↑

Properties

Label 25T7
Degree $25$
Order $100$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_5\times F_5$

Related objects

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(25, 7);
 

Group action invariants

Degree $n$:  $25$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $7$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_5\times F_5$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $5$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,20,21,8)(2,16,22,9)(3,17,23,10)(4,18,24,6)(5,19,25,7), (1,2,3,4,5)(6,19,21,15,10,18,25,14,9,17,24,13,8,16,23,12,7,20,22,11)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$
$5$:  $C_5$
$10$:  $C_{10}$
$20$:  $F_5$, 20T1

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $C_5$, $F_5$

Low degree siblings

20T29

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $100=2^{2} \cdot 5^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  100.9
magma: IdentifyGroup(G);
 
Character table:

1A 2A 4A1 4A-1 5A1 5A-1 5A2 5A-2 5B 5C1 5C-1 5C2 5C-2 10A1 10A-1 10A3 10A-3 20A1 20A-1 20A3 20A-3 20A7 20A-7 20A9 20A-9
Size 1 5 5 5 1 1 1 1 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 5
2 P 1A 1A 2A 2A 5A-2 5A2 5A1 5A-1 5B 5C-1 5C-2 5C1 5C2 5A-2 5A-1 5A2 5A1 10A1 10A1 10A3 10A-3 10A-3 10A-1 10A-1 10A3
5 P 1A 2A 4A1 4A-1 1A 1A 1A 1A 1A 1A 1A 1A 1A 2A 2A 2A 2A 4A1 4A-1 4A-1 4A1 4A-1 4A-1 4A1 4A1
Type
100.9.1a 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
100.9.1b 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
100.9.1c1 C 1 −1 −i i 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −i i i −i i −i −i i
100.9.1c2 C 1 −1 i −i 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 i −i −i i −i i i −i
100.9.1d1 C 1 1 1 1 ζ5−2 ζ52 ζ5 ζ5−1 1 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−1 ζ5 ζ52 ζ5−2 ζ52 ζ5−2 ζ5 ζ5−1 ζ5−1 ζ5 ζ5−2 ζ52
100.9.1d2 C 1 1 1 1 ζ52 ζ5−2 ζ5−1 ζ5 1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5 ζ5−1 ζ5−2 ζ52 ζ5−2 ζ52 ζ5−1 ζ5 ζ5 ζ5−1 ζ52 ζ5−2
100.9.1d3 C 1 1 1 1 ζ5−1 ζ5 ζ5−2 ζ52 1 ζ52 ζ5−2 ζ5−1 ζ5 ζ52 ζ5−2 ζ5 ζ5−1 ζ5 ζ5−1 ζ5−2 ζ52 ζ52 ζ5−2 ζ5−1 ζ5
100.9.1d4 C 1 1 1 1 ζ5 ζ5−1 ζ52 ζ5−2 1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−2 ζ52 ζ5−1 ζ5 ζ5−1 ζ5 ζ52 ζ5−2 ζ5−2 ζ52 ζ5 ζ5−1
100.9.1e1 C 1 1 −1 −1 ζ5−2 ζ52 ζ5 ζ5−1 1 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−1 ζ5 ζ52 ζ5−2 −ζ52 −ζ5−2 −ζ5 −ζ5−1 −ζ5−1 −ζ5 −ζ5−2 −ζ52
100.9.1e2 C 1 1 −1 −1 ζ52 ζ5−2 ζ5−1 ζ5 1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5 ζ5−1 ζ5−2 ζ52 −ζ5−2 −ζ52 −ζ5−1 −ζ5 −ζ5 −ζ5−1 −ζ52 −ζ5−2
100.9.1e3 C 1 1 −1 −1 ζ5−1 ζ5 ζ5−2 ζ52 1 ζ52 ζ5−2 ζ5−1 ζ5 ζ52 ζ5−2 ζ5 ζ5−1 −ζ5 −ζ5−1 −ζ5−2 −ζ52 −ζ52 −ζ5−2 −ζ5−1 −ζ5
100.9.1e4 C 1 1 −1 −1 ζ5 ζ5−1 ζ52 ζ5−2 1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−2 ζ52 ζ5−1 ζ5 −ζ5−1 −ζ5 −ζ52 −ζ5−2 −ζ5−2 −ζ52 −ζ5 −ζ5−1
100.9.1f1 C 1 −1 −ζ205 ζ205 −ζ202 ζ208 ζ204 −ζ206 1 −ζ206 ζ204 −ζ202 ζ208 ζ206 −ζ204 −ζ208 ζ202 ζ203 −ζ207 ζ209 −ζ20 ζ20 −ζ209 ζ207 −ζ203
100.9.1f2 C 1 −1 ζ205 −ζ205 ζ208 −ζ202 −ζ206 ζ204 1 ζ204 −ζ206 ζ208 −ζ202 −ζ204 ζ206 ζ202 −ζ208 −ζ207 ζ203 −ζ20 ζ209 −ζ209 ζ20 −ζ203 ζ207
100.9.1f3 C 1 −1 −ζ205 ζ205 ζ208 −ζ202 −ζ206 ζ204 1 ζ204 −ζ206 ζ208 −ζ202 −ζ204 ζ206 ζ202 −ζ208 ζ207 −ζ203 ζ20 −ζ209 ζ209 −ζ20 ζ203 −ζ207
100.9.1f4 C 1 −1 ζ205 −ζ205 −ζ202 ζ208 ζ204 −ζ206 1 −ζ206 ζ204 −ζ202 ζ208 ζ206 −ζ204 −ζ208 ζ202 −ζ203 ζ207 −ζ209 ζ20 −ζ20 ζ209 −ζ207 ζ203
100.9.1f5 C 1 −1 −ζ205 ζ205 −ζ206 ζ204 −ζ202 ζ208 1 ζ208 −ζ202 −ζ206 ζ204 −ζ208 ζ202 −ζ204 ζ206 −ζ209 ζ20 −ζ207 ζ203 −ζ203 ζ207 −ζ20 ζ209
100.9.1f6 C 1 −1 ζ205 −ζ205 ζ204 −ζ206 ζ208 −ζ202 1 −ζ202 ζ208 ζ204 −ζ206 ζ202 −ζ208 ζ206 −ζ204 ζ20 −ζ209 ζ203 −ζ207 ζ207 −ζ203 ζ209 −ζ20
100.9.1f7 C 1 −1 −ζ205 ζ205 ζ204 −ζ206 ζ208 −ζ202 1 −ζ202 ζ208 ζ204 −ζ206 ζ202 −ζ208 ζ206 −ζ204 −ζ20 ζ209 −ζ203 ζ207 −ζ207 ζ203 −ζ209 ζ20
100.9.1f8 C 1 −1 ζ205 −ζ205 −ζ206 ζ204 −ζ202 ζ208 1 ζ208 −ζ202 −ζ206 ζ204 −ζ208 ζ202 −ζ204 ζ206 ζ209 −ζ20 ζ207 −ζ203 ζ203 −ζ207 ζ20 −ζ209
100.9.4a R 4 0 0 0 4 4 4 4 −1 −1 −1 −1 −1 0 0 0 0 0 0 0 0 0 0 0 0
100.9.4b1 C 4 0 0 0 4ζ5−2 4ζ52 4ζ5 4ζ5−1 −1 −ζ5−1 −ζ5 −ζ5−2 −ζ52 0 0 0 0 0 0 0 0 0 0 0 0
100.9.4b2 C 4 0 0 0 4ζ52 4ζ5−2 4ζ5−1 4ζ5 −1 −ζ5 −ζ5−1 −ζ52 −ζ5−2 0 0 0 0 0 0 0 0 0 0 0 0
100.9.4b3 C 4 0 0 0 4ζ5−1 4ζ5 4ζ5−2 4ζ52 −1 −ζ52 −ζ5−2 −ζ5−1 −ζ5 0 0 0 0 0 0 0 0 0 0 0 0
100.9.4b4 C 4 0 0 0 4ζ5 4ζ5−1 4ζ52 4ζ5−2 −1 −ζ5−2 −ζ52 −ζ5 −ζ5−1 0 0 0 0 0 0 0 0 0 0 0 0

magma: CharacterTable(G);