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Properties

Label 42T7
Degree $42$
Order $84$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_7\times A_4$

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Show commands: Magma

magma: G := TransitiveGroup(42, 7);
 

Group action invariants

Degree $n$:  $42$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $7$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_7\times A_4$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $14$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,29,12,33,14,37,20,4,26,8,35,16,39,21,5,28,9,31,18,42,23)(2,30,11,34,13,38,19,3,25,7,36,15,40,22,6,27,10,32,17,41,24), (1,22,38,18,36,11,28,3,24,39,13,32,8,30,6,20,41,15,33,10,25)(2,21,37,17,35,12,27,4,23,40,14,31,7,29,5,19,42,16,34,9,26)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$3$:  $C_3$
$7$:  $C_7$
$12$:  $A_4$
$21$:  $C_{21}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 6: $A_4$

Degree 7: $C_7$

Degree 14: None

Degree 21: $C_{21}$

Low degree siblings

28T17

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $84=2^{2} \cdot 3 \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  84.10
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A1 3A-1 7A1 7A-1 7A2 7A-2 7A3 7A-3 14A1 14A-1 14A3 14A-3 14A5 14A-5 21A1 21A-1 21A2 21A-2 21A4 21A-4 21A5 21A-5 21A8 21A-8 21A10 21A-10
Size 1 3 4 4 1 1 1 1 1 1 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4
2 P 1A 1A 3A-1 3A1 7A2 7A1 7A-3 7A3 7A-2 7A-1 7A-3 7A1 7A2 7A-1 7A3 7A-2 21A-8 21A-2 21A5 21A4 21A-10 21A-5 21A8 21A10 21A2 21A-4 21A-1 21A1
3 P 1A 2A 1A 1A 7A3 7A-2 7A-1 7A1 7A-3 7A2 14A5 14A3 14A-1 14A-3 14A-5 14A1 7A2 7A-3 7A-3 7A-1 7A-1 7A3 7A-2 7A1 7A3 7A1 7A2 7A-2
7 P 1A 2A 3A1 3A-1 1A 1A 1A 1A 1A 1A 2A 2A 2A 2A 2A 2A 3A-1 3A-1 3A1 3A-1 3A1 3A-1 3A1 3A-1 3A1 3A1 3A1 3A-1
Type
84.10.1a 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
84.10.1b1 C 1 1 ζ3−1 ζ3 1 1 1 1 1 1 1 1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3
84.10.1b2 C 1 1 ζ3 ζ3−1 1 1 1 1 1 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
84.10.1c1 C 1 1 1 1 ζ7−3 ζ73 ζ7 ζ7−1 ζ7−2 ζ72 ζ72 ζ7−2 ζ7−1 ζ7 ζ73 ζ7−3 ζ7−3 ζ73 ζ7 ζ7−1 ζ72 ζ7−2 ζ7−1 ζ7 ζ7−3 ζ73 ζ7−2 ζ72
84.10.1c2 C 1 1 1 1 ζ73 ζ7−3 ζ7−1 ζ7 ζ72 ζ7−2 ζ7−2 ζ72 ζ7 ζ7−1 ζ7−3 ζ73 ζ73 ζ7−3 ζ7−1 ζ7 ζ7−2 ζ72 ζ7 ζ7−1 ζ73 ζ7−3 ζ72 ζ7−2
84.10.1c3 C 1 1 1 1 ζ7−2 ζ72 ζ73 ζ7−3 ζ7 ζ7−1 ζ7−1 ζ7 ζ7−3 ζ73 ζ72 ζ7−2 ζ7−2 ζ72 ζ73 ζ7−3 ζ7−1 ζ7 ζ7−3 ζ73 ζ7−2 ζ72 ζ7 ζ7−1
84.10.1c4 C 1 1 1 1 ζ72 ζ7−2 ζ7−3 ζ73 ζ7−1 ζ7 ζ7 ζ7−1 ζ73 ζ7−3 ζ7−2 ζ72 ζ72 ζ7−2 ζ7−3 ζ73 ζ7 ζ7−1 ζ73 ζ7−3 ζ72 ζ7−2 ζ7−1 ζ7
84.10.1c5 C 1 1 1 1 ζ7−1 ζ7 ζ7−2 ζ72 ζ7−3 ζ73 ζ73 ζ7−3 ζ72 ζ7−2 ζ7 ζ7−1 ζ7−1 ζ7 ζ7−2 ζ72 ζ73 ζ7−3 ζ72 ζ7−2 ζ7−1 ζ7 ζ7−3 ζ73
84.10.1c6 C 1 1 1 1 ζ7 ζ7−1 ζ72 ζ7−2 ζ73 ζ7−3 ζ7−3 ζ73 ζ7−2 ζ72 ζ7−1 ζ7 ζ7 ζ7−1 ζ72 ζ7−2 ζ7−3 ζ73 ζ7−2 ζ72 ζ7 ζ7−1 ζ73 ζ7−3
84.10.1d1 C 1 1 ζ21−7 ζ217 ζ21−9 ζ219 ζ213 ζ21−3 ζ21−6 ζ216 ζ216 ζ21−6 ζ21−3 ζ213 ζ219 ζ21−9 ζ215 ζ21−5 ζ2110 ζ21−10 ζ21−1 ζ21 ζ214 ζ21−4 ζ21−2 ζ212 ζ218 ζ21−8
84.10.1d2 C 1 1 ζ217 ζ21−7 ζ219 ζ21−9 ζ21−3 ζ213 ζ216 ζ21−6 ζ21−6 ζ216 ζ213 ζ21−3 ζ21−9 ζ219 ζ21−5 ζ215 ζ21−10 ζ2110 ζ21 ζ21−1 ζ21−4 ζ214 ζ212 ζ21−2 ζ21−8 ζ218
84.10.1d3 C 1 1 ζ21−7 ζ217 ζ219 ζ21−9 ζ21−3 ζ213 ζ216 ζ21−6 ζ21−6 ζ216 ζ213 ζ21−3 ζ21−9 ζ219 ζ212 ζ21−2 ζ214 ζ21−4 ζ218 ζ21−8 ζ2110 ζ21−10 ζ21−5 ζ215 ζ21−1 ζ21
84.10.1d4 C 1 1 ζ217 ζ21−7 ζ21−9 ζ219 ζ213 ζ21−3 ζ21−6 ζ216 ζ216 ζ21−6 ζ21−3 ζ213 ζ219 ζ21−9 ζ21−2 ζ212 ζ21−4 ζ214 ζ21−8 ζ218 ζ21−10 ζ2110 ζ215 ζ21−5 ζ21 ζ21−1
84.10.1d5 C 1 1 ζ21−7 ζ217 ζ21−6 ζ216 ζ219 ζ21−9 ζ213 ζ21−3 ζ21−3 ζ213 ζ21−9 ζ219 ζ216 ζ21−6 ζ218 ζ21−8 ζ21−5 ζ215 ζ21−10 ζ2110 ζ21−2 ζ212 ζ21 ζ21−1 ζ21−4 ζ214
84.10.1d6 C 1 1 ζ217 ζ21−7 ζ216 ζ21−6 ζ21−9 ζ219 ζ21−3 ζ213 ζ213 ζ21−3 ζ219 ζ21−9 ζ21−6 ζ216 ζ21−8 ζ218 ζ215 ζ21−5 ζ2110 ζ21−10 ζ212 ζ21−2 ζ21−1 ζ21 ζ214 ζ21−4
84.10.1d7 C 1 1 ζ21−7 ζ217 ζ216 ζ21−6 ζ21−9 ζ219 ζ21−3 ζ213 ζ213 ζ21−3 ζ219 ζ21−9 ζ21−6 ζ216 ζ21−1 ζ21 ζ21−2 ζ212 ζ21−4 ζ214 ζ21−5 ζ215 ζ21−8 ζ218 ζ21−10 ζ2110
84.10.1d8 C 1 1 ζ217 ζ21−7 ζ21−6 ζ216 ζ219 ζ21−9 ζ213 ζ21−3 ζ21−3 ζ213 ζ21−9 ζ219 ζ216 ζ21−6 ζ21 ζ21−1 ζ212 ζ21−2 ζ214 ζ21−4 ζ215 ζ21−5 ζ218 ζ21−8 ζ2110 ζ21−10
84.10.1d9 C 1 1 ζ21−7 ζ217 ζ21−3 ζ213 ζ21−6 ζ216 ζ21−9 ζ219 ζ219 ζ21−9 ζ216 ζ21−6 ζ213 ζ21−3 ζ21−10 ζ2110 ζ21 ζ21−1 ζ212 ζ21−2 ζ21−8 ζ218 ζ214 ζ21−4 ζ215 ζ21−5
84.10.1d10 C 1 1 ζ217 ζ21−7 ζ213 ζ21−3 ζ216 ζ21−6 ζ219 ζ21−9 ζ21−9 ζ219 ζ21−6 ζ216 ζ21−3 ζ213 ζ2110 ζ21−10 ζ21−1 ζ21 ζ21−2 ζ212 ζ218 ζ21−8 ζ21−4 ζ214 ζ21−5 ζ215
84.10.1d11 C 1 1 ζ21−7 ζ217 ζ213 ζ21−3 ζ216 ζ21−6 ζ219 ζ21−9 ζ21−9 ζ219 ζ21−6 ζ216 ζ21−3 ζ213 ζ21−4 ζ214 ζ21−8 ζ218 ζ215 ζ21−5 ζ21 ζ21−1 ζ2110 ζ21−10 ζ212 ζ21−2
84.10.1d12 C 1 1 ζ217 ζ21−7 ζ21−3 ζ213 ζ21−6 ζ216 ζ21−9 ζ219 ζ219 ζ21−9 ζ216 ζ21−6 ζ213 ζ21−3 ζ214 ζ21−4 ζ218 ζ21−8 ζ21−5 ζ215 ζ21−1 ζ21 ζ21−10 ζ2110 ζ21−2 ζ212
84.10.3a R 3 −1 0 0 3 3 3 3 3 3 −1 −1 −1 −1 −1 −1 0 0 0 0 0 0 0 0 0 0 0 0
84.10.3b1 C 3 −1 0 0 3ζ7−3 3ζ73 3ζ7 3ζ7−1 3ζ7−2 3ζ72 −ζ72 −ζ7−2 −ζ7−1 −ζ7 −ζ73 −ζ7−3 0 0 0 0 0 0 0 0 0 0 0 0
84.10.3b2 C 3 −1 0 0 3ζ73 3ζ7−3 3ζ7−1 3ζ7 3ζ72 3ζ7−2 −ζ7−2 −ζ72 −ζ7 −ζ7−1 −ζ7−3 −ζ73 0 0 0 0 0 0 0 0 0 0 0 0
84.10.3b3 C 3 −1 0 0 3ζ7−2 3ζ72 3ζ73 3ζ7−3 3ζ7 3ζ7−1 −ζ7−1 −ζ7 −ζ7−3 −ζ73 −ζ72 −ζ7−2 0 0 0 0 0 0 0 0 0 0 0 0
84.10.3b4 C 3 −1 0 0 3ζ72 3ζ7−2 3ζ7−3 3ζ73 3ζ7−1 3ζ7 −ζ7 −ζ7−1 −ζ73 −ζ7−3 −ζ7−2 −ζ72 0 0 0 0 0 0 0 0 0 0 0 0
84.10.3b5 C 3 −1 0 0 3ζ7−1 3ζ7 3ζ7−2 3ζ72 3ζ7−3 3ζ73 −ζ73 −ζ7−3 −ζ72 −ζ7−2 −ζ7 −ζ7−1 0 0 0 0 0 0 0 0 0 0 0 0
84.10.3b6 C 3 −1 0 0 3ζ7 3ζ7−1 3ζ72 3ζ7−2 3ζ73 3ζ7−3 −ζ7−3 −ζ73 −ζ7−2 −ζ72 −ζ7−1 −ζ7 0 0 0 0 0 0 0 0 0 0 0 0

magma: CharacterTable(G);