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Properties

Label 25T14
Degree $25$
Order $125$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $\He_5$

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Show commands: Magma

magma: G := TransitiveGroup(25, 14);
 

Group action invariants

Degree $n$:  $25$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $14$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $\He_5$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $5$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,8,11,17,21)(2,9,12,18,22)(3,10,13,19,23)(4,6,14,20,24)(5,7,15,16,25), (1,11,24,7,20)(2,12,25,8,16)(3,13,21,9,17)(4,14,22,10,18)(5,15,23,6,19)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$5$:  $C_5$ x 6
$25$:  25T2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $C_5$

Low degree siblings

25T14 x 5

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $125=5^{3}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $2$
Label:  125.3
magma: IdentifyGroup(G);
 
Character table:

1A 5A1 5A-1 5A2 5A-2 5B1 5B-1 5B2 5B-2 5C1 5C-1 5C2 5C-2 5D1 5D-1 5D2 5D-2 5E1 5E-1 5E2 5E-2 5F1 5F-1 5F2 5F-2 5G1 5G-1 5G2 5G-2
Size 1 1 1 1 1 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
5 P 1A 5A2 5A-2 5A-1 5A1 5C-2 5G-2 5D2 5E2 5G1 5C2 5F1 5E1 5E-2 5G-1 5C-1 5F2 5F-1 5B-1 5B-2 5C1 5D-1 5D1 5B1 5E-1 5B2 5F-2 5D-2 5G2
Type
125.3.1a R 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 1 1
125.3.1b1 C 1 1 1 1 1 ζ5−2 ζ52 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5 ζ5−1 ζ52 ζ5−2 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−2 ζ52 ζ5 ζ5−1 1 1 1 1
125.3.1b2 C 1 1 1 1 1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−1 ζ5 ζ5−2 ζ52 ζ52 ζ5−2 ζ5−1 ζ5 1 1 1 1
125.3.1b3 C 1 1 1 1 1 ζ5−1 ζ5 ζ5−2 ζ52 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−1 ζ5 ζ5−2 ζ52 1 1 1 1
125.3.1b4 C 1 1 1 1 1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−2 ζ52 ζ52 ζ5−2 ζ5−1 ζ5 ζ52 ζ5−2 ζ5−1 ζ5 ζ5 ζ5−1 ζ52 ζ5−2 1 1 1 1
125.3.1c1 C 1 1 1 1 1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−2 ζ52 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−2 ζ52 ζ5 ζ5−1 1 1 1 1 ζ52 ζ5−2 ζ5−1 ζ5
125.3.1c2 C 1 1 1 1 1 ζ52 ζ5−2 ζ5−1 ζ5 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−2 ζ52 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−1 ζ5 1 1 1 1 ζ5−2 ζ52 ζ5 ζ5−1
125.3.1c3 C 1 1 1 1 1 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−1 ζ5 ζ5−2 ζ52 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−2 ζ52 1 1 1 1 ζ5 ζ5−1 ζ52 ζ5−2
125.3.1c4 C 1 1 1 1 1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−2 ζ52 ζ5 ζ5−1 ζ52 ζ5−2 1 1 1 1 ζ5−1 ζ5 ζ5−2 ζ52
125.3.1d1 C 1 1 1 1 1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−2 ζ52 ζ5 ζ5−1 1 1 1 1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−1 ζ5 ζ5−2 ζ52
125.3.1d2 C 1 1 1 1 1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5 ζ5−1 ζ52 ζ5−2 ζ52 ζ5−2 ζ5−1 ζ5 1 1 1 1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5 ζ5−1 ζ52 ζ5−2
125.3.1d3 C 1 1 1 1 1 ζ5−1 ζ5 ζ5−2 ζ52 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−1 ζ5 ζ5−2 ζ52 1 1 1 1 ζ5 ζ5−1 ζ52 ζ5−2 ζ52 ζ5−2 ζ5−1 ζ5
125.3.1d4 C 1 1 1 1 1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−2 ζ52 ζ5 ζ5−1 ζ5 ζ5−1 ζ52 ζ5−2 1 1 1 1 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−2 ζ52 ζ5 ζ5−1
125.3.1e1 C 1 1 1 1 1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5 ζ5−1 ζ52 ζ5−2 1 1 1 1 ζ5−1 ζ5 ζ5−2 ζ52 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−2 ζ52 ζ5 ζ5−1
125.3.1e2 C 1 1 1 1 1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−1 ζ5 ζ5−2 ζ52 1 1 1 1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−2 ζ52 ζ52 ζ5−2 ζ5−1 ζ5
125.3.1e3 C 1 1 1 1 1 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−2 ζ52 ζ5 ζ5−1 1 1 1 1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−1 ζ5 ζ5−2 ζ52
125.3.1e4 C 1 1 1 1 1 ζ5 ζ5−1 ζ52 ζ5−2 ζ52 ζ5−2 ζ5−1 ζ5 1 1 1 1 ζ5−2 ζ52 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5 ζ5−1 ζ52 ζ5−2
125.3.1f1 C 1 1 1 1 1 ζ5−2 ζ52 ζ5 ζ5−1 1 1 1 1 ζ5−1 ζ5 ζ5−2 ζ52 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−1 ζ5 ζ5−2 ζ52 ζ5 ζ5−1 ζ52 ζ5−2
125.3.1f2 C 1 1 1 1 1 ζ52 ζ5−2 ζ5−1 ζ5 1 1 1 1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−2 ζ52 ζ5 ζ5−1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−2 ζ52
125.3.1f3 C 1 1 1 1 1 ζ5−1 ζ5 ζ5−2 ζ52 1 1 1 1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5 ζ5−1 ζ52 ζ5−2 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−2 ζ52 ζ5 ζ5−1
125.3.1f4 C 1 1 1 1 1 ζ5 ζ5−1 ζ52 ζ5−2 1 1 1 1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−2 ζ52 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−1 ζ5
125.3.1g1 C 1 1 1 1 1 1 1 1 1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5 ζ5−1 ζ52 ζ5−2 ζ5 ζ5−1 ζ52 ζ5−2
125.3.1g2 C 1 1 1 1 1 1 1 1 1 ζ52 ζ5−2 ζ5−1 ζ5 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−1 ζ5 ζ5−2 ζ52
125.3.1g3 C 1 1 1 1 1 1 1 1 1 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−2 ζ52 ζ5 ζ5−1
125.3.1g4 C 1 1 1 1 1 1 1 1 1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5 ζ5−1 ζ52 ζ5−2 ζ52 ζ5−2 ζ5−1 ζ5 ζ52 ζ5−2 ζ5−1 ζ5 ζ52 ζ5−2 ζ5−1 ζ5
125.3.5a1 C 5 5ζ5−2 5ζ52 5ζ5 5ζ5−1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
125.3.5a2 C 5 5ζ52 5ζ5−2 5ζ5−1 5ζ5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
125.3.5a3 C 5 5ζ5−1 5ζ5 5ζ5−2 5ζ52 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
125.3.5a4 C 5 5ζ5 5ζ5−1 5ζ52 5ζ5−2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

magma: CharacterTable(G);