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Properties

Label 42T25
Degree $42$
Order $168$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3\times F_8$

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Show commands: Magma

magma: G := TransitiveGroup(42, 25);
 

Group action invariants

Degree $n$:  $42$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $25$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3\times F_8$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $6$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,35,23,8,41,26,18,3,31,20,9,37,27,14,5,33,22,11,39,30,16)(2,36,24,7,42,25,17,4,32,19,10,38,28,13,6,34,21,12,40,29,15), (1,33,19,7,39,28,18)(2,34,20,8,40,27,17)(3,35,21,10,41,29,14)(4,36,22,9,42,30,13)(5,31,24,12,37,25,16)(6,32,23,11,38,26,15)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$3$:  $C_3$
$7$:  $C_7$
$21$:  $C_{21}$
$56$:  $C_2^3:C_7$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 6: None

Degree 7: $C_7$

Degree 14: 14T6

Degree 21: $C_{21}$

Low degree siblings

24T282

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $168=2^{3} \cdot 3 \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  168.44
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A1 3A-1 6A1 6A-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
Size 1 7 1 1 7 7 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8
2 P 1A 1A 3A-1 3A1 3A-1 3A1 7A1 7A3 7A-3 7A-2 7A-1 7A2 21A8 21A-1 21A-5 21A10 21A2 21A-8 21A5 21A-10 21A1 21A-4 21A-2 21A4
3 P 1A 2A 1A 1A 2A 2A 7A-2 7A1 7A-1 7A-3 7A2 7A3 7A-3 7A3 7A1 7A-2 7A1 7A3 7A-1 7A2 7A-3 7A-2 7A-1 7A2
7 P 1A 2A 3A1 3A-1 6A1 6A-1 1A 1A 1A 1A 1A 1A 3A1 3A1 3A-1 3A-1 3A1 3A-1 3A1 3A1 3A-1 3A1 3A-1 3A-1
Type
168.44.1a 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
168.44.1b1 C 1 1 ζ3−1 ζ3 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
168.44.1b2 C 1 1 ζ3 ζ3−1 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
168.44.1c1 C 1 1 1 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
168.44.1c2 C 1 1 1 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
168.44.1c3 C 1 1 1 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
168.44.1c4 C 1 1 1 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
168.44.1c5 C 1 1 1 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
168.44.1c6 C 1 1 1 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
168.44.1d1 C 1 1 ζ21−7 ζ217 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
168.44.1d2 C 1 1 ζ217 ζ21−7 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
168.44.1d3 C 1 1 ζ21−7 ζ217 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
168.44.1d4 C 1 1 ζ217 ζ21−7 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
168.44.1d5 C 1 1 ζ21−7 ζ217 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
168.44.1d6 C 1 1 ζ217 ζ21−7 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
168.44.1d7 C 1 1 ζ21−7 ζ217 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
168.44.1d8 C 1 1 ζ217 ζ21−7 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
168.44.1d9 C 1 1 ζ21−7 ζ217 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
168.44.1d10 C 1 1 ζ217 ζ21−7 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
168.44.1d11 C 1 1 ζ21−7 ζ217 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
168.44.1d12 C 1 1 ζ217 ζ21−7 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
168.44.7a R 7 −1 7 7 −1 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
168.44.7b1 C 7 −1 7ζ3−1 7ζ3 −ζ3−1 −ζ3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
168.44.7b2 C 7 −1 7ζ3 7ζ3−1 −ζ3 −ζ3−1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

magma: CharacterTable(G);