Properties

Label 32T397
Degree 3232
Order 9696
Cyclic no
Abelian no
Solvable yes
Primitive no
pp-group no
Group: C4:S4C_4:S_4

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Copy content magma:G := TransitiveGroup(32, 397);
 

Group invariants

Abstract group:  C4:S4C_4:S_4
Copy content magma:IdentifyGroup(G);
 
Order:  96=25396=2^{5} \cdot 3
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
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Group action invariants

Degree nn:  3232
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number tt:  397397
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)}:  88
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,28,5,16)(2,27,6,15)(3,26,8,13)(4,25,7,14)(9,22,29,17)(10,21,30,18)(11,24,32,20)(12,23,31,19)(1,28,5,16)(2,27,6,15)(3,26,8,13)(4,25,7,14)(9,22,29,17)(10,21,30,18)(11,24,32,20)(12,23,31,19), (1,7,12,29)(2,8,11,30)(3,6,10,32)(4,5,9,31)(13,25,21,17)(14,26,22,18)(15,28,24,19)(16,27,23,20)(1,7,12,29)(2,8,11,30)(3,6,10,32)(4,5,9,31)(13,25,21,17)(14,26,22,18)(15,28,24,19)(16,27,23,20)
Copy content magma:Generators(G);
 

Low degree resolvents

#(G/N)\card{(G/N)}Galois groups for stem field(s)
22C2C_2 x 3
44C22C_2^2
66S3S_3
88D4D_{4}
1212D6D_{6}
2424S4S_4, D12D_{12}
4848S4×C2S_4\times C_2

Resolvents shown for degrees 47\leq 47

Subfields

Degree 2: C2C_2 x 3

Degree 4: C22C_2^2, D4D_{4} x 2, S4S_4

Degree 8: D4D_4, S4S_4, S4×C2S_4\times C_2 x 2

Degree 16: 16T61, 16T191 x 2

Low degree siblings

12T54 x 2, 16T191 x 2, 24T128 x 2, 24T170, 24T171 x 2, 24T172 x 2

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 1321^{32} 11 11 00 ()()
2A 2162^{16} 11 22 1616 (1,3)(2,4)(5,8)(6,7)(9,11)(10,12)(13,16)(14,15)(17,20)(18,19)(21,23)(22,24)(25,27)(26,28)(29,32)(30,31)( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,11)(10,12)(13,16)(14,15)(17,20)(18,19)(21,23)(22,24)(25,27)(26,28)(29,32)(30,31)
2B 2162^{16} 33 22 1616 (1,8)(2,7)(3,5)(4,6)(9,32)(10,31)(11,29)(12,30)(13,28)(14,27)(15,25)(16,26)(17,24)(18,23)(19,21)(20,22)( 1, 8)( 2, 7)( 3, 5)( 4, 6)( 9,32)(10,31)(11,29)(12,30)(13,28)(14,27)(15,25)(16,26)(17,24)(18,23)(19,21)(20,22)
2C 2162^{16} 33 22 1616 (1,12)(2,11)(3,10)(4,9)(5,31)(6,32)(7,29)(8,30)(13,21)(14,22)(15,24)(16,23)(17,25)(18,26)(19,28)(20,27)( 1,12)( 2,11)( 3,10)( 4, 9)( 5,31)( 6,32)( 7,29)( 8,30)(13,21)(14,22)(15,24)(16,23)(17,25)(18,26)(19,28)(20,27)
2D 2162^{16} 1212 22 1616 (1,9)(2,10)(3,11)(4,12)(5,7)(6,8)(13,22)(14,21)(15,23)(16,24)(17,18)(19,20)(25,26)(27,28)(29,31)(30,32)( 1, 9)( 2,10)( 3,11)( 4,12)( 5, 7)( 6, 8)(13,22)(14,21)(15,23)(16,24)(17,18)(19,20)(25,26)(27,28)(29,31)(30,32)
2E 2162^{16} 1212 22 1616 (1,23)(2,24)(3,21)(4,22)(5,16)(6,15)(7,14)(8,13)(9,17)(10,18)(11,20)(12,19)(25,29)(26,30)(27,32)(28,31)( 1,23)( 2,24)( 3,21)( 4,22)( 5,16)( 6,15)( 7,14)( 8,13)( 9,17)(10,18)(11,20)(12,19)(25,29)(26,30)(27,32)(28,31)
3A 38,183^{8},1^{8} 88 33 1616 (5,12,31)(6,11,32)(7,9,29)(8,10,30)(13,26,18)(14,25,17)(15,27,20)(16,28,19)( 5,12,31)( 6,11,32)( 7, 9,29)( 8,10,30)(13,26,18)(14,25,17)(15,27,20)(16,28,19)
4A 484^{8} 22 44 2424 (1,22,3,24)(2,21,4,23)(5,17,8,20)(6,18,7,19)(9,16,11,13)(10,15,12,14)(25,30,27,31)(26,29,28,32)( 1,22, 3,24)( 2,21, 4,23)( 5,17, 8,20)( 6,18, 7,19)( 9,16,11,13)(10,15,12,14)(25,30,27,31)(26,29,28,32)
4B 484^{8} 66 44 2424 (1,17,3,20)(2,18,4,19)(5,22,8,24)(6,21,7,23)(9,28,11,26)(10,27,12,25)(13,29,16,32)(14,30,15,31)( 1,17, 3,20)( 2,18, 4,19)( 5,22, 8,24)( 6,21, 7,23)( 9,28,11,26)(10,27,12,25)(13,29,16,32)(14,30,15,31)
4C 484^{8} 1212 44 2424 (1,7,12,29)(2,8,11,30)(3,6,10,32)(4,5,9,31)(13,25,21,17)(14,26,22,18)(15,28,24,19)(16,27,23,20)( 1, 7,12,29)( 2, 8,11,30)( 3, 6,10,32)( 4, 5, 9,31)(13,25,21,17)(14,26,22,18)(15,28,24,19)(16,27,23,20)
4D 484^{8} 1212 44 2424 (1,19,12,28)(2,20,11,27)(3,18,10,26)(4,17,9,25)(5,16,31,23)(6,15,32,24)(7,14,29,22)(8,13,30,21)( 1,19,12,28)( 2,20,11,27)( 3,18,10,26)( 4,17, 9,25)( 5,16,31,23)( 6,15,32,24)( 7,14,29,22)( 8,13,30,21)
6A 64,246^{4},2^{4} 88 66 2424 (1,3)(2,4)(5,30,12,8,31,10)(6,29,11,7,32,9)(13,19,26,16,18,28)(14,20,25,15,17,27)(21,23)(22,24)( 1, 3)( 2, 4)( 5,30,12, 8,31,10)( 6,29,11, 7,32, 9)(13,19,26,16,18,28)(14,20,25,15,17,27)(21,23)(22,24)
12A1 122,4212^{2},4^{2} 88 1212 2828 (1,24,3,22)(2,23,4,21)(5,15,30,17,12,27,8,14,31,20,10,25)(6,16,29,18,11,28,7,13,32,19,9,26)( 1,24, 3,22)( 2,23, 4,21)( 5,15,30,17,12,27, 8,14,31,20,10,25)( 6,16,29,18,11,28, 7,13,32,19, 9,26)
12A5 122,4212^{2},4^{2} 88 1212 2828 (1,24,3,22)(2,23,4,21)(5,27,10,17,31,15,8,25,12,20,30,14)(6,28,9,18,32,16,7,26,11,19,29,13)( 1,24, 3,22)( 2,23, 4,21)( 5,27,10,17,31,15, 8,25,12,20,30,14)( 6,28, 9,18,32,16, 7,26,11,19,29,13)

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

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Character table

1A 2A 2B 2C 2D 2E 3A 4A 4B 4C 4D 6A 12A1 12A5
Size 1 1 3 3 12 12 8 2 6 12 12 8 8 8
2 P 1A 1A 1A 1A 1A 1A 3A 2A 2A 2C 2C 3A 6A 6A
3 P 1A 2A 2B 2C 2D 2E 1A 4A 4B 4C 4D 2A 4A 4A
Type
96.187.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
96.187.1b R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
96.187.1c R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
96.187.1d R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
96.187.2a R 2 2 2 2 0 0 1 2 2 0 0 1 1 1
96.187.2b R 2 2 2 2 0 0 2 0 0 0 0 2 0 0
96.187.2c R 2 2 2 2 0 0 1 2 2 0 0 1 1 1
96.187.2d1 R 2 2 2 2 0 0 1 0 0 0 0 1 ζ121ζ12 ζ121+ζ12
96.187.2d2 R 2 2 2 2 0 0 1 0 0 0 0 1 ζ121+ζ12 ζ121ζ12
96.187.3a R 3 3 1 1 1 1 0 3 1 1 1 0 0 0
96.187.3b R 3 3 1 1 1 1 0 3 1 1 1 0 0 0
96.187.3c R 3 3 1 1 1 1 0 3 1 1 1 0 0 0
96.187.3d R 3 3 1 1 1 1 0 3 1 1 1 0 0 0
96.187.6a R 6 6 2 2 0 0 0 0 0 0 0 0 0 0

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Regular extensions

Data not computed