↑

Properties

Label 25T1
Degree $25$
Order $25$
Cyclic yes
Abelian yes
Solvable yes
Primitive no
$p$-group yes
Group: $C_{25}$

Related objects

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(25, 1);
 

Group action invariants

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

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$5$:  $C_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $C_5$

Low degree siblings

There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{25}$ $1$ $1$ $0$ $()$
5A1 $5^{5}$ $1$ $5$ $20$ $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)(11,14,12,15,13)(16,19,17,20,18)(21,24,22,25,23)$
5A-1 $5^{5}$ $1$ $5$ $20$ $( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)(11,15,14,13,12)(16,20,19,18,17)(21,25,24,23,22)$
5A2 $5^{5}$ $1$ $5$ $20$ $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19)(21,23,25,22,24)$
5A-2 $5^{5}$ $1$ $5$ $20$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)$
25A1 $25$ $1$ $25$ $24$ $( 1,14,25, 8,19, 2,15,21, 9,20, 3,11,22,10,16, 4,12,23, 6,17, 5,13,24, 7,18)$
25A-1 $25$ $1$ $25$ $24$ $( 1,16, 8,23,15, 5,20, 7,22,14, 4,19, 6,21,13, 3,18,10,25,12, 2,17, 9,24,11)$
25A2 $25$ $1$ $25$ $24$ $( 1, 8,15,20,22, 4, 6,13,18,25, 2, 9,11,16,23, 5, 7,14,19,21, 3,10,12,17,24)$
25A-2 $25$ $1$ $25$ $24$ $( 1, 9,12,18,21, 4, 7,15,16,24, 2,10,13,19,22, 5, 8,11,17,25, 3, 6,14,20,23)$
25A3 $25$ $1$ $25$ $24$ $( 1,12,21, 7,16, 2,13,22, 8,17, 3,14,23, 9,18, 4,15,24,10,19, 5,11,25, 6,20)$
25A-3 $25$ $1$ $25$ $24$ $( 1,23,20,14, 6, 3,25,17,11, 8, 5,22,19,13,10, 2,24,16,15, 7, 4,21,18,12, 9)$
25A4 $25$ $1$ $25$ $24$ $( 1,18, 7,24,13, 5,17, 6,23,12, 4,16,10,22,11, 3,20, 9,21,15, 2,19, 8,25,14)$
25A-4 $25$ $1$ $25$ $24$ $( 1, 7,13,17,23, 4,10,11,20,21, 2, 8,14,18,24, 5, 6,12,16,22, 3, 9,15,19,25)$
25A6 $25$ $1$ $25$ $24$ $( 1,13,23,10,20, 2,14,24, 6,16, 3,15,25, 7,17, 4,11,21, 8,18, 5,12,22, 9,19)$
25A-6 $25$ $1$ $25$ $24$ $( 1,24,17,12,10, 3,21,19,14, 7, 5,23,16,11, 9, 2,25,18,13, 6, 4,22,20,15, 8)$
25A7 $25$ $1$ $25$ $24$ $( 1,17,10,21,14, 5,16, 9,25,13, 4,20, 8,24,12, 3,19, 7,23,11, 2,18, 6,22,15)$
25A-7 $25$ $1$ $25$ $24$ $( 1, 6,11,19,24, 4, 9,14,17,22, 2, 7,12,20,25, 5,10,15,18,23, 3, 8,13,16,21)$
25A8 $25$ $1$ $25$ $24$ $( 1,19, 9,22,12, 5,18, 8,21,11, 4,17, 7,25,15, 3,16, 6,24,14, 2,20,10,23,13)$
25A-8 $25$ $1$ $25$ $24$ $( 1,10,14,16,25, 4, 8,12,19,23, 2, 6,15,17,21, 5, 9,13,20,24, 3, 7,11,18,22)$
25A9 $25$ $1$ $25$ $24$ $( 1,11,24, 9,17, 2,12,25,10,18, 3,13,21, 6,19, 4,14,22, 7,20, 5,15,23, 8,16)$
25A-9 $25$ $1$ $25$ $24$ $( 1,25,19,15, 9, 3,22,16,12, 6, 5,24,18,14, 8, 2,21,20,11,10, 4,23,17,13, 7)$
25A11 $25$ $1$ $25$ $24$ $( 1,20, 6,25,11, 5,19,10,24,15, 4,18, 9,23,14, 3,17, 8,22,13, 2,16, 7,21,12)$
25A-11 $25$ $1$ $25$ $24$ $( 1,21,16,13, 8, 3,23,18,15,10, 5,25,20,12, 7, 2,22,17,14, 9, 4,24,19,11, 6)$
25A12 $25$ $1$ $25$ $24$ $( 1,15,22, 6,18, 2,11,23, 7,19, 3,12,24, 8,20, 4,13,25, 9,16, 5,14,21,10,17)$
25A-12 $25$ $1$ $25$ $24$ $( 1,22,18,11, 7, 3,24,20,13, 9, 5,21,17,15, 6, 2,23,19,12, 8, 4,25,16,14,10)$

Malle's constant $a(G)$:     $1/20$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $25=5^{2}$
magma: Order(G);
 
Cyclic:  yes
magma: IsCyclic(G);
 
Abelian:  yes
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $1$
Label:  25.1
magma: IdentifyGroup(G);
 
Character table:

1A 5A1 5A-1 5A2 5A-2 25A1 25A-1 25A2 25A-2 25A3 25A-3 25A4 25A-4 25A6 25A-6 25A7 25A-7 25A8 25A-8 25A9 25A-9 25A11 25A-11 25A12 25A-12
Size 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 P 1A 5A2 5A1 5A-2 5A-1 25A4 25A6 25A12 25A7 25A-11 25A-7 25A-4 25A-8 25A9 25A-12 25A1 25A-3 25A-9 25A2 25A-6 25A8 25A11 25A3 25A-1 25A-2
Type
25.1.1a 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
25.1.1b1 C 1 1 1 1 1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−1 ζ5 ζ52 ζ5−2 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−1 ζ5 ζ52 ζ5−2 ζ5−2 ζ52 ζ5 ζ5−1
25.1.1b2 C 1 1 1 1 1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5 ζ5−1 ζ5−2 ζ52 ζ52 ζ5−2 ζ5−1 ζ5 ζ5 ζ5−1 ζ5−2 ζ52 ζ52 ζ5−2 ζ5−1 ζ5
25.1.1b3 C 1 1 1 1 1 ζ5−1 ζ5 ζ5−2 ζ52 ζ52 ζ5−2 ζ5 ζ5−1 ζ5−1 ζ5 ζ5−2 ζ52 ζ52 ζ5−2 ζ5 ζ5−1 ζ5−1 ζ5 ζ5−2 ζ52
25.1.1b4 C 1 1 1 1 1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−2 ζ52 ζ5−1 ζ5 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−2 ζ52 ζ5−1 ζ5 ζ5 ζ5−1 ζ52 ζ5−2
25.1.1c1 C 1 ζ25−10 ζ2510 ζ255 ζ25−5 ζ25−12 ζ2512 ζ25 ζ25−1 ζ25−11 ζ2511 ζ252 ζ25−2 ζ253 ζ25−3 ζ25−9 ζ259 ζ254 ζ25−4 ζ25−8 ζ258 ζ25−7 ζ257 ζ256 ζ25−6
25.1.1c2 C 1 ζ2510 ζ25−10 ζ25−5 ζ255 ζ2512 ζ25−12 ζ25−1 ζ25 ζ2511 ζ25−11 ζ25−2 ζ252 ζ25−3 ζ253 ζ259 ζ25−9 ζ25−4 ζ254 ζ258 ζ25−8 ζ257 ζ25−7 ζ25−6 ζ256
25.1.1c3 C 1 ζ25−10 ζ2510 ζ255 ζ25−5 ζ258 ζ25−8 ζ25−9 ζ259 ζ25−1 ζ25 ζ257 ζ25−7 ζ25−2 ζ252 ζ256 ζ25−6 ζ25−11 ζ2511 ζ25−3 ζ253 ζ25−12 ζ2512 ζ25−4 ζ254
25.1.1c4 C 1 ζ2510 ζ25−10 ζ25−5 ζ255 ζ25−8 ζ258 ζ259 ζ25−9 ζ25 ζ25−1 ζ25−7 ζ257 ζ252 ζ25−2 ζ25−6 ζ256 ζ2511 ζ25−11 ζ253 ζ25−3 ζ2512 ζ25−12 ζ254 ζ25−4
25.1.1c5 C 1 ζ25−10 ζ2510 ζ255 ζ25−5 ζ25−7 ζ257 ζ2511 ζ25−11 ζ254 ζ25−4 ζ25−3 ζ253 ζ258 ζ25−8 ζ25 ζ25−1 ζ25−6 ζ256 ζ2512 ζ25−12 ζ25−2 ζ252 ζ25−9 ζ259
25.1.1c6 C 1 ζ2510 ζ25−10 ζ25−5 ζ255 ζ257 ζ25−7 ζ25−11 ζ2511 ζ25−4 ζ254 ζ253 ζ25−3 ζ25−8 ζ258 ζ25−1 ζ25 ζ256 ζ25−6 ζ25−12 ζ2512 ζ252 ζ25−2 ζ259 ζ25−9
25.1.1c7 C 1 ζ25−10 ζ2510 ζ255 ζ25−5 ζ253 ζ25−3 ζ256 ζ25−6 ζ259 ζ25−9 ζ2512 ζ25−12 ζ25−7 ζ257 ζ25−4 ζ254 ζ25−1 ζ25 ζ252 ζ25−2 ζ258 ζ25−8 ζ2511 ζ25−11
25.1.1c8 C 1 ζ2510 ζ25−10 ζ25−5 ζ255 ζ25−3 ζ253 ζ25−6 ζ256 ζ25−9 ζ259 ζ25−12 ζ2512 ζ257 ζ25−7 ζ254 ζ25−4 ζ25 ζ25−1 ζ25−2 ζ252 ζ25−8 ζ258 ζ25−11 ζ2511
25.1.1c9 C 1 ζ25−10 ζ2510 ζ255 ζ25−5 ζ25−2 ζ252 ζ25−4 ζ254 ζ25−6 ζ256 ζ25−8 ζ258 ζ25−12 ζ2512 ζ2511 ζ25−11 ζ259 ζ25−9 ζ257 ζ25−7 ζ253 ζ25−3 ζ25 ζ25−1
25.1.1c10 C 1 ζ2510 ζ25−10 ζ25−5 ζ255 ζ252 ζ25−2 ζ254 ζ25−4 ζ256 ζ25−6 ζ258 ζ25−8 ζ2512 ζ25−12 ζ25−11 ζ2511 ζ25−9 ζ259 ζ25−7 ζ257 ζ25−3 ζ253 ζ25−1 ζ25
25.1.1c11 C 1 ζ25−5 ζ255 ζ25−10 ζ2510 ζ25−11 ζ2511 ζ253 ζ25−3 ζ25−8 ζ258 ζ256 ζ25−6 ζ259 ζ25−9 ζ25−2 ζ252 ζ2512 ζ25−12 ζ25 ζ25−1 ζ254 ζ25−4 ζ25−7 ζ257
25.1.1c12 C 1 ζ255 ζ25−5 ζ2510 ζ25−10 ζ2511 ζ25−11 ζ25−3 ζ253 ζ258 ζ25−8 ζ25−6 ζ256 ζ25−9 ζ259 ζ252 ζ25−2 ζ25−12 ζ2512 ζ25−1 ζ25 ζ25−4 ζ254 ζ257 ζ25−7
25.1.1c13 C 1 ζ25−5 ζ255 ζ25−10 ζ2510 ζ259 ζ25−9 ζ25−7 ζ257 ζ252 ζ25−2 ζ2511 ζ25−11 ζ254 ζ25−4 ζ25−12 ζ2512 ζ25−3 ζ253 ζ256 ζ25−6 ζ25−1 ζ25 ζ258 ζ25−8
25.1.1c14 C 1 ζ255 ζ25−5 ζ2510 ζ25−10 ζ25−9 ζ259 ζ257 ζ25−7 ζ25−2 ζ252 ζ25−11 ζ2511 ζ25−4 ζ254 ζ2512 ζ25−12 ζ253 ζ25−3 ζ25−6 ζ256 ζ25 ζ25−1 ζ25−8 ζ258
25.1.1c15 C 1 ζ25−5 ζ255 ζ25−10 ζ2510 ζ25−6 ζ256 ζ25−12 ζ2512 ζ257 ζ25−7 ζ25 ζ25−1 ζ25−11 ζ2511 ζ258 ζ25−8 ζ252 ζ25−2 ζ25−4 ζ254 ζ259 ζ25−9 ζ253 ζ25−3
25.1.1c16 C 1 ζ255 ζ25−5 ζ2510 ζ25−10 ζ256 ζ25−6 ζ2512 ζ25−12 ζ25−7 ζ257 ζ25−1 ζ25 ζ2511 ζ25−11 ζ25−8 ζ258 ζ25−2 ζ252 ζ254 ζ25−4 ζ25−9 ζ259 ζ25−3 ζ253
25.1.1c17 C 1 ζ25−5 ζ255 ζ25−10 ζ2510 ζ254 ζ25−4 ζ258 ζ25−8 ζ2512 ζ25−12 ζ25−9 ζ259 ζ25−1 ζ25 ζ253 ζ25−3 ζ257 ζ25−7 ζ2511 ζ25−11 ζ25−6 ζ256 ζ25−2 ζ252
25.1.1c18 C 1 ζ255 ζ25−5 ζ2510 ζ25−10 ζ25−4 ζ254 ζ25−8 ζ258 ζ25−12 ζ2512 ζ259 ζ25−9 ζ25 ζ25−1 ζ25−3 ζ253 ζ25−7 ζ257 ζ25−11 ζ2511 ζ256 ζ25−6 ζ252 ζ25−2
25.1.1c19 C 1 ζ25−5 ζ255 ζ25−10 ζ2510 ζ25−1 ζ25 ζ25−2 ζ252 ζ25−3 ζ253 ζ25−4 ζ254 ζ25−6 ζ256 ζ25−7 ζ257 ζ25−8 ζ258 ζ25−9 ζ259 ζ25−11 ζ2511 ζ25−12 ζ2512
25.1.1c20 C 1 ζ255 ζ25−5 ζ2510 ζ25−10 ζ25 ζ25−1 ζ252 ζ25−2 ζ253 ζ25−3 ζ254 ζ25−4 ζ256 ζ25−6 ζ257 ζ25−7 ζ258 ζ25−8 ζ259 ζ25−9 ζ2511 ζ25−11 ζ2512 ζ25−12

magma: CharacterTable(G);