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Properties

Label 25T30
Degree $25$
Order $400$
Cyclic no
Abelian no
Solvable yes
Primitive yes
$p$-group no
Group: $D_5^2.C_2^2$

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Show commands: Magma

magma: G := TransitiveGroup(25, 30);
 

Group action invariants

Degree $n$:  $25$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $30$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_5^2.C_2^2$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  yes
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,20,22,11,18,7,14,3,10,24)(2,15,23,6,19)(4,5,25,21,16,17,12,13,8,9), (1,7,25,19)(2,10,24,16)(3,8,23,18)(4,6,22,20)(5,9,21,17)(11,12,15,14), (1,15)(2,25)(3,10)(4,20)(6,13)(7,23)(9,18)(12,21)(14,16)(17,24)
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$8$:  $C_2^3$
$16$:  $Q_8:C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: None

Low degree siblings

10T27 x 3, 20T90 x 3, 20T96 x 3, 20T97 x 3, 40T393 x 3, 40T394 x 3, 40T395 x 3

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 $2^{10},1^{5}$ $10$ $2$ $10$ $( 1, 9)( 2,24)( 3,14)( 5,19)( 7,21)( 8,11)(10,16)(12,23)(15,18)(17,25)$
2B $2^{10},1^{5}$ $10$ $2$ $10$ $( 6,21)( 7,22)( 8,23)( 9,24)(10,25)(11,16)(12,17)(13,18)(14,19)(15,20)$
2C $2^{10},1^{5}$ $10$ $2$ $10$ $( 1,22)( 3, 7)( 4,12)( 5,17)( 6,23)( 9,13)(10,18)(11,24)(15,19)(16,25)$
2D $2^{12},1$ $25$ $2$ $12$ $( 1, 3)( 4, 5)( 6,23)( 7,22)( 8,21)( 9,25)(10,24)(11,18)(12,17)(13,16)(14,20)(15,19)$
4A1 $4^{6},1$ $25$ $4$ $18$ $( 1, 2, 4, 3)( 6,12,24,18)( 7,14,23,16)( 8,11,22,19)( 9,13,21,17)(10,15,25,20)$
4A-1 $4^{6},1$ $25$ $4$ $18$ $( 2, 4, 5, 3)( 6,16,21,11)( 7,19,25,13)( 8,17,24,15)( 9,20,23,12)(10,18,22,14)$
4B $4^{6},1$ $50$ $4$ $18$ $( 1, 3, 4, 2)( 6,13,24,17)( 7,11,23,19)( 8,14,22,16)( 9,12,21,18)(10,15,25,20)$
4C $4^{6},1$ $50$ $4$ $18$ $( 1,24,15,17)( 2, 4,14,12)( 3, 9,13, 7)( 5,19,11,22)( 6,23,10,18)(16,21,25,20)$
4D $4^{6},1$ $50$ $4$ $18$ $( 1,17, 3,12)( 4,22, 5, 7)( 6,19,23,15)( 8,14,21,20)( 9,24,25,10)(11,16,18,13)$
5A $5^{5}$ $8$ $5$ $20$ $( 1,17, 8,24,15)( 2,18, 9,25,11)( 3,19,10,21,12)( 4,20, 6,22,13)( 5,16, 7,23,14)$
5B $5^{5}$ $8$ $5$ $20$ $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)(11,14,12,15,13)(16,19,17,20,18)(21,24,22,25,23)$
5C $5^{5}$ $8$ $5$ $20$ $( 1,19, 7,25,13)( 2,20, 8,21,14)( 3,16, 9,22,15)( 4,17,10,23,11)( 5,18, 6,24,12)$
10A $10^{2},5$ $40$ $10$ $22$ $( 1, 4,24,22,17,20,15,13, 8, 6)( 2,19,25,12,18,10,11, 3, 9,21)( 5,14,23, 7,16)$
10B $10^{2},5$ $40$ $10$ $22$ $( 1, 4, 2, 5, 3)( 6,24, 7,25, 8,21, 9,22,10,23)(11,19,12,20,13,16,14,17,15,18)$
10C $10^{2},5$ $40$ $10$ $22$ $( 1,12,19, 5, 7,18,25, 6,13,24)( 2,17,20,10, 8,23,21,11,14, 4)( 3,22,16,15, 9)$

Malle's constant $a(G)$:     $1/10$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $400=2^{4} \cdot 5^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  400.207
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 4A1 4A-1 4B 4C 4D 5A 5B 5C 10A 10B 10C
Size 1 10 10 10 25 25 25 50 50 50 8 8 8 40 40 40
2 P 1A 1A 1A 1A 1A 2D 2D 2D 2D 2D 5A 5B 5C 5A 5B 5C
5 P 1A 2A 2B 2C 2D 4A1 4A-1 4B 4C 4D 1A 1A 1A 2A 2B 2C
Type
400.207.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
400.207.1b R 1 −1 −1 −1 1 −1 −1 1 1 1 1 1 1 −1 −1 −1
400.207.1c R 1 −1 −1 1 1 1 1 −1 1 −1 1 1 1 −1 −1 1
400.207.1d R 1 −1 1 −1 1 1 1 1 −1 −1 1 1 1 −1 1 −1
400.207.1e R 1 −1 1 1 1 −1 −1 −1 −1 1 1 1 1 −1 1 1
400.207.1f R 1 1 −1 −1 1 1 1 −1 −1 1 1 1 1 1 −1 −1
400.207.1g R 1 1 −1 1 1 −1 −1 1 −1 −1 1 1 1 1 −1 1
400.207.1h R 1 1 1 −1 1 −1 −1 −1 1 −1 1 1 1 1 1 −1
400.207.2a1 C 2 0 0 0 −2 −2i 2i 0 0 0 2 2 2 0 0 0
400.207.2a2 C 2 0 0 0 −2 2i −2i 0 0 0 2 2 2 0 0 0
400.207.8a R 8 0 0 4 0 0 0 0 0 0 −2 −2 3 0 0 −1
400.207.8b R 8 0 4 0 0 0 0 0 0 0 −2 3 −2 0 −1 0
400.207.8c R 8 4 0 0 0 0 0 0 0 0 3 −2 −2 −1 0 0
400.207.8d R 8 −4 0 0 0 0 0 0 0 0 3 −2 −2 1 0 0
400.207.8e R 8 0 −4 0 0 0 0 0 0 0 −2 3 −2 0 1 0
400.207.8f R 8 0 0 −4 0 0 0 0 0 0 −2 −2 3 0 0 1

magma: CharacterTable(G);