,30,23,34,29,43,21,31)$ 21A-2 $21^{2},1$ $43$ $21$ $40$ $( 2,11,15,12,25,26,36, 7,18,42,24,16,22,39,37,17,32,10, 5,41,14)( 3,21,29,23, 6, 8,28,13,35,40, 4,31,43,34,30,33,20,19, 9,38,27)$ 21A4 $21^{2},1$ $43$ $21$ $40$ $( 2,39,26, 5,24,15,17, 7,14,22,25,10,42,11,37,36,41,16,12,32,18)( 3,34, 8, 9, 4,29,33,13,27,43, 6,19,40,21,30,28,38,31,23,20,35)$ 21A-4 $21^{2},1$ $43$ $21$ $40$ $( 2,16,11,22,15,39,12,37,25,17,26,32,36,10, 7, 5,18,41,42,14,24)( 3,31,21,43,29,34,23,30, 6,33, 8,20,28,19,13, 9,35,38,40,27, 4)$ 21A5 $21^{2},1$ $43$ $21$ $40$ $( 2,10,39,42,26,11, 5,37,24,36,15,41,17,16, 7,12,14,32,22,18,25)( 3,19,34,40, 8,21, 9,30, 4,28,29,38,33,31,13,23,27,20,43,35, 6)$ 21A-5 $21^{2},1$ $43$ $21$ $40$ $( 2,41,10,17,39,16,42, 7,26,12,11,14, 5,32,37,22,24,18,36,25,15)( 3,38,19,33,34,31,40,13, 8,23,21,27, 9,20,30,43, 4,35,28, 6,29)$ 21A8 $21^{2},1$ $43$ $21$ $40$ $( 2,26,24,17,14,25,42,37,41,12,18,39, 5,15, 7,22,10,11,36,16,32)( 3, 8, 4,33,27, 6,40,30,38,23,35,34, 9,29,13,43,19,21,28,31,20)$ 21A-8 $21^{2},1$ $43$ $21$ $40$ $( 2,25,18,22,32,14,12, 7,16,17,41,15,36,24,37, 5,11,26,42,39,10)( 3, 6,35,43,20,27,23,13,31,33,38,29,28, 4,30, 9,21, 8,40,34,19)$ 21A10 $21^{2},1$ $43$ $21$ $40$ $( 2,14,41, 5,10,32,17,37,39,22,16,24,42,18, 7,36,26,25,12,15,11)( 3,27,38, 9,19,20,33,30,34,43,31, 4,40,35,13,28, 8, 6,23,29,21)$ 21A-10 $21^{2},1$ $43$ $21$ $40$ $( 2,15,25,36,18,24,22,37,32, 5,14,11,12,26, 7,42,16,39,17,10,41)( 3,29, 6,28,35, 4,43,30,20, 9,27,21,23, 8,13,40,31,34,33,19,38)$ 43A1 $43$ $21$ $43$ $42$ $( 1,23, 2,24, 3,25, 4,26, 5,27, 6,28, 7,29, 8,30, 9,31,10,32,11,33,12,34,13,35,14,36,15,37,16,38,17,39,18,40,19,41,20,42,21,43,22)$ 43A-1 $43$ $21$ $43$ $42$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43)$

magma: ConjugacyClasses(G);

Malle's constant $a(G)$:     $1/28$

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);