Properties

Label 36T34
Degree 3636
Order 7272
Cyclic no
Abelian no
Solvable yes
Primitive no
pp-group no
Group: S3×D6S_3\times D_6

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Copy content magma:G := TransitiveGroup(36, 34);
 

Group invariants

Abstract group:  S3×D6S_3\times D_6
Copy content magma:IdentifyGroup(G);
 
Order:  72=233272=2^{3} \cdot 3^{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:  3636
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number tt:  3434
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)}:  44
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,3)(2,4)(5,7)(6,8)(9,21)(10,22)(11,23)(12,24)(13,26)(14,25)(15,28)(16,27)(17,29)(18,30)(19,31)(20,32)(33,36)(34,35)(1,3)(2,4)(5,7)(6,8)(9,21)(10,22)(11,23)(12,24)(13,26)(14,25)(15,28)(16,27)(17,29)(18,30)(19,31)(20,32)(33,36)(34,35), (1,21)(2,22)(3,24)(4,23)(5,32)(6,31)(7,30)(8,29)(9,16)(10,15)(11,14)(12,13)(25,35)(26,36)(27,33)(28,34)(1,21)(2,22)(3,24)(4,23)(5,32)(6,31)(7,30)(8,29)(9,16)(10,15)(11,14)(12,13)(25,35)(26,36)(27,33)(28,34), (1,8,33,2,7,34)(3,6,36,4,5,35)(9,15,18,10,16,17)(11,13,19,12,14,20)(21,28,30,22,27,29)(23,26,31,24,25,32)(1,8,33,2,7,34)(3,6,36,4,5,35)(9,15,18,10,16,17)(11,13,19,12,14,20)(21,28,30,22,27,29)(23,26,31,24,25,32)
Copy content magma:Generators(G);
 

Low degree resolvents

#(G/N)\card{(G/N)}Galois groups for stem field(s)
22C2C_2 x 7
44C22C_2^2 x 7
66S3S_3 x 2
88C23C_2^3
1212D6D_{6} x 6
2424S3×C22S_3 \times C_2^2 x 2
3636S32S_3^2

Resolvents shown for degrees 47\leq 47

Subfields

Degree 2: C2C_2 x 3

Degree 3: S3S_3 x 2

Degree 4: C22C_2^2

Degree 6: D6D_{6} x 6, S32S_3^2

Degree 9: S32S_3^2

Degree 12: S3×C22S_3 \times C_2^2 x 2, 12T37

Degree 18: S32S_3^2, 18T29 x 2

Low degree siblings

12T37 x 2, 18T29 x 4, 24T73, 36T34, 36T40 x 4

Siblings are shown with degree 47\leq 47

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A 1361^{36} 11 11 00 ()()
2A 2182^{18} 11 22 1818 (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)
2B 2182^{18} 33 22 1818 (1,3)(2,4)(5,7)(6,8)(9,21)(10,22)(11,23)(12,24)(13,26)(14,25)(15,28)(16,27)(17,29)(18,30)(19,31)(20,32)(33,36)(34,35)( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,21)(10,22)(11,23)(12,24)(13,26)(14,25)(15,28)(16,27)(17,29)(18,30)(19,31)(20,32)(33,36)(34,35)
2C 2182^{18} 33 22 1818 (1,15)(2,16)(3,14)(4,13)(5,19)(6,20)(7,17)(8,18)(9,34)(10,33)(11,36)(12,35)(21,23)(22,24)(25,27)(26,28)(29,32)(30,31)( 1,15)( 2,16)( 3,14)( 4,13)( 5,19)( 6,20)( 7,17)( 8,18)( 9,34)(10,33)(11,36)(12,35)(21,23)(22,24)(25,27)(26,28)(29,32)(30,31)
2D 2182^{18} 33 22 1818 (1,3)(2,4)(5,33)(6,34)(7,36)(8,35)(9,20)(10,19)(11,17)(12,18)(13,16)(14,15)(21,32)(22,31)(23,29)(24,30)(25,28)(26,27)( 1, 3)( 2, 4)( 5,33)( 6,34)( 7,36)( 8,35)( 9,20)(10,19)(11,17)(12,18)(13,16)(14,15)(21,32)(22,31)(23,29)(24,30)(25,28)(26,27)
2E 2182^{18} 33 22 1818 (1,4)(2,3)(5,34)(6,33)(7,35)(8,36)(9,19)(10,20)(11,18)(12,17)(13,15)(14,16)(21,31)(22,32)(23,30)(24,29)(25,27)(26,28)( 1, 4)( 2, 3)( 5,34)( 6,33)( 7,35)( 8,36)( 9,19)(10,20)(11,18)(12,17)(13,15)(14,16)(21,31)(22,32)(23,30)(24,29)(25,27)(26,28)
2F 2182^{18} 99 22 1818 (1,34)(2,33)(3,35)(4,36)(5,6)(7,8)(9,25)(10,26)(11,27)(12,28)(13,22)(14,21)(15,24)(16,23)(17,32)(18,31)(19,30)(20,29)( 1,34)( 2,33)( 3,35)( 4,36)( 5, 6)( 7, 8)( 9,25)(10,26)(11,27)(12,28)(13,22)(14,21)(15,24)(16,23)(17,32)(18,31)(19,30)(20,29)
2G 216,142^{16},1^{4} 99 22 1616 (1,21)(2,22)(3,24)(4,23)(5,32)(6,31)(7,30)(8,29)(9,16)(10,15)(11,14)(12,13)(25,35)(26,36)(27,33)(28,34)( 1,21)( 2,22)( 3,24)( 4,23)( 5,32)( 6,31)( 7,30)( 8,29)( 9,16)(10,15)(11,14)(12,13)(25,35)(26,36)(27,33)(28,34)
3A 3123^{12} 22 33 2424 (1,27,13)(2,28,14)(3,26,16)(4,25,15)(5,32,18)(6,31,17)(7,30,20)(8,29,19)(9,36,24)(10,35,23)(11,34,22)(12,33,21)( 1,27,13)( 2,28,14)( 3,26,16)( 4,25,15)( 5,32,18)( 6,31,17)( 7,30,20)( 8,29,19)( 9,36,24)(10,35,23)(11,34,22)(12,33,21)
3B 3123^{12} 22 33 2424 (1,33,7)(2,34,8)(3,36,5)(4,35,6)(9,18,16)(10,17,15)(11,19,14)(12,20,13)(21,30,27)(22,29,28)(23,31,25)(24,32,26)( 1,33, 7)( 2,34, 8)( 3,36, 5)( 4,35, 6)( 9,18,16)(10,17,15)(11,19,14)(12,20,13)(21,30,27)(22,29,28)(23,31,25)(24,32,26)
3C 3123^{12} 44 33 2424 (1,21,20)(2,22,19)(3,24,18)(4,23,17)(5,26,9)(6,25,10)(7,27,12)(8,28,11)(13,33,30)(14,34,29)(15,35,31)(16,36,32)( 1,21,20)( 2,22,19)( 3,24,18)( 4,23,17)( 5,26, 9)( 6,25,10)( 7,27,12)( 8,28,11)(13,33,30)(14,34,29)(15,35,31)(16,36,32)
6A 666^{6} 22 66 3030 (1,14,27,2,13,28)(3,15,26,4,16,25)(5,17,32,6,18,31)(7,19,30,8,20,29)(9,23,36,10,24,35)(11,21,34,12,22,33)( 1,14,27, 2,13,28)( 3,15,26, 4,16,25)( 5,17,32, 6,18,31)( 7,19,30, 8,20,29)( 9,23,36,10,24,35)(11,21,34,12,22,33)
6B 666^{6} 22 66 3030 (1,8,33,2,7,34)(3,6,36,4,5,35)(9,15,18,10,16,17)(11,13,19,12,14,20)(21,28,30,22,27,29)(23,26,31,24,25,32)( 1, 8,33, 2, 7,34)( 3, 6,36, 4, 5,35)( 9,15,18,10,16,17)(11,13,19,12,14,20)(21,28,30,22,27,29)(23,26,31,24,25,32)
6C 666^{6} 44 66 3030 (1,19,21,2,20,22)(3,17,24,4,18,23)(5,10,26,6,9,25)(7,11,27,8,12,28)(13,29,33,14,30,34)(15,32,35,16,31,36)( 1,19,21, 2,20,22)( 3,17,24, 4,18,23)( 5,10,26, 6, 9,25)( 7,11,27, 8,12,28)(13,29,33,14,30,34)(15,32,35,16,31,36)
6D 666^{6} 66 66 3030 (1,36,7,3,33,5)(2,35,8,4,34,6)(9,30,16,21,18,27)(10,29,15,22,17,28)(11,31,14,23,19,25)(12,32,13,24,20,26)( 1,36, 7, 3,33, 5)( 2,35, 8, 4,34, 6)( 9,30,16,21,18,27)(10,29,15,22,17,28)(11,31,14,23,19,25)(12,32,13,24,20,26)
6E 666^{6} 66 66 3030 (1,10,7,15,33,17)(2,9,8,16,34,18)(3,11,5,14,36,19)(4,12,6,13,35,20)(21,31,27,23,30,25)(22,32,28,24,29,26)( 1,10, 7,15,33,17)( 2, 9, 8,16,34,18)( 3,11, 5,14,36,19)( 4,12, 6,13,35,20)(21,31,27,23,30,25)(22,32,28,24,29,26)
6F 666^{6} 66 66 3030 (1,26,13,3,27,16)(2,25,14,4,28,15)(5,21,18,33,32,12)(6,22,17,34,31,11)(7,24,20,36,30,9)(8,23,19,35,29,10)( 1,26,13, 3,27,16)( 2,25,14, 4,28,15)( 5,21,18,33,32,12)( 6,22,17,34,31,11)( 7,24,20,36,30, 9)( 8,23,19,35,29,10)
6G 666^{6} 66 66 3030 (1,15,27,4,13,25)(2,16,28,3,14,26)(5,11,32,34,18,22)(6,12,31,33,17,21)(7,10,30,35,20,23)(8,9,29,36,19,24)( 1,15,27, 4,13,25)( 2,16,28, 3,14,26)( 5,11,32,34,18,22)( 6,12,31,33,17,21)( 7,10,30,35,20,23)( 8, 9,29,36,19,24)

Malle's constant a(G)a(G):     1/161/16

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 6A 6B 6C 6D 6E 6F 6G
Size 1 1 3 3 3 3 9 9 2 2 4 2 2 4 6 6 6 6
2 P 1A 1A 1A 1A 1A 1A 1A 1A 3A 3B 3C 3A 3B 3C 3B 3B 3A 3A
3 P 1A 2A 2B 2C 2D 2E 2F 2G 1A 1A 1A 2A 2A 2A 2B 2C 2D 2E
Type
72.46.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
72.46.1b R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
72.46.1c R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
72.46.1d R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
72.46.1e R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
72.46.1f R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
72.46.1g R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
72.46.1h R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
72.46.2a R 2 2 0 0 2 2 0 0 1 2 1 1 2 1 0 0 1 1
72.46.2b R 2 2 2 2 0 0 0 0 2 1 1 2 1 1 1 1 0 0
72.46.2c R 2 2 2 2 0 0 0 0 2 1 1 2 1 1 1 1 0 0
72.46.2d R 2 2 0 0 2 2 0 0 1 2 1 1 2 1 0 0 1 1
72.46.2e R 2 2 0 0 2 2 0 0 1 2 1 1 2 1 0 0 1 1
72.46.2f R 2 2 2 2 0 0 0 0 2 1 1 2 1 1 1 1 0 0
72.46.2g R 2 2 2 2 0 0 0 0 2 1 1 2 1 1 1 1 0 0
72.46.2h R 2 2 0 0 2 2 0 0 1 2 1 1 2 1 0 0 1 1
72.46.4a R 4 4 0 0 0 0 0 0 2 2 1 2 2 1 0 0 0 0
72.46.4b R 4 4 0 0 0 0 0 0 2 2 1 2 2 1 0 0 0 0

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Regular extensions

Data not computed