↑

Properties

Label 28T29
Degree $28$
Order $168$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $A_4\times D_7$

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(28, 29);
 

Group action invariants

Degree $n$:  $28$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $29$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $A_4\times D_7$
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:  (1,22,15,5,26,19,9,2,23,13,6,27,17,10,3,21,14,7,25,18,11)(4,24,16,8,28,20,12), (1,28,3,25,4,27)(2,26)(5,24,7,21,8,23)(6,22)(9,20,11,17,12,19)(10,18)(13,16,15)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$12$:  $A_4$
$14$:  $D_{7}$
$24$:  $A_4\times C_2$
$42$:  21T3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: $A_4$

Degree 7: $D_{7}$

Degree 14: None

Low degree siblings

42T28, 42T29

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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $7$ $2$ $( 5,25)( 6,26)( 7,27)( 8,28)( 9,21)(10,22)(11,23)(12,24)(13,17)(14,18)(15,19) (16,20)$
$ 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1 $ $4$ $3$ $( 2, 3, 4)( 6, 7, 8)(10,11,12)(14,15,16)(18,19,20)(22,23,24)(26,27,28)$
$ 6, 6, 6, 3, 2, 2, 2, 1 $ $28$ $6$ $( 2, 3, 4)( 5,25)( 6,27, 8,26, 7,28)( 9,21)(10,23,12,22,11,24)(13,17) (14,19,16,18,15,20)$
$ 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1 $ $4$ $3$ $( 2, 4, 3)( 6, 8, 7)(10,12,11)(14,16,15)(18,20,19)(22,24,23)(26,28,27)$
$ 6, 6, 6, 3, 2, 2, 2, 1 $ $28$ $6$ $( 2, 4, 3)( 5,25)( 6,28, 7,26, 8,27)( 9,21)(10,24,11,22,12,23)(13,17) (14,20,15,18,16,19)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $21$ $2$ $( 1, 2)( 3, 4)( 5,26)( 6,25)( 7,28)( 8,27)( 9,22)(10,21)(11,24)(12,23)(13,18) (14,17)(15,20)(16,19)$
$ 7, 7, 7, 7 $ $2$ $7$ $( 1, 5, 9,13,17,21,25)( 2, 6,10,14,18,22,26)( 3, 7,11,15,19,23,27) ( 4, 8,12,16,20,24,28)$
$ 21, 7 $ $8$ $21$ $( 1, 5, 9,13,17,21,25)( 2, 7,12,14,19,24,26, 3, 8,10,15,20,22,27, 4, 6,11,16, 18,23,28)$
$ 21, 7 $ $8$ $21$ $( 1, 5, 9,13,17,21,25)( 2, 8,11,14,20,23,26, 4, 7,10,16,19,22,28, 3, 6,12,15, 18,24,27)$
$ 14, 14 $ $6$ $14$ $( 1, 6, 9,14,17,22,25, 2, 5,10,13,18,21,26)( 3, 8,11,16,19,24,27, 4, 7,12,15, 20,23,28)$
$ 7, 7, 7, 7 $ $2$ $7$ $( 1, 9,17,25, 5,13,21)( 2,10,18,26, 6,14,22)( 3,11,19,27, 7,15,23) ( 4,12,20,28, 8,16,24)$
$ 21, 7 $ $8$ $21$ $( 1, 9,17,25, 5,13,21)( 2,11,20,26, 7,16,22, 3,12,18,27, 8,14,23, 4,10,19,28, 6,15,24)$
$ 21, 7 $ $8$ $21$ $( 1, 9,17,25, 5,13,21)( 2,12,19,26, 8,15,22, 4,11,18,28, 7,14,24, 3,10,20,27, 6,16,23)$
$ 14, 14 $ $6$ $14$ $( 1,10,17,26, 5,14,21, 2, 9,18,25, 6,13,22)( 3,12,19,28, 7,16,23, 4,11,20,27, 8,15,24)$
$ 7, 7, 7, 7 $ $2$ $7$ $( 1,13,25, 9,21, 5,17)( 2,14,26,10,22, 6,18)( 3,15,27,11,23, 7,19) ( 4,16,28,12,24, 8,20)$
$ 21, 7 $ $8$ $21$ $( 1,13,25, 9,21, 5,17)( 2,15,28,10,23, 8,18, 3,16,26,11,24, 6,19, 4,14,27,12, 22, 7,20)$
$ 21, 7 $ $8$ $21$ $( 1,13,25, 9,21, 5,17)( 2,16,27,10,24, 7,18, 4,15,26,12,23, 6,20, 3,14,28,11, 22, 8,19)$
$ 14, 14 $ $6$ $14$ $( 1,14,25,10,21, 6,17, 2,13,26, 9,22, 5,18)( 3,16,27,12,23, 8,19, 4,15,28,11, 24, 7,20)$

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:  168.48
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A1 3A-1 6A1 6A-1 7A1 7A2 7A3 14A1 14A3 14A5 21A1 21A-1 21A2 21A-2 21A4 21A-4
Size 1 3 7 21 4 4 28 28 2 2 2 6 6 6 8 8 8 8 8 8
2 P 1A 1A 1A 1A 3A-1 3A1 3A1 3A-1 7A2 7A3 7A1 7A2 7A1 7A3 21A4 21A-1 21A-4 21A2 21A1 21A-2
3 P 1A 2A 2B 2C 1A 1A 2B 2B 7A3 7A1 7A2 14A1 14A3 14A5 7A2 7A3 7A2 7A1 7A3 7A1
7 P 1A 2A 2B 2C 3A1 3A-1 6A1 6A-1 1A 1A 1A 2A 2A 2A 3A1 3A-1 3A-1 3A-1 3A1 3A1
Type
168.48.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
168.48.1b R 1 1 −1 −1 1 1 −1 −1 1 1 1 1 1 1 1 1 1 1 1 1
168.48.1c1 C 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1
168.48.1c2 C 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3
168.48.1d1 C 1 1 −1 −1 ζ3−1 ζ3 −ζ3 −ζ3−1 1 1 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1
168.48.1d2 C 1 1 −1 −1 ζ3 ζ3−1 −ζ3−1 −ζ3 1 1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3
168.48.2a1 R 2 2 0 0 2 2 0 0 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73
168.48.2a2 R 2 2 0 0 2 2 0 0 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72
168.48.2a3 R 2 2 0 0 2 2 0 0 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7
168.48.2b1 C 2 2 0 0 2ζ21−7 2ζ217 0 0 ζ21−9+ζ219 ζ21−3+ζ213 ζ21−6+ζ216 ζ21−6+ζ216 ζ21−3+ζ213 ζ21−9+ζ219 ζ214+ζ2110 ζ21−10−ζ213−ζ2110 −ζ21−10+1−ζ212+ζ213−ζ215+ζ217+ζ2110 ζ21−10−1+ζ21+ζ212−ζ213+ζ215−ζ216−ζ217+ζ218−ζ2110 −ζ21−10+1−ζ21−ζ212+ζ213−ζ214−ζ215+ζ216−ζ218 ζ212+ζ215
168.48.2b2 C 2 2 0 0 2ζ217 2ζ21−7 0 0 ζ21−9+ζ219 ζ21−3+ζ213 ζ21−6+ζ216 ζ21−6+ζ216 ζ21−3+ζ213 ζ21−9+ζ219 ζ21−10−ζ213−ζ2110 ζ214+ζ2110 ζ21−10−1+ζ21+ζ212−ζ213+ζ215−ζ216−ζ217+ζ218−ζ2110 −ζ21−10+1−ζ212+ζ213−ζ215+ζ217+ζ2110 ζ212+ζ215 −ζ21−10+1−ζ21−ζ212+ζ213−ζ214−ζ215+ζ216−ζ218
168.48.2b3 C 2 2 0 0 2ζ21−7 2ζ217 0 0 ζ21−6+ζ216 ζ21−9+ζ219 ζ21−3+ζ213 ζ21−3+ζ213 ζ21−9+ζ219 ζ21−6+ζ216 −ζ21−10+1−ζ21−ζ212+ζ213−ζ214−ζ215+ζ216−ζ218 ζ212+ζ215 ζ21−10−ζ213−ζ2110 ζ214+ζ2110 ζ21−10−1+ζ21+ζ212−ζ213+ζ215−ζ216−ζ217+ζ218−ζ2110 −ζ21−10+1−ζ212+ζ213−ζ215+ζ217+ζ2110
168.48.2b4 C 2 2 0 0 2ζ217 2ζ21−7 0 0 ζ21−6+ζ216 ζ21−9+ζ219 ζ21−3+ζ213 ζ21−3+ζ213 ζ21−9+ζ219 ζ21−6+ζ216 ζ212+ζ215 −ζ21−10+1−ζ21−ζ212+ζ213−ζ214−ζ215+ζ216−ζ218 ζ214+ζ2110 ζ21−10−ζ213−ζ2110 −ζ21−10+1−ζ212+ζ213−ζ215+ζ217+ζ2110 ζ21−10−1+ζ21+ζ212−ζ213+ζ215−ζ216−ζ217+ζ218−ζ2110
168.48.2b5 C 2 2 0 0 2ζ21−7 2ζ217 0 0 ζ21−3+ζ213 ζ21−6+ζ216 ζ21−9+ζ219 ζ21−9+ζ219 ζ21−6+ζ216 ζ21−3+ζ213 ζ21−10−1+ζ21+ζ212−ζ213+ζ215−ζ216−ζ217+ζ218−ζ2110 −ζ21−10+1−ζ212+ζ213−ζ215+ζ217+ζ2110 ζ212+ζ215 −ζ21−10+1−ζ21−ζ212+ζ213−ζ214−ζ215+ζ216−ζ218 ζ214+ζ2110 ζ21−10−ζ213−ζ2110
168.48.2b6 C 2 2 0 0 2ζ217 2ζ21−7 0 0 ζ21−3+ζ213 ζ21−6+ζ216 ζ21−9+ζ219 ζ21−9+ζ219 ζ21−6+ζ216 ζ21−3+ζ213 −ζ21−10+1−ζ212+ζ213−ζ215+ζ217+ζ2110 ζ21−10−1+ζ21+ζ212−ζ213+ζ215−ζ216−ζ217+ζ218−ζ2110 −ζ21−10+1−ζ21−ζ212+ζ213−ζ214−ζ215+ζ216−ζ218 ζ212+ζ215 ζ21−10−ζ213−ζ2110 ζ214+ζ2110
168.48.3a R 3 −1 3 −1 0 0 0 0 3 3 3 −1 −1 −1 0 0 0 0 0 0
168.48.3b R 3 −1 −3 1 0 0 0 0 3 3 3 −1 −1 −1 0 0 0 0 0 0
168.48.6a1 R 6 −2 0 0 0 0 0 0 3ζ7−3+3ζ73 3ζ7−1+3ζ7 3ζ7−2+3ζ72 −ζ7−2−ζ72 −ζ7−1−ζ7 −ζ7−3−ζ73 0 0 0 0 0 0
168.48.6a2 R 6 −2 0 0 0 0 0 0 3ζ7−2+3ζ72 3ζ7−3+3ζ73 3ζ7−1+3ζ7 −ζ7−1−ζ7 −ζ7−3−ζ73 −ζ7−2−ζ72 0 0 0 0 0 0
168.48.6a3 R 6 −2 0 0 0 0 0 0 3ζ7−1+3ζ7 3ζ7−2+3ζ72 3ζ7−3+3ζ73 −ζ7−3−ζ73 −ζ7−2−ζ72 −ζ7−1−ζ7 0 0 0 0 0 0

magma: CharacterTable(G);