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Properties

Label 28T17
Degree $28$
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(28, 17);
 

Group action invariants

Degree $n$:  $28$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $17$
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);
 
$\card{\Aut(F/K)}$:  $7$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,23,16,5,27,20,9,3,24,13,7,28,17,11,4,21,15,8,25,19,12)(2,22,14,6,26,18,10), (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)
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 4: $A_4$

Degree 7: $C_7$

Degree 14: None

Low degree siblings

42T7

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

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);