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Properties

Label 25T43
Degree $25$
Order $600$
Cyclic no
Abelian no
Solvable yes
Primitive yes
$p$-group no
Group: $C_5^2:(C_4\times S_3)$

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Show commands: Magma

magma: G := TransitiveGroup(25, 43);
 

Group action invariants

Degree $n$:  $25$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $43$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_5^2:(C_4\times S_3)$
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:  (1,2,4,3)(6,20,24,15)(7,17,23,13)(8,19,22,11)(9,16,21,14)(10,18,25,12), (1,11)(2,17)(3,23)(5,10)(7,12)(8,18)(9,24)(14,19)(15,25)(16,21)
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, $C_2^2$
$6$:  $S_3$
$8$:  $C_4\times C_2$
$12$:  $D_{6}$
$24$:  $S_3 \times C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: None

Low degree siblings

15T27 x 2, 30T150 x 2, 30T153 x 2, 30T155 x 2

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 $2^{10},1^{5}$ $15$ $2$ $10$ $( 1,11)( 2,17)( 3,23)( 5,10)( 7,12)( 8,18)( 9,24)(14,19)(15,25)(16,21)$
2B $2^{10},1^{5}$ $15$ $2$ $10$ $( 1,24)( 2,18)( 3,12)( 4, 6)( 7,23)( 8,17)( 9,11)(13,22)(14,16)(19,21)$
2C $2^{12},1$ $25$ $2$ $12$ $( 1, 2)( 3, 5)( 6,22)( 7,21)( 8,25)( 9,24)(10,23)(11,17)(12,16)(13,20)(14,19)(15,18)$
3A $3^{8},1$ $50$ $3$ $16$ $( 1, 3,13)( 2, 8, 7)( 4,18,25)( 5,23,19)( 6,22,14)( 9,12,21)(10,17,20)(11,16,15)$
4A1 $4^{6},1$ $25$ $4$ $18$ $( 1, 3, 2, 5)( 6,13,22,20)( 7,15,21,18)( 8,12,25,16)( 9,14,24,19)(10,11,23,17)$
4A-1 $4^{6},1$ $25$ $4$ $18$ $( 1, 5, 2, 3)( 6,20,22,13)( 7,18,21,15)( 8,16,25,12)( 9,19,24,14)(10,17,23,11)$
4B1 $4^{6},1$ $75$ $4$ $18$ $( 1,19,18, 5)( 2, 7,17,12)( 3,25,16,24)( 4,13,20, 6)( 8,10,11,14)( 9,23,15,21)$
4B-1 $4^{6},1$ $75$ $4$ $18$ $( 1, 7, 9, 3)( 2,19, 8,16)( 4,13, 6,22)( 5,25,10,15)(11,12,24,23)(14,18,21,17)$
5A $5^{5}$ $12$ $5$ $20$ $( 1,20, 9,23,12)( 2,16,10,24,13)( 3,17, 6,25,14)( 4,18, 7,21,15)( 5,19, 8,22,11)$
5B $5^{5}$ $12$ $5$ $20$ $( 1, 6,11,16,21)( 2, 7,12,17,22)( 3, 8,13,18,23)( 4, 9,14,19,24)( 5,10,15,20,25)$
6A $6^{4},1$ $50$ $6$ $20$ $( 1, 5,10,11,12, 7)( 2,25, 4,15,17,13)( 3,20,23,14,22,19)( 8,21,24, 9,16,18)$
10A $10^{2},5$ $60$ $10$ $22$ $( 1,18,15, 2,24,11, 8,25,17, 9)( 3, 5,12,14,21,23,10, 7,19,16)( 4, 6,13,20,22)$
10B $10^{2},5$ $60$ $10$ $22$ $( 1,20,11, 5,21,15, 6,25,16,10)( 2,14,12,24,22, 9, 7,19,17, 4)( 3, 8,13,18,23)$
12A1 $12^{2},1$ $50$ $12$ $22$ $( 2,16, 7, 3, 6,13, 5,11,25, 4,21,19)( 8,18,12,15,10,23,24,14,20,17,22, 9)$
12A-1 $12^{2},1$ $50$ $12$ $22$ $( 1, 2,12,18,16,21,14,13, 3,22,24,19)( 4, 7, 5,17, 6,20,11, 8,15,23, 9,25)$

Malle's constant $a(G)$:     $1/10$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $600=2^{3} \cdot 3 \cdot 5^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  600.151
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 4A1 4A-1 4B1 4B-1 5A 5B 6A 10A 10B 12A1 12A-1
Size 1 15 15 25 50 25 25 75 75 12 12 50 60 60 50 50
2 P 1A 1A 1A 1A 3A 2C 2C 2C 2C 5A 5B 3A 5A 5B 6A 6A
3 P 1A 2A 2B 2C 1A 4A-1 4A1 4B-1 4B1 5A 5B 2C 10A 10B 4A1 4A-1
5 P 1A 2A 2B 2C 3A 4A1 4A-1 4B1 4B-1 1A 1A 6A 2A 2B 12A1 12A-1
Type
600.151.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
600.151.1b R 1 −1 −1 1 1 −1 −1 1 1 1 1 1 −1 −1 −1 −1
600.151.1c R 1 −1 −1 1 1 1 1 −1 −1 1 1 1 −1 −1 1 1
600.151.1d R 1 1 1 1 1 −1 −1 −1 −1 1 1 1 1 1 −1 −1
600.151.1e1 C 1 −1 1 −1 1 −i i −i i 1 1 −1 −1 1 i −i
600.151.1e2 C 1 −1 1 −1 1 i −i i −i 1 1 −1 −1 1 −i i
600.151.1f1 C 1 1 −1 −1 1 −i i i −i 1 1 −1 1 −1 i −i
600.151.1f2 C 1 1 −1 −1 1 i −i −i i 1 1 −1 1 −1 −i i
600.151.2a R 2 0 0 2 −1 2 2 0 0 2 2 −1 0 0 −1 −1
600.151.2b R 2 0 0 2 −1 −2 −2 0 0 2 2 −1 0 0 1 1
600.151.2c1 C 2 0 0 −2 −1 −2i 2i 0 0 2 2 1 0 0 −i i
600.151.2c2 C 2 0 0 −2 −1 2i −2i 0 0 2 2 1 0 0 i −i
600.151.12a R 12 0 4 0 0 0 0 0 0 2 −3 0 0 −1 0 0
600.151.12b R 12 4 0 0 0 0 0 0 0 −3 2 0 −1 0 0 0
600.151.12c R 12 −4 0 0 0 0 0 0 0 −3 2 0 1 0 0 0
600.151.12d R 12 0 −4 0 0 0 0 0 0 2 −3 0 0 1 0 0

magma: CharacterTable(G);