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Properties

Label 32T408
Degree $32$
Order $96$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $\SL(2,3):C_2^2$

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Show commands: Magma

magma: G := TransitiveGroup(32, 408);
 

Group action invariants

Degree $n$:  $32$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $408$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $\SL(2,3):C_2^2$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $8$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,22,2,21)(3,23,4,24)(5,16,29,20,12,28,6,15,30,19,11,27)(7,13,31,17,10,25,8,14,32,18,9,26), (1,28,2,27)(3,26,4,25)(5,14,6,13)(7,16,8,15)(9,19,10,20)(11,17,12,18)(21,30,22,29)(23,32,24,31)
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_2^2$
$6$:  $C_6$ x 3
$12$:  $A_4$, $C_6\times C_2$
$24$:  $A_4\times C_2$ x 3
$48$:  $C_2^2 \times A_4$, 16T60 x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$, $A_4$

Degree 8: $A_4\times C_2$ x 3

Degree 16: 16T58, 16T60 x 2

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 $1^{32}$ $1$ $1$ $0$ $()$
2A $2^{16}$ $1$ $2$ $16$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)$
2B $2^{16}$ $1$ $2$ $16$ $( 1,23)( 2,24)( 3,21)( 4,22)( 5,17)( 6,18)( 7,19)( 8,20)( 9,15)(10,16)(11,13)(12,14)(25,29)(26,30)(27,31)(28,32)$
2C $2^{16}$ $1$ $2$ $16$ $( 1,24)( 2,23)( 3,22)( 4,21)( 5,18)( 6,17)( 7,20)( 8,19)( 9,16)(10,15)(11,14)(12,13)(25,30)(26,29)(27,32)(28,31)$
2D $2^{16}$ $6$ $2$ $16$ $( 1,30)( 2,29)( 3,31)( 4,32)( 5, 9)( 6,10)( 7,12)( 8,11)(13,20)(14,19)(15,17)(16,18)(21,27)(22,28)(23,26)(24,25)$
2E $2^{16}$ $6$ $2$ $16$ $( 1,13)( 2,14)( 3,16)( 4,15)( 5,27)( 6,28)( 7,26)( 8,25)( 9,22)(10,21)(11,23)(12,24)(17,31)(18,32)(19,30)(20,29)$
3A1 $3^{8},1^{8}$ $4$ $3$ $16$ $( 5,30,12)( 6,29,11)( 7,32,10)( 8,31, 9)(13,18,25)(14,17,26)(15,20,27)(16,19,28)$
3A-1 $3^{8},1^{8}$ $4$ $3$ $16$ $( 5,12,30)( 6,11,29)( 7,10,32)( 8, 9,31)(13,25,18)(14,26,17)(15,27,20)(16,28,19)$
4A1 $4^{8}$ $1$ $4$ $24$ $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,12,10,11)(13,15,14,16)(17,19,18,20)(21,23,22,24)(25,27,26,28)(29,31,30,32)$
4A-1 $4^{8}$ $1$ $4$ $24$ $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,11,10,12)(13,16,14,15)(17,20,18,19)(21,24,22,23)(25,28,26,27)(29,32,30,31)$
4B1 $4^{8}$ $1$ $4$ $24$ $( 1,22, 2,21)( 3,23, 4,24)( 5,19, 6,20)( 7,18, 8,17)( 9,14,10,13)(11,15,12,16)(25,31,26,32)(27,30,28,29)$
4B-1 $4^{8}$ $1$ $4$ $24$ $( 1,21, 2,22)( 3,24, 4,23)( 5,20, 6,19)( 7,17, 8,18)( 9,13,10,14)(11,16,12,15)(25,32,26,31)(27,29,28,30)$
4C $4^{8}$ $6$ $4$ $24$ $( 1,31, 2,32)( 3,29, 4,30)( 5,11, 6,12)( 7, 9, 8,10)(13,18,14,17)(15,20,16,19)(21,25,22,26)(23,27,24,28)$
4D $4^{8}$ $6$ $4$ $24$ $( 1,15, 2,16)( 3,13, 4,14)( 5,26, 6,25)( 7,28, 8,27)( 9,24,10,23)(11,22,12,21)(17,30,18,29)(19,32,20,31)$
6A1 $6^{4},2^{4}$ $4$ $6$ $24$ $( 1,24)( 2,23)( 3,22)( 4,21)( 5,13,30,18,12,25)( 6,14,29,17,11,26)( 7,15,32,20,10,27)( 8,16,31,19, 9,28)$
6A-1 $6^{4},2^{4}$ $4$ $6$ $24$ $( 1,23)( 2,24)( 3,21)( 4,22)( 5,26,12,17,30,14)( 6,25,11,18,29,13)( 7,28,10,19,32,16)( 8,27, 9,20,31,15)$
6B1 $6^{4},2^{4}$ $4$ $6$ $24$ $( 1,23)( 2,24)( 3,21)( 4,22)( 5,14,30,17,12,26)( 6,13,29,18,11,25)( 7,16,32,19,10,28)( 8,15,31,20, 9,27)$
6B-1 $6^{4},2^{4}$ $4$ $6$ $24$ $( 1, 2)( 3, 4)( 5,29,12, 6,30,11)( 7,31,10, 8,32, 9)(13,17,25,14,18,26)(15,19,27,16,20,28)(21,22)(23,24)$
6C1 $6^{4},2^{4}$ $4$ $6$ $24$ $( 1,24)( 2,23)( 3,22)( 4,21)( 5,25,12,18,30,13)( 6,26,11,17,29,14)( 7,27,10,20,32,15)( 8,28, 9,19,31,16)$
6C-1 $6^{4},2^{4}$ $4$ $6$ $24$ $( 1, 2)( 3, 4)( 5,11,30, 6,12,29)( 7, 9,32, 8,10,31)(13,26,18,14,25,17)(15,28,20,16,27,19)(21,22)(23,24)$
12A1 $12^{2},4^{2}$ $4$ $12$ $28$ $( 1, 4, 2, 3)( 5,32,11, 8,30,10, 6,31,12, 7,29, 9)(13,20,26,16,18,27,14,19,25,15,17,28)(21,23,22,24)$
12A-1 $12^{2},4^{2}$ $4$ $12$ $28$ $( 1, 3, 2, 4)( 5,31,11, 7,30, 9, 6,32,12, 8,29,10)(13,19,26,15,18,28,14,20,25,16,17,27)(21,24,22,23)$
12A5 $12^{2},4^{2}$ $4$ $12$ $28$ $( 1, 3, 2, 4)( 5, 9,29, 7,12,31, 6,10,30, 8,11,32)(13,28,17,15,25,19,14,27,18,16,26,20)(21,24,22,23)$
12A-5 $12^{2},4^{2}$ $4$ $12$ $28$ $( 1, 4, 2, 3)( 5,10,29, 8,12,32, 6, 9,30, 7,11,31)(13,27,17,16,25,20,14,28,18,15,26,19)(21,23,22,24)$
12B1 $12^{2},4^{2}$ $4$ $12$ $28$ $( 1,13,32,21,10,26, 2,14,31,22, 9,25)( 3,16,30,24,12,27, 4,15,29,23,11,28)( 5,19, 6,20)( 7,18, 8,17)$
12B-1 $12^{2},4^{2}$ $4$ $12$ $28$ $( 1,14,32,22,10,25, 2,13,31,21, 9,26)( 3,15,30,23,12,28, 4,16,29,24,11,27)( 5,20, 6,19)( 7,17, 8,18)$
12B5 $12^{2},4^{2}$ $4$ $12$ $28$ $( 1,25, 9,22,31,14, 2,26,10,21,32,13)( 3,28,11,23,29,15, 4,27,12,24,30,16)( 5,20, 6,19)( 7,17, 8,18)$
12B-5 $12^{2},4^{2}$ $4$ $12$ $28$ $( 1,26, 9,21,31,13, 2,25,10,22,32,14)( 3,27,11,24,29,16, 4,28,12,23,30,15)( 5,19, 6,20)( 7,18, 8,17)$

Malle's constant $a(G)$:     $1/16$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $96=2^{5} \cdot 3$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  96.200
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 3A1 3A-1 4A1 4A-1 4B1 4B-1 4C 4D 6A1 6A-1 6B1 6B-1 6C1 6C-1 12A1 12A-1 12A5 12A-5 12B1 12B-1 12B5 12B-5
Size 1 1 1 1 6 6 4 4 1 1 1 1 6 6 4 4 4 4 4 4 4 4 4 4 4 4 4 4
2 P 1A 1A 1A 1A 1A 1A 3A-1 3A1 2A 2A 2A 2A 2A 2A 3A1 3A-1 3A1 3A-1 3A-1 3A1 6A1 6A1 6A-1 6A-1 6A1 6A1 6A-1 6A-1
3 P 1A 2A 2B 2C 2D 2E 1A 1A 4A-1 4A1 4B-1 4B1 4C 4D 2C 2B 2B 2A 2C 2A 4A-1 4A1 4A1 4A-1 4B-1 4B1 4B1 4B-1
Type
96.200.1a 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
96.200.1b 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
96.200.1c 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
96.200.1d 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
96.200.1e1 C 1 1 1 1 1 1 ζ3−1 ζ3 1 1 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3
96.200.1e2 C 1 1 1 1 1 1 ζ3 ζ3−1 1 1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1
96.200.1f1 C 1 1 −1 −1 −1 1 ζ3−1 ζ3 1 1 −1 −1 1 −1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 ζ3−1 ζ3 ζ3 ζ3−1 −ζ3 −ζ3−1 −ζ3−1 −ζ3
96.200.1f2 C 1 1 −1 −1 −1 1 ζ3 ζ3−1 1 1 −1 −1 1 −1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3 −ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 −ζ3−1 −ζ3 −ζ3 −ζ3−1
96.200.1g1 C 1 1 −1 −1 1 −1 ζ3−1 ζ3 −1 −1 1 1 1 −1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3 −ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3
96.200.1g2 C 1 1 −1 −1 1 −1 ζ3 ζ3−1 −1 −1 1 1 1 −1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 ζ3−1 ζ3 ζ3 ζ3−1
96.200.1h1 C 1 1 1 1 −1 −1 ζ3−1 ζ3 −1 −1 −1 −1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 −ζ3−1 −ζ3 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3−1 −ζ3
96.200.1h2 C 1 1 1 1 −1 −1 ζ3 ζ3−1 −1 −1 −1 −1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3 −ζ3−1
96.200.2a1 C 2 −2 −2 2 0 0 −1 −1 −2i 2i 2i −2i 0 0 1 1 1 1 −1 −1 −i i −i i i −i i −i
96.200.2a2 C 2 −2 −2 2 0 0 −1 −1 2i −2i −2i 2i 0 0 1 1 1 1 −1 −1 i −i i −i −i i −i i
96.200.2b1 C 2 −2 2 −2 0 0 −1 −1 −2i 2i −2i 2i 0 0 1 1 −1 −1 1 1 −i i −i i −i i −i i
96.200.2b2 C 2 −2 2 −2 0 0 −1 −1 2i −2i 2i −2i 0 0 1 1 −1 −1 1 1 i −i i −i i −i i −i
96.200.2c1 C 2 −2 −2 2 0 0 −ζ124 ζ122 −2ζ123 2ζ123 2ζ123 −2ζ123 0 0 −ζ122 ζ124 ζ124 −ζ122 −ζ124 ζ122 ζ12 −ζ125 ζ125 −ζ12 −ζ125 ζ12 −ζ12 ζ125
96.200.2c2 C 2 −2 −2 2 0 0 ζ122 −ζ124 2ζ123 −2ζ123 −2ζ123 2ζ123 0 0 ζ124 −ζ122 −ζ122 ζ124 ζ122 −ζ124 −ζ125 ζ12 −ζ12 ζ125 ζ12 −ζ125 ζ125 −ζ12
96.200.2c3 C 2 −2 −2 2 0 0 −ζ124 ζ122 2ζ123 −2ζ123 −2ζ123 2ζ123 0 0 −ζ122 ζ124 ζ124 −ζ122 −ζ124 ζ122 −ζ12 ζ125 −ζ125 ζ12 ζ125 −ζ12 ζ12 −ζ125
96.200.2c4 C 2 −2 −2 2 0 0 ζ122 −ζ124 −2ζ123 2ζ123 2ζ123 −2ζ123 0 0 ζ124 −ζ122 −ζ122 ζ124 ζ122 −ζ124 ζ125 −ζ12 ζ12 −ζ125 −ζ12 ζ125 −ζ125 ζ12
96.200.2d1 C 2 −2 2 −2 0 0 −ζ124 ζ122 −2ζ123 2ζ123 −2ζ123 2ζ123 0 0 −ζ122 ζ124 −ζ124 ζ122 ζ124 −ζ122 ζ12 −ζ125 ζ125 −ζ12 ζ125 −ζ12 ζ12 −ζ125
96.200.2d2 C 2 −2 2 −2 0 0 ζ122 −ζ124 2ζ123 −2ζ123 2ζ123 −2ζ123 0 0 ζ124 −ζ122 ζ122 −ζ124 −ζ122 ζ124 −ζ125 ζ12 −ζ12 ζ125 −ζ12 ζ125 −ζ125 ζ12
96.200.2d3 C 2 −2 2 −2 0 0 −ζ124 ζ122 2ζ123 −2ζ123 2ζ123 −2ζ123 0 0 −ζ122 ζ124 −ζ124 ζ122 ζ124 −ζ122 −ζ12 ζ125 −ζ125 ζ12 −ζ125 ζ12 −ζ12 ζ125
96.200.2d4 C 2 −2 2 −2 0 0 ζ122 −ζ124 −2ζ123 2ζ123 −2ζ123 2ζ123 0 0 ζ124 −ζ122 ζ122 −ζ124 −ζ122 ζ124 ζ125 −ζ12 ζ12 −ζ125 ζ12 −ζ125 ζ125 −ζ12
96.200.3a R 3 3 3 3 −1 −1 0 0 3 3 3 3 −1 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
96.200.3b R 3 3 −3 −3 −1 1 0 0 −3 −3 3 3 −1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
96.200.3c R 3 3 −3 −3 1 −1 0 0 3 3 −3 −3 −1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
96.200.3d R 3 3 3 3 1 1 0 0 −3 −3 −3 −3 −1 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

magma: CharacterTable(G);