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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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Show commands: Magma

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

$\card{(G/N)}$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 TypeSizeOrderIndexRepresentative
1A $1^{25}$ $1$ $1$ $0$ $()$
2A $2^{10},1^{5}$ $5$ $2$ $10$ $( 6,21)( 7,22)( 8,23)( 9,24)(10,25)(11,16)(12,17)(13,18)(14,19)(15,20)$
2B $2^{10},1^{5}$ $5$ $2$ $10$ $( 2, 5)( 3, 4)( 7,10)( 8, 9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)$
2C $2^{12},1$ $25$ $2$ $12$ $( 1, 5)( 2, 4)( 6,25)( 7,24)( 8,23)( 9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)$
4A1 $4^{5},1^{5}$ $5$ $4$ $15$ $( 6,16,21,11)( 7,17,22,12)( 8,18,23,13)( 9,19,24,14)(10,20,25,15)$
4A-1 $4^{5},1^{5}$ $5$ $4$ $15$ $( 6,11,21,16)( 7,12,22,17)( 8,13,23,18)( 9,14,24,19)(10,15,25,20)$
4B1 $4^{5},2^{2},1$ $25$ $4$ $17$ $( 1, 2)( 3, 5)( 6,12,21,17)( 7,11,22,16)( 8,15,23,20)( 9,14,24,19)(10,13,25,18)$
4B-1 $4^{5},2^{2},1$ $25$ $4$ $17$ $( 1, 3)( 4, 5)( 6,18,21,13)( 7,17,22,12)( 8,16,23,11)( 9,20,24,15)(10,19,25,14)$
5A1 $5^{5}$ $2$ $5$ $20$ $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19)(21,23,25,22,24)$
5A2 $5^{5}$ $2$ $5$ $20$ $( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)(11,15,14,13,12)(16,20,19,18,17)(21,25,24,23,22)$
5B $5^{5}$ $4$ $5$ $20$ $( 1,21,16,11, 6)( 2,22,17,12, 7)( 3,23,18,13, 8)( 4,24,19,14, 9)( 5,25,20,15,10)$
5C1 $5^{5}$ $8$ $5$ $20$ $( 1,23,20,12, 9)( 2,24,16,13,10)( 3,25,17,14, 6)( 4,21,18,15, 7)( 5,22,19,11, 8)$
5C2 $5^{5}$ $8$ $5$ $20$ $( 1,25,19,13, 7)( 2,21,20,14, 8)( 3,22,16,15, 9)( 4,23,17,11,10)( 5,24,18,12, 6)$
10A1 $10^{2},5$ $10$ $10$ $22$ $( 1, 2, 3, 4, 5)( 6,22, 8,24,10,21, 7,23, 9,25)(11,17,13,19,15,16,12,18,14,20)$
10A3 $10^{2},5$ $10$ $10$ $22$ $( 1, 4, 2, 5, 3)( 6,24, 7,25, 8,21, 9,22,10,23)(11,19,12,20,13,16,14,17,15,18)$
10B $10^{2},5$ $20$ $10$ $22$ $( 1,21,16,11, 6)( 2,25,17,15, 7, 5,22,20,12,10)( 3,24,18,14, 8, 4,23,19,13, 9)$
20A1 $20,5$ $10$ $20$ $23$ $( 1, 4, 2, 5, 3)( 6,19,22,15, 8,16,24,12,10,18,21,14, 7,20,23,11, 9,17,25,13)$
20A-1 $20,5$ $10$ $20$ $23$ $( 1, 4, 2, 5, 3)( 6,14,22,20, 8,11,24,17,10,13,21,19, 7,15,23,16, 9,12,25,18)$
20A3 $20,5$ $10$ $20$ $23$ $( 1, 5, 4, 3, 2)( 6,15,24,18, 7,11,25,19, 8,12,21,20, 9,13,22,16,10,14,23,17)$
20A-3 $20,5$ $10$ $20$ $23$ $( 1, 5, 4, 3, 2)( 6,20,24,13, 7,16,25,14, 8,17,21,15, 9,18,22,11,10,19,23,12)$

Malle's constant $a(G)$:     $1/10$

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);