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Properties

Label 36T460
Degree $36$
Order $288$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $\PSOPlus(4,3)$

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Show commands: Magma

magma: G := TransitiveGroup(36, 460);
 

Group action invariants

Degree $n$:  $36$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $460$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $\PSOPlus(4,3)$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,22,3,23)(2,21,4,24)(5,17,8,19)(6,18,7,20)(9,13,11,15)(10,14,12,16)(25,33,26,34)(27,35,28,36)(29,32,30,31), (1,5,3,7)(2,6,4,8)(9,10)(13,29,14,30)(15,32,16,31)(17,36)(18,35)(19,34)(20,33)(21,28,24,25)(22,27,23,26), (1,35,15,2,36,16)(3,33,14,4,34,13)(5,32,18,7,30,20)(6,31,17,8,29,19)(9,25,24,11,28,22)(10,26,23,12,27,21)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$6$:  $S_3$ x 4
$18$:  $C_3^2:C_2$
$24$:  $S_4$ x 2
$72$:  12T44 x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$ x 4

Degree 4: None

Degree 6: None

Degree 9: $C_3^2:C_2$

Degree 12: 12T127

Degree 18: 18T117

Low degree siblings

12T127 x 2, 16T710, 18T116, 18T117, 24T636 x 2, 24T637 x 2, 24T693 x 2, 32T9308, 36T324, 36T403 x 2, 36T404 x 2, 36T405, 36T420, 36T460

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

Conjugacy classes

LabelCycle TypeSizeOrderRepresentative
$ 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, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $9$ $2$ $( 5, 6)( 7, 8)( 9,11)(10,12)(13,16)(14,15)(17,19)(18,20)(25,27)(26,28)(33,34) (35,36)$
$ 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 1, 1 $ $36$ $4$ $( 3, 4)( 5, 9, 8,12)( 6,10, 7,11)(13,33,15,35)(14,34,16,36)(17,28,18,27) (19,25,20,26)(21,29)(22,30)(23,31)(24,32)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $36$ $2$ $( 3, 4)( 5,10)( 6, 9)( 7,12)( 8,11)(13,36)(14,35)(15,34)(16,33)(17,25)(18,26) (19,28)(20,27)(21,29)(22,30)(23,31)(24,32)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,16)(14,15)(17,20)(18,19)(21,23) (22,24)(25,27)(26,28)(29,31)(30,32)(33,36)(34,35)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1, 2)( 3, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,14)(15,16)(17,19)(18,20)(21,23) (22,24)(25,28)(26,27)(29,31)(30,32)(33,34)(35,36)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4 $ $36$ $4$ $( 1, 3, 2, 4)( 5, 9, 6,10)( 7,11, 8,12)(13,33,16,36)(14,34,15,35)(17,25,20,27) (18,26,19,28)(21,30,23,32)(22,29,24,31)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4 $ $36$ $4$ $( 1, 3, 2, 4)( 5,10, 7,12)( 6, 9, 8,11)(13,36,14,35)(15,34,16,33)(17,28,19,25) (18,27,20,26)(21,30,23,32)(22,29,24,31)$
$ 6, 6, 6, 6, 6, 6 $ $24$ $6$ $( 1, 5, 9, 4, 7,12)( 2, 6,10, 3, 8,11)(13,17,21,14,18,22)(15,19,24,16,20,23) (25,36,29,28,34,32)(26,35,30,27,33,31)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $8$ $3$ $( 1, 5,10)( 2, 6, 9)( 3, 8,12)( 4, 7,11)(13,20,22)(14,19,21)(15,18,23) (16,17,24)(25,33,32)(26,34,31)(27,36,30)(28,35,29)$
$ 6, 6, 6, 6, 6, 6 $ $24$ $6$ $( 1,13,35, 2,14,36)( 3,16,33, 4,15,34)( 5,17,29, 7,19,31)( 6,18,30, 8,20,32) ( 9,24,27,11,22,26)(10,23,28,12,21,25)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $8$ $3$ $( 1,13,34)( 2,14,33)( 3,16,36)( 4,15,35)( 5,20,31)( 6,19,32)( 7,18,29) ( 8,17,30)( 9,21,25)(10,22,26)(11,23,28)(12,24,27)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $32$ $3$ $( 1,17,28)( 2,18,27)( 3,20,25)( 4,19,26)( 5,21,36)( 6,22,35)( 7,24,33) ( 8,23,34)( 9,16,31)(10,15,32)(11,13,30)(12,14,29)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $32$ $3$ $( 1,21,29)( 2,22,30)( 3,23,32)( 4,24,31)( 5,13,26)( 6,14,25)( 7,15,28) ( 8,16,27)( 9,17,35)(10,18,36)(11,20,33)(12,19,34)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $288=2^{5} \cdot 3^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  288.1026
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 3A 3B 3C 3D 4A 4B 4C 6A 6B
Size 1 3 3 9 36 8 8 32 32 36 36 36 24 24
2 P 1A 1A 1A 1A 1A 3A 3B 3C 3D 2A 2C 2B 3A 3B
3 P 1A 2A 2B 2C 2D 1A 1A 1A 1A 4A 4B 4C 2A 2B
Type
288.1026.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
288.1026.1b R 1 1 1 1 −1 1 1 1 1 −1 −1 −1 1 1
288.1026.2a R 2 2 2 2 0 −1 −1 −1 2 0 0 0 −1 −1
288.1026.2b R 2 2 2 2 0 −1 −1 2 −1 0 0 0 −1 −1
288.1026.2c R 2 2 2 2 0 −1 2 −1 −1 0 0 0 −1 2
288.1026.2d R 2 2 2 2 0 2 −1 −1 −1 0 0 0 2 −1
288.1026.3a R 3 −1 3 −1 1 3 0 0 0 −1 −1 1 −1 0
288.1026.3b R 3 3 −1 −1 1 0 3 0 0 1 −1 −1 0 −1
288.1026.3c R 3 −1 3 −1 −1 3 0 0 0 1 1 −1 −1 0
288.1026.3d R 3 3 −1 −1 −1 0 3 0 0 −1 1 1 0 −1
288.1026.6a R 6 −2 6 −2 0 −3 0 0 0 0 0 0 1 0
288.1026.6b R 6 6 −2 −2 0 0 −3 0 0 0 0 0 0 1
288.1026.9a R 9 −3 −3 1 1 0 0 0 0 −1 1 −1 0 0
288.1026.9b R 9 −3 −3 1 −1 0 0 0 0 1 −1 1 0 0

magma: CharacterTable(G);