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Properties

Label 25T38
Degree $25$
Order $500$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_5^2:D_{10}$

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Show commands: Magma

magma: G := TransitiveGroup(25, 38);
 

Group action invariants

Degree $n$:  $25$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $38$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_5^2:D_{10}$
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:  (1,4,2,5,3)(6,25,10,21,9,22,8,23,7,24)(11,18,14,20,12,17,15,19,13,16), (1,14,10,23,20,4,15,7,25,16)(2,11,9,22,17,3,13,8,21,19)(5,12,6,24,18)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$10$:  $D_{5}$ x 2
$20$:  $D_{10}$ x 2
$100$:  $D_5^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $D_{5}$

Low degree siblings

25T38

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

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:  500.27
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 5A1 5A2 5B1 5B2 5C1 5C2 5D1 5D2 5E1 5E2 10A1 10A3 10B1 10B3 10C1 10C3
Size 1 25 25 25 2 2 10 10 10 10 20 20 20 20 50 50 50 50 50 50
2 P 1A 1A 1A 1A 5A2 5A1 5B2 5B1 5C2 5C1 5D2 5D1 5E2 5E1 5B1 5A2 5A1 5C1 5B2 5C2
5 P 1A 2A 2B 2C 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 2A 2B 2B 2C 2A 2C
Type
500.27.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
500.27.1b R 1 −1 −1 1 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 1 1
500.27.1c R 1 −1 1 −1 1 1 1 1 1 1 1 1 1 1 −1 −1 1 1 −1 −1
500.27.1d R 1 1 −1 −1 1 1 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1
500.27.2a1 R 2 0 0 2 2 2 2 2 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 0 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52
500.27.2a2 R 2 0 0 2 2 2 2 2 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 0 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5
500.27.2b1 R 2 2 0 0 2 2 ζ5−2+ζ52 ζ5−1+ζ5 2 2 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 0 0 0 0
500.27.2b2 R 2 2 0 0 2 2 ζ5−1+ζ5 ζ5−2+ζ52 2 2 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 0 0 0 0
500.27.2c1 R 2 −2 0 0 2 2 ζ5−2+ζ52 ζ5−1+ζ5 2 2 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 0 0 0 0
500.27.2c2 R 2 −2 0 0 2 2 ζ5−1+ζ5 ζ5−2+ζ52 2 2 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 0 0 0 0
500.27.2d1 R 2 0 0 −2 2 2 2 2 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 0 0 0 0 −ζ5−1−ζ5 −ζ5−2−ζ52
500.27.2d2 R 2 0 0 −2 2 2 2 2 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 0 0 0 0 −ζ5−2−ζ52 −ζ5−1−ζ5
500.27.4a1 R 4 0 0 0 4 4 2ζ5−2+2ζ52 2ζ5−1+2ζ5 2ζ5−2+2ζ52 2ζ5−1+2ζ5 −1 −1 ζ5−2+2+ζ52 −ζ5−2+1−ζ52 0 0 0 0 0 0
500.27.4a2 R 4 0 0 0 4 4 2ζ5−1+2ζ5 2ζ5−2+2ζ52 2ζ5−1+2ζ5 2ζ5−2+2ζ52 −1 −1 −ζ5−2+1−ζ52 ζ5−2+2+ζ52 0 0 0 0 0 0
500.27.4b1 R 4 0 0 0 4 4 2ζ5−2+2ζ52 2ζ5−1+2ζ5 2ζ5−1+2ζ5 2ζ5−2+2ζ52 ζ5−2+2+ζ52 −ζ5−2+1−ζ52 −1 −1 0 0 0 0 0 0
500.27.4b2 R 4 0 0 0 4 4 2ζ5−1+2ζ5 2ζ5−2+2ζ52 2ζ5−2+2ζ52 2ζ5−1+2ζ5 −ζ5−2+1−ζ52 ζ5−2+2+ζ52 −1 −1 0 0 0 0 0 0
500.27.10a1 R 10 0 2 0 5ζ5−2+5ζ52 5ζ5−1+5ζ5 0 0 0 0 0 0 0 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52 0 0
500.27.10a2 R 10 0 2 0 5ζ5−1+5ζ5 5ζ5−2+5ζ52 0 0 0 0 0 0 0 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5 0 0
500.27.10b1 R 10 0 −2 0 5ζ5−2+5ζ52 5ζ5−1+5ζ5 0 0 0 0 0 0 0 0 0 0 −ζ5−1−ζ5 −ζ5−2−ζ52 0 0
500.27.10b2 R 10 0 −2 0 5ζ5−1+5ζ5 5ζ5−2+5ζ52 0 0 0 0 0 0 0 0 0 0 −ζ5−2−ζ52 −ζ5−1−ζ5 0 0

magma: CharacterTable(G);