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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

|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 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, 5 $ $12$ $5$ $( 1, 6,11,16,21)( 2, 7,12,17,22)( 3, 8,13,18,23)( 4, 9,14,19,24) ( 5,10,15,20,25)$
$ 5, 5, 5, 5, 5 $ $12$ $5$ $( 1,24,17,15, 8)( 2,25,18,11, 9)( 3,21,19,12,10)( 4,22,20,13, 6) ( 5,23,16,14, 7)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $25$ $2$ $( 2, 5)( 3, 4)( 6,21)( 7,25)( 8,24)( 9,23)(10,22)(11,16)(12,20)(13,19)(14,18) (15,17)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1 $ $50$ $3$ $( 2, 6,25)( 3,11,19)( 4,16,13)( 5,21, 7)( 8,10,20)( 9,15,14)(12,24,22) (17,18,23)$
$ 6, 6, 6, 6, 1 $ $50$ $6$ $( 2,21,25, 5, 6, 7)( 3,16,19, 4,11,13)( 8,22,20,24,10,12)( 9,17,14,23,15,18)$
$ 4, 4, 4, 4, 4, 4, 1 $ $75$ $4$ $( 2, 3, 5, 4)( 6,19,21,13)( 7,16,25,11)( 8,18,24,14)( 9,20,23,12)(10,17,22,15)$
$ 4, 4, 4, 4, 4, 4, 1 $ $75$ $4$ $( 2, 4, 5, 3)( 6,13,21,19)( 7,11,25,16)( 8,14,24,18)( 9,12,23,20)(10,15,22,17)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $15$ $2$ $( 6,25)( 7,21)( 8,22)( 9,23)(10,24)(11,19)(12,20)(13,16)(14,17)(15,18)$
$ 10, 10, 5 $ $60$ $10$ $( 1, 6, 5,10, 4, 9, 3, 8, 2, 7)(11,24,15,23,14,22,13,21,12,25)(16,18,20,17,19)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $15$ $2$ $( 2, 5)( 3, 4)( 6, 7)( 8,10)(11,13)(14,15)(16,19)(17,18)(21,25)(22,24)$
$ 10, 10, 5 $ $60$ $10$ $( 1, 6,12,17,23, 3, 9,14,20,25)( 2,10,13,16,24)( 4, 8,15,19,21, 5, 7,11,18,22)$
$ 4, 4, 4, 4, 4, 4, 1 $ $25$ $4$ $( 2, 3, 5, 4)( 6,11,21,16)( 7,13,25,19)( 8,15,24,17)( 9,12,23,20)(10,14,22,18)$
$ 4, 4, 4, 4, 4, 4, 1 $ $25$ $4$ $( 2, 4, 5, 3)( 6,16,21,11)( 7,19,25,13)( 8,17,24,15)( 9,20,23,12)(10,18,22,14)$
$ 12, 12, 1 $ $50$ $12$ $( 2,11, 7, 4, 6,19, 5,16,25, 3,21,13)( 8,14,12,17,10, 9,24,18,20,15,22,23)$
$ 12, 12, 1 $ $50$ $12$ $( 2,16, 7, 3, 6,13, 5,11,25, 4,21,19)( 8,18,12,15,10,23,24,14,20,17,22, 9)$

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);