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Properties

Label 28T45
Degree $28$
Order $336$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_7\times S_4$

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Show commands: Magma

magma: G := TransitiveGroup(28, 45);
 

Group action invariants

Degree $n$:  $28$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $45$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_7\times S_4$
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:  (1,23,2,24)(3,22,4,21)(5,19,6,20)(7,18,8,17)(9,15,10,16)(11,14,12,13)(25,27,26,28), (1,7,12,14,17,23,28,2,5,11,16,18,21,27,4,6,9,15,20,22,25,3,8,10,13,19,24,26)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$
$12$:  $D_{6}$
$14$:  $D_{7}$
$24$:  $S_4$
$28$:  $D_{14}$
$48$:  $S_4\times C_2$
$84$:  21T8

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: $S_4$

Degree 7: $D_{7}$

Degree 14: None

Low degree siblings

42T74, 42T75, 42T76, 42T77

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 $ $7$ $2$ $( 5,25)( 6,26)( 7,27)( 8,28)( 9,21)(10,22)(11,23)(12,24)(13,17)(14,18)(15,19) (16,20)$
$ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $2$ $( 3, 4)( 7, 8)(11,12)(15,16)(19,20)(23,24)(27,28)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $42$ $2$ $( 3, 4)( 5,25)( 6,26)( 7,28)( 8,27)( 9,21)(10,22)(11,24)(12,23)(13,17)(14,18) (15,20)(16,19)$
$ 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1 $ $8$ $3$ $( 2, 3, 4)( 6, 7, 8)(10,11,12)(14,15,16)(18,19,20)(22,23,24)(26,27,28)$
$ 6, 6, 6, 3, 2, 2, 2, 1 $ $56$ $6$ $( 2, 3, 4)( 5,25)( 6,27, 8,26, 7,28)( 9,21)(10,23,12,22,11,24)(13,17) (14,19,16,18,15,20)$
$ 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 $ $21$ $2$ $( 1, 2)( 3, 4)( 5,26)( 6,25)( 7,28)( 8,27)( 9,22)(10,21)(11,24)(12,23)(13,18) (14,17)(15,20)(16,19)$
$ 4, 4, 4, 4, 4, 4, 4 $ $6$ $4$ $( 1, 2, 3, 4)( 5, 6, 7, 8)( 9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24) (25,26,27,28)$
$ 4, 4, 4, 4, 4, 4, 4 $ $42$ $4$ $( 1, 2, 3, 4)( 5,26, 7,28)( 6,27, 8,25)( 9,22,11,24)(10,23,12,21)(13,18,15,20) (14,19,16,17)$
$ 7, 7, 7, 7 $ $2$ $7$ $( 1, 5, 9,13,17,21,25)( 2, 6,10,14,18,22,26)( 3, 7,11,15,19,23,27) ( 4, 8,12,16,20,24,28)$
$ 14, 7, 7 $ $12$ $14$ $( 1, 5, 9,13,17,21,25)( 2, 6,10,14,18,22,26)( 3, 8,11,16,19,24,27, 4, 7,12,15, 20,23,28)$
$ 21, 7 $ $16$ $21$ $( 1, 5, 9,13,17,21,25)( 2, 7,12,14,19,24,26, 3, 8,10,15,20,22,27, 4, 6,11,16, 18,23,28)$
$ 14, 14 $ $6$ $14$ $( 1, 6, 9,14,17,22,25, 2, 5,10,13,18,21,26)( 3, 8,11,16,19,24,27, 4, 7,12,15, 20,23,28)$
$ 28 $ $12$ $28$ $( 1, 6,11,16,17,22,27, 4, 5,10,15,20,21,26, 3, 8, 9,14,19,24,25, 2, 7,12,13, 18,23,28)$
$ 7, 7, 7, 7 $ $2$ $7$ $( 1, 9,17,25, 5,13,21)( 2,10,18,26, 6,14,22)( 3,11,19,27, 7,15,23) ( 4,12,20,28, 8,16,24)$
$ 14, 7, 7 $ $12$ $14$ $( 1, 9,17,25, 5,13,21)( 2,10,18,26, 6,14,22)( 3,12,19,28, 7,16,23, 4,11,20,27, 8,15,24)$
$ 21, 7 $ $16$ $21$ $( 1, 9,17,25, 5,13,21)( 2,11,20,26, 7,16,22, 3,12,18,27, 8,14,23, 4,10,19,28, 6,15,24)$
$ 14, 14 $ $6$ $14$ $( 1,10,17,26, 5,14,21, 2, 9,18,25, 6,13,22)( 3,12,19,28, 7,16,23, 4,11,20,27, 8,15,24)$
$ 28 $ $12$ $28$ $( 1,10,19,28, 5,14,23, 4, 9,18,27, 8,13,22, 3,12,17,26, 7,16,21, 2,11,20,25, 6,15,24)$
$ 7, 7, 7, 7 $ $2$ $7$ $( 1,13,25, 9,21, 5,17)( 2,14,26,10,22, 6,18)( 3,15,27,11,23, 7,19) ( 4,16,28,12,24, 8,20)$
$ 14, 7, 7 $ $12$ $14$ $( 1,13,25, 9,21, 5,17)( 2,14,26,10,22, 6,18)( 3,16,27,12,23, 8,19, 4,15,28,11, 24, 7,20)$
$ 21, 7 $ $16$ $21$ $( 1,13,25, 9,21, 5,17)( 2,15,28,10,23, 8,18, 3,16,26,11,24, 6,19, 4,14,27,12, 22, 7,20)$
$ 14, 14 $ $6$ $14$ $( 1,14,25,10,21, 6,17, 2,13,26, 9,22, 5,18)( 3,16,27,12,23, 8,19, 4,15,28,11, 24, 7,20)$
$ 28 $ $12$ $28$ $( 1,14,27,12,21, 6,19, 4,13,26,11,24, 5,18, 3,16,25,10,23, 8,17, 2,15,28, 9, 22, 7,20)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $336=2^{4} \cdot 3 \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  336.212
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 3A 4A 4B 6A 7A1 7A2 7A3 14A1 14A3 14A5 14B1 14B3 14B5 21A1 21A2 21A4 28A1 28A3 28A5
Size 1 3 6 7 21 42 8 6 42 56 2 2 2 6 6 6 12 12 12 16 16 16 12 12 12
2 P 1A 1A 1A 1A 1A 1A 3A 2A 2A 3A 7A2 7A3 7A1 7A1 7A3 7A2 7A3 7A2 7A1 21A2 21A4 21A1 14A5 14A1 14A3
3 P 1A 2A 2B 2C 2D 2E 1A 4A 4B 2C 7A3 7A1 7A2 14A3 14A5 14A1 14B3 14B5 14B1 7A2 7A3 7A1 28A1 28A3 28A5
7 P 1A 2A 2B 2C 2D 2E 3A 4A 4B 6A 1A 1A 1A 2A 2A 2A 2B 2B 2B 3A 3A 3A 4A 4A 4A
Type
336.212.1a 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
336.212.1b 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
336.212.1c 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
336.212.1d 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
336.212.2a R 2 2 0 2 2 0 −1 0 0 −1 2 2 2 2 2 2 0 0 0 −1 −1 −1 0 0 0
336.212.2b R 2 2 0 −2 −2 0 −1 0 0 1 2 2 2 2 2 2 0 0 0 −1 −1 −1 0 0 0
336.212.2c1 R 2 2 2 0 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 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72
336.212.2c2 R 2 2 2 0 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 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7
336.212.2c3 R 2 2 2 0 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 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73
336.212.2d1 R 2 2 −2 0 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 −ζ7−3−ζ73 −ζ7−2−ζ72 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 −ζ7−1−ζ7 −ζ7−3−ζ73 −ζ7−2−ζ72
336.212.2d2 R 2 2 −2 0 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 −ζ7−2−ζ72 −ζ7−1−ζ7 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 −ζ7−3−ζ73 −ζ7−2−ζ72 −ζ7−1−ζ7
336.212.2d3 R 2 2 −2 0 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 −ζ7−1−ζ7 −ζ7−3−ζ73 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 −ζ7−2−ζ72 −ζ7−1−ζ7 −ζ7−3−ζ73
336.212.3a R 3 −1 1 3 −1 1 0 −1 −1 0 3 3 3 −1 −1 −1 1 1 1 0 0 0 −1 −1 −1
336.212.3b R 3 −1 −1 3 −1 −1 0 1 1 0 3 3 3 −1 −1 −1 −1 −1 −1 0 0 0 1 1 1
336.212.3c R 3 −1 −1 −3 1 1 0 1 −1 0 3 3 3 −1 −1 −1 −1 −1 −1 0 0 0 1 1 1
336.212.3d R 3 −1 1 −3 1 −1 0 −1 1 0 3 3 3 −1 −1 −1 1 1 1 0 0 0 −1 −1 −1
336.212.4a1 R 4 4 0 0 0 0 −2 0 0 0 2ζ7−3+2ζ73 2ζ7−1+2ζ7 2ζ7−2+2ζ72 2ζ7−2+2ζ72 2ζ7−1+2ζ7 2ζ7−3+2ζ73 0 0 0 −ζ7−2−ζ72 −ζ7−3−ζ73 −ζ7−1−ζ7 0 0 0
336.212.4a2 R 4 4 0 0 0 0 −2 0 0 0 2ζ7−2+2ζ72 2ζ7−3+2ζ73 2ζ7−1+2ζ7 2ζ7−1+2ζ7 2ζ7−3+2ζ73 2ζ7−2+2ζ72 0 0 0 −ζ7−1−ζ7 −ζ7−2−ζ72 −ζ7−3−ζ73 0 0 0
336.212.4a3 R 4 4 0 0 0 0 −2 0 0 0 2ζ7−1+2ζ7 2ζ7−2+2ζ72 2ζ7−3+2ζ73 2ζ7−3+2ζ73 2ζ7−2+2ζ72 2ζ7−1+2ζ7 0 0 0 −ζ7−3−ζ73 −ζ7−1−ζ7 −ζ7−2−ζ72 0 0 0
336.212.6a1 R 6 −2 2 0 0 0 0 −2 0 0 3ζ7−3+3ζ73 3ζ7−1+3ζ7 3ζ7−2+3ζ72 −ζ7−2−ζ72 −ζ7−1−ζ7 −ζ7−3−ζ73 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 0 0 0 −ζ7−1−ζ7 −ζ7−3−ζ73 −ζ7−2−ζ72
336.212.6a2 R 6 −2 2 0 0 0 0 −2 0 0 3ζ7−2+3ζ72 3ζ7−3+3ζ73 3ζ7−1+3ζ7 −ζ7−1−ζ7 −ζ7−3−ζ73 −ζ7−2−ζ72 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 0 0 0 −ζ7−3−ζ73 −ζ7−2−ζ72 −ζ7−1−ζ7
336.212.6a3 R 6 −2 2 0 0 0 0 −2 0 0 3ζ7−1+3ζ7 3ζ7−2+3ζ72 3ζ7−3+3ζ73 −ζ7−3−ζ73 −ζ7−2−ζ72 −ζ7−1−ζ7 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 0 0 0 −ζ7−2−ζ72 −ζ7−1−ζ7 −ζ7−3−ζ73
336.212.6b1 R 6 −2 −2 0 0 0 0 2 0 0 3ζ7−3+3ζ73 3ζ7−1+3ζ7 3ζ7−2+3ζ72 −ζ7−2−ζ72 −ζ7−1−ζ7 −ζ7−3−ζ73 −ζ7−1−ζ7 −ζ7−3−ζ73 −ζ7−2−ζ72 0 0 0 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72
336.212.6b2 R 6 −2 −2 0 0 0 0 2 0 0 3ζ7−2+3ζ72 3ζ7−3+3ζ73 3ζ7−1+3ζ7 −ζ7−1−ζ7 −ζ7−3−ζ73 −ζ7−2−ζ72 −ζ7−3−ζ73 −ζ7−2−ζ72 −ζ7−1−ζ7 0 0 0 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7
336.212.6b3 R 6 −2 −2 0 0 0 0 2 0 0 3ζ7−1+3ζ7 3ζ7−2+3ζ72 3ζ7−3+3ζ73 −ζ7−3−ζ73 −ζ7−2−ζ72 −ζ7−1−ζ7 −ζ7−2−ζ72 −ζ7−1−ζ7 −ζ7−3−ζ73 0 0 0 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73

magma: CharacterTable(G);