Properties

Label 25T4
25T4 1 17 1->17 24 1->24 2 16 2->16 23 2->23 3 20 3->20 22 3->22 4 19 4->19 21 4->21 5 18 5->18 25 5->25 6 12 6->12 6->17 7 11 7->11 7->16 8 15 8->15 8->20 9 14 9->14 9->19 10 13 10->13 10->18 11->12 13->15 21->22 23->25
Degree 2525
Order 5050
Cyclic no
Abelian no
Solvable yes
Primitive no
pp-group no
Group: D25D_{25}

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Copy content magma:G := TransitiveGroup(25, 4);
 

Group invariants

Abstract group:  D25D_{25}
Copy content magma:IdentifyGroup(G);
 
Order:  50=25250=2 \cdot 5^{2}
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  yes
Copy content magma:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree nn:  2525
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number tt:  44
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  11
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
#Aut(F/K)\card{\Aut(F/K)}:  11
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,17)(2,16)(3,20)(4,19)(5,18)(6,12)(7,11)(8,15)(9,14)(10,13)(21,22)(23,25)(1,17)(2,16)(3,20)(4,19)(5,18)(6,12)(7,11)(8,15)(9,14)(10,13)(21,22)(23,25), (1,24)(2,23)(3,22)(4,21)(5,25)(6,17)(7,16)(8,20)(9,19)(10,18)(11,12)(13,15)(1,24)(2,23)(3,22)(4,21)(5,25)(6,17)(7,16)(8,20)(9,19)(10,18)(11,12)(13,15)
Copy content magma:Generators(G);
 

Low degree resolvents

#(G/N)\card{(G/N)}Galois groups for stem field(s)
22C2C_2
1010D5D_{5}

Resolvents shown for degrees 47\leq 47

Subfields

Degree 5: D5D_{5}

Low degree siblings

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A 1251^{25} 11 11 00 ()()
2A 212,12^{12},1 2525 22 1212 (2,5)(3,4)(6,24)(7,23)(8,22)(9,21)(10,25)(11,17)(12,16)(13,20)(14,19)(15,18)( 2, 5)( 3, 4)( 6,24)( 7,23)( 8,22)( 9,21)(10,25)(11,17)(12,16)(13,20)(14,19)(15,18)
5A1 555^{5} 22 55 2020 (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)( 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)
5A2 555^{5} 22 55 2020 (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)( 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)
25A1 2525 22 2525 2424 (1,23,20,14,8,2,24,16,15,9,3,25,17,11,10,4,21,18,12,6,5,22,19,13,7)( 1,23,20,14, 8, 2,24,16,15, 9, 3,25,17,11,10, 4,21,18,12, 6, 5,22,19,13, 7)
25A2 2525 22 2525 2424 (1,20,8,24,15,3,17,10,21,12,5,19,7,23,14,2,16,9,25,11,4,18,6,22,13)( 1,20, 8,24,15, 3,17,10,21,12, 5,19, 7,23,14, 2,16, 9,25,11, 4,18, 6,22,13)
25A3 2525 22 2525 2424 (1,14,24,9,17,4,12,22,7,20,2,15,25,10,18,5,13,23,8,16,3,11,21,6,19)( 1,14,24, 9,17, 4,12,22, 7,20, 2,15,25,10,18, 5,13,23, 8,16, 3,11,21, 6,19)
25A4 2525 22 2525 2424 (1,8,15,17,21,5,7,14,16,25,4,6,13,20,24,3,10,12,19,23,2,9,11,18,22)( 1, 8,15,17,21, 5, 7,14,16,25, 4, 6,13,20,24, 3,10,12,19,23, 2, 9,11,18,22)
25A6 2525 22 2525 2424 (1,24,17,12,7,2,25,18,13,8,3,21,19,14,9,4,22,20,15,10,5,23,16,11,6)( 1,24,17,12, 7, 2,25,18,13, 8, 3,21,19,14, 9, 4,22,20,15,10, 5,23,16,11, 6)
25A7 2525 22 2525 2424 (1,16,10,22,14,3,18,7,24,11,5,20,9,21,13,2,17,6,23,15,4,19,8,25,12)( 1,16,10,22,14, 3,18, 7,24,11, 5,20, 9,21,13, 2,17, 6,23,15, 4,19, 8,25,12)
25A8 2525 22 2525 2424 (1,15,21,7,16,4,13,24,10,19,2,11,22,8,17,5,14,25,6,20,3,12,23,9,18)( 1,15,21, 7,16, 4,13,24,10,19, 2,11,22, 8,17, 5,14,25, 6,20, 3,12,23, 9,18)
25A9 2525 22 2525 2424 (1,21,16,13,10,2,22,17,14,6,3,23,18,15,7,4,24,19,11,8,5,25,20,12,9)( 1,21,16,13,10, 2,22,17,14, 6, 3,23,18,15, 7, 4,24,19,11, 8, 5,25,20,12, 9)
25A11 2525 22 2525 2424 (1,25,19,15,6,2,21,20,11,7,3,22,16,12,8,4,23,17,13,9,5,24,18,14,10)( 1,25,19,15, 6, 2,21,20,11, 7, 3,22,16,12, 8, 4,23,17,13, 9, 5,24,18,14,10)
25A12 2525 22 2525 2424 (1,17,7,25,13,3,19,9,22,15,5,16,6,24,12,2,18,8,21,14,4,20,10,23,11)( 1,17, 7,25,13, 3,19, 9,22,15, 5,16, 6,24,12, 2,18, 8,21,14, 4,20,10,23,11)

Malle's constant a(G)a(G):     1/121/12

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Character table

1A 2A 5A1 5A2 25A1 25A2 25A3 25A4 25A6 25A7 25A8 25A9 25A11 25A12
Size 1 25 2 2 2 2 2 2 2 2 2 2 2 2
2 P 1A 1A 5A2 5A1 25A2 25A4 25A6 25A8 25A12 25A11 25A9 25A7 25A3 25A1
5 P 1A 2A 1A 1A 5A1 5A2 5A2 5A1 5A1 5A2 5A2 5A1 5A1 5A2
Type
50.1.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
50.1.1b R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
50.1.2a1 R 2 0 2 2 ζ52+ζ52 ζ52+ζ52 ζ52+ζ52 ζ51+ζ5 ζ51+ζ5 ζ52+ζ52 ζ51+ζ5 ζ52+ζ52 ζ51+ζ5 ζ51+ζ5
50.1.2a2 R 2 0 2 2 ζ51+ζ5 ζ51+ζ5 ζ51+ζ5 ζ52+ζ52 ζ52+ζ52 ζ51+ζ5 ζ52+ζ52 ζ51+ζ5 ζ52+ζ52 ζ52+ζ52
50.1.2b1 R 2 0 ζ2510+ζ2510 ζ255+ζ255 ζ2511+ζ2511 ζ254+ζ254 ζ256+ζ256 ζ253+ζ253 ζ252+ζ252 ζ251+ζ25 ζ257+ζ257 ζ259+ζ259 ζ2512+ζ2512 ζ258+ζ258
50.1.2b2 R 2 0 ζ2510+ζ2510 ζ255+ζ255 ζ259+ζ259 ζ251+ζ25 ζ2511+ζ2511 ζ257+ζ257 ζ2512+ζ2512 ζ256+ζ256 ζ258+ζ258 ζ254+ζ254 ζ253+ζ253 ζ252+ζ252
50.1.2b3 R 2 0 ζ2510+ζ2510 ζ255+ζ255 ζ256+ζ256 ζ259+ζ259 ζ251+ζ25 ζ2512+ζ2512 ζ258+ζ258 ζ254+ζ254 ζ253+ζ253 ζ2511+ζ2511 ζ252+ζ252 ζ257+ζ257
50.1.2b4 R 2 0 ζ2510+ζ2510 ζ255+ζ255 ζ254+ζ254 ζ256+ζ256 ζ259+ζ259 ζ258+ζ258 ζ253+ζ253 ζ2511+ζ2511 ζ252+ζ252 ζ251+ζ25 ζ257+ζ257 ζ2512+ζ2512
50.1.2b5 R 2 0 ζ2510+ζ2510 ζ255+ζ255 ζ251+ζ25 ζ2511+ζ2511 ζ254+ζ254 ζ252+ζ252 ζ257+ζ257 ζ259+ζ259 ζ2512+ζ2512 ζ256+ζ256 ζ258+ζ258 ζ253+ζ253
50.1.2b6 R 2 0 ζ255+ζ255 ζ2510+ζ2510 ζ2512+ζ2512 ζ257+ζ257 ζ252+ζ252 ζ251+ζ25 ζ259+ζ259 ζ258+ζ258 ζ256+ζ256 ζ253+ζ253 ζ254+ζ254 ζ2511+ζ2511
50.1.2b7 R 2 0 ζ255+ζ255 ζ2510+ζ2510 ζ258+ζ258 ζ2512+ζ2512 ζ257+ζ257 ζ259+ζ259 ζ256+ζ256 ζ253+ζ253 ζ254+ζ254 ζ252+ζ252 ζ2511+ζ2511 ζ251+ζ25
50.1.2b8 R 2 0 ζ255+ζ255 ζ2510+ζ2510 ζ257+ζ257 ζ252+ζ252 ζ253+ζ253 ζ2511+ζ2511 ζ251+ζ25 ζ2512+ζ2512 ζ259+ζ259 ζ258+ζ258 ζ256+ζ256 ζ254+ζ254
50.1.2b9 R 2 0 ζ255+ζ255 ζ2510+ζ2510 ζ253+ζ253 ζ258+ζ258 ζ2512+ζ2512 ζ256+ζ256 ζ254+ζ254 ζ252+ζ252 ζ2511+ζ2511 ζ257+ζ257 ζ251+ζ25 ζ259+ζ259
50.1.2b10 R 2 0 ζ255+ζ255 ζ2510+ζ2510 ζ252+ζ252 ζ253+ζ253 ζ258+ζ258 ζ254+ζ254 ζ2511+ζ2511 ζ257+ζ257 ζ251+ζ25 ζ2512+ζ2512 ζ259+ζ259 ζ256+ζ256

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Regular extensions

Data not computed