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Properties

Label 25T18
Degree $25$
Order $200$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_5\times F_5$

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magma: G := TransitiveGroup(25, 18);
 

Group action invariants

Degree $n$:  $25$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $18$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_5\times F_5$
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:  (1,2)(3,5)(6,17,21,12)(7,16,22,11)(8,20,23,15)(9,19,24,14)(10,18,25,13), (1,3)(4,5)(6,23)(7,22)(8,21)(9,25)(10,24)(11,18)(12,17)(13,16)(14,20)(15,19), (1,7,13,19,25)(2,8,14,20,21)(3,9,15,16,22)(4,10,11,17,23)(5,6,12,18,24)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $C_4\times C_2$
$10$:  $D_{5}$
$20$:  $F_5$, $D_{10}$
$40$:  $F_{5}\times C_2$, 20T6

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $D_{5}$, $F_5$

Low degree siblings

20T51, 40T168

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $200=2^{3} \cdot 5^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  200.41
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 4A1 4A-1 4B1 4B-1 5A1 5A2 5B 5C1 5C2 10A1 10A3 10B 20A1 20A-1 20A3 20A-3
Size 1 5 5 25 5 5 25 25 2 2 4 8 8 10 10 20 10 10 10 10
2 P 1A 1A 1A 1A 2A 2A 2A 2A 5A2 5A1 5B 5C2 5C1 5A1 5A2 5B 10A1 10A1 10A3 10A3
5 P 1A 2A 2B 2C 4A1 4A-1 4B1 4B-1 1A 1A 1A 1A 1A 2A 2A 2B 4A1 4A-1 4A-1 4A1
Type
200.41.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
200.41.1b R 1 1 −1 −1 −1 −1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1
200.41.1c R 1 1 −1 −1 1 1 −1 −1 1 1 1 1 1 1 1 −1 1 1 1 1
200.41.1d R 1 1 1 1 −1 −1 −1 −1 1 1 1 1 1 1 1 1 −1 −1 −1 −1
200.41.1e1 C 1 −1 −1 1 −i i i −i 1 1 1 1 1 −1 −1 −1 −i i i −i
200.41.1e2 C 1 −1 −1 1 i −i −i i 1 1 1 1 1 −1 −1 −1 i −i −i i
200.41.1f1 C 1 −1 1 −1 −i i −i i 1 1 1 1 1 −1 −1 1 −i i i −i
200.41.1f2 C 1 −1 1 −1 i −i i −i 1 1 1 1 1 −1 −1 1 i −i −i i
200.41.2a1 R 2 2 0 0 2 2 0 0 ζ5−2+ζ52 ζ5−1+ζ5 2 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 0 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5
200.41.2a2 R 2 2 0 0 2 2 0 0 ζ5−1+ζ5 ζ5−2+ζ52 2 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 0 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52
200.41.2b1 R 2 2 0 0 −2 −2 0 0 ζ5−2+ζ52 ζ5−1+ζ5 2 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 0 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5
200.41.2b2 R 2 2 0 0 −2 −2 0 0 ζ5−1+ζ5 ζ5−2+ζ52 2 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 0 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52
200.41.2c1 C 2 −2 0 0 −2ζ205 2ζ205 0 0 −ζ20−2−ζ202 ζ20−4+ζ204 2 ζ20−4+ζ204 −ζ20−2−ζ202 −ζ20−4−ζ204 ζ20−2+ζ202 0 ζ203+ζ207 −ζ203−ζ207 ζ203−ζ205+ζ207 −ζ203+ζ205−ζ207
200.41.2c2 C 2 −2 0 0 2ζ205 −2ζ205 0 0 −ζ20−2−ζ202 ζ20−4+ζ204 2 ζ20−4+ζ204 −ζ20−2−ζ202 −ζ20−4−ζ204 ζ20−2+ζ202 0 −ζ203−ζ207 ζ203+ζ207 −ζ203+ζ205−ζ207 ζ203−ζ205+ζ207
200.41.2c3 C 2 −2 0 0 −2ζ205 2ζ205 0 0 ζ20−4+ζ204 −ζ20−2−ζ202 2 −ζ20−2−ζ202 ζ20−4+ζ204 ζ20−2+ζ202 −ζ20−4−ζ204 0 −ζ203+ζ205−ζ207 ζ203−ζ205+ζ207 −ζ203−ζ207 ζ203+ζ207
200.41.2c4 C 2 −2 0 0 2ζ205 −2ζ205 0 0 ζ20−4+ζ204 −ζ20−2−ζ202 2 −ζ20−2−ζ202 ζ20−4+ζ204 ζ20−2+ζ202 −ζ20−4−ζ204 0 ζ203−ζ205+ζ207 −ζ203+ζ205−ζ207 ζ203+ζ207 −ζ203−ζ207
200.41.4a R 4 0 4 0 0 0 0 0 4 4 −1 −1 −1 0 0 −1 0 0 0 0
200.41.4b R 4 0 −4 0 0 0 0 0 4 4 −1 −1 −1 0 0 1 0 0 0 0
200.41.8a1 R 8 0 0 0 0 0 0 0 4ζ5−2+4ζ52 4ζ5−1+4ζ5 −2 −ζ5−1−ζ5 −ζ5−2−ζ52 0 0 0 0 0 0 0
200.41.8a2 R 8 0 0 0 0 0 0 0 4ζ5−1+4ζ5 4ζ5−2+4ζ52 −2 −ζ5−2−ζ52 −ζ5−1−ζ5 0 0 0 0 0 0 0

magma: CharacterTable(G);