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Properties

Label 25T39
Degree $25$
Order $500$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_5^2:F_5$

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Show commands: Magma

magma: G := TransitiveGroup(25, 39);
 

Group action invariants

Degree $n$:  $25$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $39$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_5^2:F_5$
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:  (1,16,12,21)(2,18,11,24)(3,20,15,22)(4,17,14,25)(5,19,13,23)(6,7,9,8), (1,15,21,8,20)(2,11,22,9,16)(3,12,23,10,17)(4,13,24,6,18)(5,14,25,7,19)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$
$10$:  $D_{5}$
$20$:  $F_5$, 20T2
$100$:  20T26

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $F_5$

Low degree siblings

25T36

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 $ $1$ $1$ $()$
$ 5, 5, 5, 5, 1, 1, 1, 1, 1 $ $10$ $5$ $( 6, 7, 8, 9,10)(11,13,15,12,14)(16,19,17,20,18)(21,25,24,23,22)$
$ 5, 5, 5, 5, 1, 1, 1, 1, 1 $ $10$ $5$ $( 6, 8,10, 7, 9)(11,15,14,13,12)(16,17,18,19,20)(21,24,22,25,23)$
$ 4, 4, 4, 4, 4, 4, 1 $ $125$ $4$ $( 2, 3, 5, 4)( 6,16,21,11)( 7,18,25,14)( 8,20,24,12)( 9,17,23,15)(10,19,22,13)$
$ 4, 4, 4, 4, 4, 4, 1 $ $125$ $4$ $( 2, 4, 5, 3)( 6,11,21,16)( 7,14,25,18)( 8,12,24,20)( 9,15,23,17)(10,13,22,19)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $25$ $2$ $( 2, 5)( 3, 4)( 6,21)( 7,25)( 8,24)( 9,23)(10,22)(11,16)(12,20)(13,19)(14,18) (15,17)$
$ 10, 10, 2, 2, 1 $ $50$ $10$ $( 2, 5)( 3, 4)( 6,22, 9,24, 7,21,10,23, 8,25)(11,18,12,17,13,16,14,20,15,19)$
$ 10, 10, 2, 2, 1 $ $50$ $10$ $( 2, 5)( 3, 4)( 6,23, 7,22, 8,21, 9,25,10,24)(11,20,13,18,15,16,12,19,14,17)$
$ 5, 5, 5, 5, 5 $ $4$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)$
$ 5, 5, 5, 5, 5 $ $20$ $5$ $( 1, 6,11,19,23)( 2, 7,12,20,24)( 3, 8,13,16,25)( 4, 9,14,17,21) ( 5,10,15,18,22)$
$ 5, 5, 5, 5, 5 $ $20$ $5$ $( 1, 6,12,17,24)( 2, 7,13,18,25)( 3, 8,14,19,21)( 4, 9,15,20,22) ( 5,10,11,16,23)$
$ 5, 5, 5, 5, 5 $ $20$ $5$ $( 1, 6,13,20,25)( 2, 7,14,16,21)( 3, 8,15,17,22)( 4, 9,11,18,23) ( 5,10,12,19,24)$
$ 5, 5, 5, 5, 5 $ $20$ $5$ $( 1, 6,14,18,21)( 2, 7,15,19,22)( 3, 8,11,20,23)( 4, 9,12,16,24) ( 5,10,13,17,25)$
$ 5, 5, 5, 5, 5 $ $20$ $5$ $( 1, 6,15,16,22)( 2, 7,11,17,23)( 3, 8,12,18,24)( 4, 9,13,19,25) ( 5,10,14,20,21)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $500=2^{2} \cdot 5^{3}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  500.21
magma: IdentifyGroup(G);
 
Character table:

1A 2A 4A1 4A-1 5A 5B1 5B2 5C 5D1 5D-1 5D2 5D-2 10A1 10A3
Size 1 25 125 125 4 10 10 20 20 20 20 20 50 50
2 P 1A 1A 2A 2A 5A 5B2 5B1 5D2 5C 5D-2 5D1 5D-1 5B1 5B2
5 P 1A 2A 4A1 4A-1 1A 1A 1A 1A 1A 1A 1A 1A 2A 2A
Type
500.21.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
500.21.1b R 1 1 −1 −1 1 1 1 1 1 1 1 1 1 1
500.21.1c1 C 1 −1 −i i 1 1 1 1 1 1 1 1 −1 −1
500.21.1c2 C 1 −1 i −i 1 1 1 1 1 1 1 1 −1 −1
500.21.2a1 R 2 2 0 0 2 ζ5−2+ζ52 ζ5−1+ζ5 2 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52
500.21.2a2 R 2 2 0 0 2 ζ5−1+ζ5 ζ5−2+ζ52 2 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5
500.21.2b1 S 2 −2 0 0 2 ζ5−2+ζ52 ζ5−1+ζ5 2 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52
500.21.2b2 S 2 −2 0 0 2 ζ5−1+ζ5 ζ5−2+ζ52 2 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5
500.21.4a R 4 0 0 0 4 4 4 −1 −1 −1 −1 −1 0 0
500.21.4b1 C 4 0 0 0 4 2ζ5−2+2ζ52 2ζ5−1+2ζ5 −1 ζ5−2−ζ5−ζ52 1+ζ5+2ζ52 ζ5−2+1+2ζ5 −2ζ5−2−1−2ζ5−ζ52 0 0
500.21.4b2 C 4 0 0 0 4 2ζ5−2+2ζ52 2ζ5−1+2ζ5 −1 1+ζ5+2ζ52 ζ5−2−ζ5−ζ52 −2ζ5−2−1−2ζ5−ζ52 ζ5−2+1+2ζ5 0 0
500.21.4b3 C 4 0 0 0 4 2ζ5−1+2ζ5 2ζ5−2+2ζ52 −1 −2ζ5−2−1−2ζ5−ζ52 ζ5−2+1+2ζ5 ζ5−2−ζ5−ζ52 1+ζ5+2ζ52 0 0
500.21.4b4 C 4 0 0 0 4 2ζ5−1+2ζ5 2ζ5−2+2ζ52 −1 ζ5−2+1+2ζ5 −2ζ5−2−1−2ζ5−ζ52 1+ζ5+2ζ52 ζ5−2−ζ5−ζ52 0 0
500.21.20a R 20 0 0 0 −5 0 0 0 0 0 0 0 0 0

magma: CharacterTable(G);