↑

Properties

Label 32T395
Degree $32$
Order $96$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $\GL(2,3):C_2$

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(32, 395);
 

Group action invariants

Degree $n$:  $32$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $395$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $\GL(2,3):C_2$
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:  (1,14,31,22,12,28,3,16,29,24,10,26)(2,13,32,21,11,27,4,15,30,23,9,25)(5,19,8,17)(6,20,7,18), (1,18,3,20)(2,17,4,19)(5,21,8,23)(6,22,7,24)(9,14,11,16)(10,13,12,15)(25,29,27,31)(26,30,28,32)
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$
$12$:  $D_{6}$
$24$:  $S_4$
$48$:  $S_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$, $S_4$

Degree 8: $S_4$, $S_4\times C_2$ x 2

Degree 16: 16T61, 16T189 x 2

Low degree siblings

16T189 x 2

Siblings are shown 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 $2^{16}$ $1$ $2$ $16$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)$
2B $2^{16}$ $6$ $2$ $16$ $( 1,16)( 2,15)( 3,14)( 4,13)( 5,26)( 6,25)( 7,27)( 8,28)( 9,21)(10,22)(11,23)(12,24)(17,31)(18,32)(19,29)(20,30)$
2C $2^{16}$ $12$ $2$ $16$ $( 1, 2)( 3, 4)( 5, 9)( 6,10)( 7,12)( 8,11)(13,17)(14,18)(15,19)(16,20)(21,22)(23,24)(25,28)(26,27)(29,32)(30,31)$
3A $3^{8},1^{8}$ $8$ $3$ $16$ $( 1,12,29)( 2,11,30)( 3,10,31)( 4, 9,32)(13,27,23)(14,28,24)(15,25,21)(16,26,22)$
4A1 $4^{8}$ $1$ $4$ $24$ $( 1,24, 3,22)( 2,23, 4,21)( 5,19, 8,17)( 6,20, 7,18)( 9,15,11,13)(10,16,12,14)(25,30,27,32)(26,29,28,31)$
4A-1 $4^{8}$ $1$ $4$ $24$ $( 1,22, 3,24)( 2,21, 4,23)( 5,17, 8,19)( 6,18, 7,20)( 9,13,11,15)(10,14,12,16)(25,32,27,30)(26,31,28,29)$
4B $4^{8}$ $6$ $4$ $24$ $( 1,10, 3,12)( 2, 9, 4,11)( 5,31, 8,29)( 6,32, 7,30)(13,23,15,21)(14,24,16,22)(17,28,19,26)(18,27,20,25)$
4C $4^{8}$ $12$ $4$ $24$ $( 1,21, 3,23)( 2,22, 4,24)( 5,13, 8,15)( 6,14, 7,16)( 9,17,11,19)(10,18,12,20)(25,29,27,31)(26,30,28,32)$
6A $6^{4},2^{4}$ $8$ $6$ $24$ $( 1,31,12, 3,29,10)( 2,32,11, 4,30, 9)( 5, 8)( 6, 7)(13,21,27,15,23,25)(14,22,28,16,24,26)(17,19)(18,20)$
8A1 $8^{4}$ $6$ $8$ $28$ $( 1, 6,12,30, 3, 7,10,32)( 2, 5,11,29, 4, 8, 9,31)(13,28,21,17,15,26,23,19)(14,27,22,18,16,25,24,20)$
8A3 $8^{4}$ $6$ $8$ $28$ $( 1,18,10,27, 3,20,12,25)( 2,17, 9,28, 4,19,11,26)( 5,15,31,21, 8,13,29,23)( 6,16,32,22, 7,14,30,24)$
8B1 $8^{4}$ $6$ $8$ $28$ $( 1, 7,12,32, 3, 6,10,30)( 2, 8,11,31, 4, 5, 9,29)(13,26,21,19,15,28,23,17)(14,25,22,20,16,27,24,18)$
8B-1 $8^{4}$ $6$ $8$ $28$ $( 1,27,12,18, 3,25,10,20)( 2,28,11,17, 4,26, 9,19)( 5,21,29,15, 8,23,31,13)( 6,22,30,16, 7,24,32,14)$
12A1 $12^{2},4^{2}$ $8$ $12$ $28$ $( 1,16,31,24,12,26, 3,14,29,22,10,28)( 2,15,32,23,11,25, 4,13,30,21, 9,27)( 5,17, 8,19)( 6,18, 7,20)$
12A-1 $12^{2},4^{2}$ $8$ $12$ $28$ $( 1,14,31,22,12,28, 3,16,29,24,10,26)( 2,13,32,21,11,27, 4,15,30,23, 9,25)( 5,19, 8,17)( 6,20, 7,18)$

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:  96.192
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 4A1 4A-1 4B 4C 6A 8A1 8A3 8B1 8B-1 12A1 12A-1
Size 1 1 6 12 8 1 1 6 12 8 6 6 6 6 8 8
2 P 1A 1A 1A 1A 3A 2A 2A 2A 2A 3A 4B 4B 4B 4B 6A 6A
3 P 1A 2A 2B 2C 1A 4A-1 4A1 4B 4C 2A 8B-1 8A3 8B1 8A1 4A1 4A-1
Type
96.192.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
96.192.1b R 1 1 −1 −1 1 −1 −1 1 1 1 −1 −1 1 1 −1 −1
96.192.1c R 1 1 −1 1 1 −1 −1 1 −1 1 1 1 −1 −1 −1 −1
96.192.1d R 1 1 1 −1 1 1 1 1 −1 1 −1 −1 −1 −1 1 1
96.192.2a R 2 2 2 0 −1 2 2 2 0 −1 0 0 0 0 −1 −1
96.192.2b R 2 2 −2 0 −1 −2 −2 2 0 −1 0 0 0 0 1 1
96.192.2c1 C 2 −2 0 0 −1 −2ζ82 2ζ82 0 0 1 −ζ8−ζ83 ζ8+ζ83 ζ8−1+ζ8 −ζ8−1−ζ8 −ζ82 ζ82
96.192.2c2 C 2 −2 0 0 −1 2ζ82 −2ζ82 0 0 1 ζ8+ζ83 −ζ8−ζ83 ζ8−1+ζ8 −ζ8−1−ζ8 ζ82 −ζ82
96.192.2c3 C 2 −2 0 0 −1 −2ζ82 2ζ82 0 0 1 ζ8+ζ83 −ζ8−ζ83 −ζ8−1−ζ8 ζ8−1+ζ8 −ζ82 ζ82
96.192.2c4 C 2 −2 0 0 −1 2ζ82 −2ζ82 0 0 1 −ζ8−ζ83 ζ8+ζ83 −ζ8−1−ζ8 ζ8−1+ζ8 ζ82 −ζ82
96.192.3a R 3 3 −1 1 0 3 3 −1 1 0 −1 −1 −1 −1 0 0
96.192.3b R 3 3 −1 −1 0 3 3 −1 −1 0 1 1 1 1 0 0
96.192.3c R 3 3 1 −1 0 −3 −3 −1 1 0 1 1 −1 −1 0 0
96.192.3d R 3 3 1 1 0 −3 −3 −1 −1 0 −1 −1 1 1 0 0
96.192.4a1 C 4 −4 0 0 1 −4i 4i 0 0 −1 0 0 0 0 i −i
96.192.4a2 C 4 −4 0 0 1 4i −4i 0 0 −1 0 0 0 0 −i i

magma: CharacterTable(G);