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Properties

Label 40T14345
Degree $40$
Order $25920$
Cyclic no
Abelian no
Solvable no
Primitive yes
$p$-group no
Group: $\PSp(4,3)$

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Copy content magma:G := TransitiveGroup(40, 14345);
 

Group invariants

Abstract group:  $\PSp(4,3)$
Copy content magma:IdentifyGroup(G);
 
Order:  $25920=2^{6} \cdot 3^{4} \cdot 5$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  no
Copy content magma:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $40$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $14345$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  yes
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,2,4,7,10)(3,6,12,21,20)(5,9,16,27,38)(8,14,24,32,34)(11,19,23,31,17)(13,22,30,40,37)(15,25,28,18,29)(26,33,39,36,35)$, $(1,3,4,7,12)(2,5,10,18,22)(6,9,16,27,38)(8,15,26,36,37)(11,20,30,40,34)(13,23,19,29,17)(14,25,28,21,31)(24,33,39,32,35)$
Copy content magma:Generators(G);
 

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Degree 5: None

Degree 8: None

Degree 10: None

Degree 20: None

Low degree siblings

27T993, 36T12781, 40T14344, 45T666

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{40}$ $1$ $1$ $0$ $()$
2A $2^{12},1^{16}$ $45$ $2$ $12$ $( 1,28)( 2,31)( 3,29)( 5,34)( 6,37)( 7,17)( 8,38)( 9,30)(10,22)(12,20)(16,33)(25,27)$
2B $2^{18},1^{4}$ $270$ $2$ $18$ $( 1,34)( 2,27)( 3,35)( 4,37)( 5,12)( 6,26)( 7,15)( 8,14)( 9,28)(10,19)(11,33)(13,38)(16,32)(17,36)(20,30)(21,29)(22,40)(23,24)$
3A1 $3^{12},1^{4}$ $40$ $3$ $24$ $( 1,31,11)( 2, 9,15)( 3,24,16)( 4,13,19)( 5,20,21)( 7,23,29)(10,37,38)(12,27,32)(14,40,26)(17,33,39)(25,34,36)(28,35,30)$
3A-1 $3^{12},1^{4}$ $40$ $3$ $24$ $( 1,11,31)( 2,15, 9)( 3,16,24)( 4,19,13)( 5,21,20)( 7,29,23)(10,38,37)(12,32,27)(14,26,40)(17,39,33)(25,36,34)(28,30,35)$
3B $3^{13},1$ $240$ $3$ $26$ $( 1,12,15)( 2,40,38)( 3, 8, 4)( 5,27,10)( 6,30,13)( 7,37,26)( 9,17,21)(11,14,39)(16,28,18)(19,24,35)(20,33,23)(22,25,34)(29,32,31)$
3C $3^{11},1^{7}$ $480$ $3$ $22$ $( 1,10,33)( 3, 8,25)( 4,13,18)( 5,37, 9)( 6,30,34)(11,15,26)(14,21,32)(16,28,22)(23,39,40)(24,35,36)(27,29,38)$
4A $4^{6},2^{8}$ $540$ $4$ $26$ $( 1,35)( 2,37)( 3,39,27,21)( 4,34,14, 7)( 5,24,17,32)( 6, 8)( 9,38)(10,15)(11,28)(12,20,16,33)(13,36,40,23)(18,22)(19,25,26,29)(30,31)$
4B $4^{9},2,1^{2}$ $3240$ $4$ $28$ $( 1,20,34,30)( 2,24,27,23)( 3,28,35, 9)( 4,17,37,36)( 5,21,12,29)( 6, 7,26,15)( 8,38,14,13)(10,11,19,33)(16,40,32,22)(18,39)$
5A $5^{8}$ $5184$ $5$ $32$ $( 1, 9,33,34,13)( 2, 8,36,32,40)( 3,29,38, 4,31)( 5,35,27,26,28)( 6,14,37,21,19)( 7,16,30,10,23)(11,20,24,12,18)(15,17,22,39,25)$
6A1 $6^{4},3^{4},1^{4}$ $360$ $6$ $28$ $( 1,11,31)( 2,15, 9)( 3,12,24,27,16,32)( 4,26,13,14,19,40)( 5,39,20,17,21,33)( 7,25,23,34,29,36)(10,38,37)(28,30,35)$
6A-1 $6^{4},3^{4},1^{4}$ $360$ $6$ $28$ $( 1,31,11)( 2, 9,15)( 3,32,16,27,24,12)( 4,40,19,14,13,26)( 5,33,21,17,20,39)( 7,36,29,34,23,25)(10,37,38)(28,35,30)$
6B1 $6^{4},3^{5},1$ $720$ $6$ $30$ $( 1,21,19)( 2, 9,15)( 3,10,37,33,39,16)( 4,13, 5,20,31,11)( 6,23,18,29, 8, 7)(12,35,40)(14,30,28,27,32,26)(17,38,24)(25,34,36)$
6B-1 $6^{4},3^{5},1$ $720$ $6$ $30$ $( 1,19,21)( 2,15, 9)( 3,16,39,33,37,10)( 4,11,31,20, 5,13)( 6, 7, 8,29,18,23)(12,40,35)(14,26,32,27,28,30)(17,24,38)(25,36,34)$
6C $6^{3},3^{5},2^{3},1$ $1440$ $6$ $28$ $( 1,16,10,28,33,22)( 2,31)( 3,27, 8,29,25,38)( 4,18,13)( 5,30,37,34, 9, 6)( 7,17)(11,26,15)(12,20)(14,32,21)(23,40,39)(24,36,35)$
6D $6^{6},3,1$ $2160$ $6$ $32$ $( 1, 6,12,30,15,13)( 2, 4,40, 3,38, 8)( 5,11,27,14,10,39)( 7,20,37,33,26,23)( 9,18,17,16,21,28)(19,25,24,34,35,22)(29,31,32)$
9A1 $9^{4},3,1$ $2880$ $9$ $34$ $( 1,33,13,31,39,19,11,17, 4)( 2,23,25, 9,29,34,15, 7,36)( 3,30,38,24,28,10,16,35,37)( 5,26,32,20,14,12,21,40,27)( 6, 8,18)$
9A-1 $9^{4},3,1$ $2880$ $9$ $34$ $( 1, 4,17,11,19,39,31,13,33)( 2,36, 7,15,34,29, 9,25,23)( 3,37,35,16,10,28,24,38,30)( 5,27,40,21,12,14,20,32,26)( 6,18, 8)$
12A1 $12^{2},6^{2},2^{2}$ $2160$ $12$ $34$ $( 1,30,11,35,31,28)( 2,38,15,37, 9,10)( 3, 5,12,39,24,20,27,17,16,21,32,33)( 4,23,26,34,13,29,14,36,19, 7,40,25)( 6, 8)(18,22)$
12A-1 $12^{2},6^{2},2^{2}$ $2160$ $12$ $34$ $( 1,28,31,35,11,30)( 2,10, 9,37,15,38)( 3,33,32,21,16,17,27,20,24,39,12, 5)( 4,25,40, 7,19,36,14,29,13,34,26,23)( 6, 8)(18,22)$

Malle's constant $a(G)$:     $1/12$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 3A1 3A-1 3B 3C 4A 4B 5A 6A1 6A-1 6B1 6B-1 6C 6D 9A1 9A-1 12A1 12A-1
Size 1 45 270 40 40 240 480 540 3240 5184 360 360 720 720 1440 2160 2880 2880 2160 2160
2 P 1A 1A 1A 3A-1 3A1 3B 3C 2A 2B 5A 3A1 3A-1 3B 3B 3C 3B 9A-1 9A1 6A1 6A-1
3 P 1A 2A 2B 1A 1A 1A 1A 4A 4B 5A 2A 2A 2A 2A 2A 2B 3A1 3A-1 4A 4A
5 P 1A 2A 2B 3A-1 3A1 3B 3C 4A 4B 1A 6A-1 6A1 6B-1 6B1 6C 6D 9A-1 9A1 12A-1 12A1
Type
25920.a.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
25920.a.5a1 C 5 −3 1 −1−3ζ3 2+3ζ3 −1 2 1 −1 0 −2−ζ3 −1+ζ3 1+2ζ3 −1−2ζ3 0 1 −ζ3−1 −ζ3 ζ3−1 ζ3
25920.a.5a2 C 5 −3 1 2+3ζ3 −1−3ζ3 −1 2 1 −1 0 −1+ζ3 −2−ζ3 −1−2ζ3 1+2ζ3 0 1 −ζ3 −ζ3−1 ζ3 ζ3−1
25920.a.6a R 6 −2 2 −3 −3 3 0 2 0 1 1 1 1 1 −2 −1 0 0 −1 −1
25920.a.10a1 C 10 2 −2 −5−3ζ3 −2+3ζ3 1 1 2 0 0 2+3ζ3 −1−3ζ3 −1 −1 −1 1 ζ3 ζ3−1 −ζ3−1 −ζ3
25920.a.10a2 C 10 2 −2 −2+3ζ3 −5−3ζ3 1 1 2 0 0 −1−3ζ3 2+3ζ3 −1 −1 −1 1 ζ3−1 ζ3 −ζ3 −ζ3−1
25920.a.15a R 15 7 3 −3 −3 0 3 −1 1 0 1 1 −2 −2 1 0 0 0 −1 −1
25920.a.15b R 15 −1 −1 6 6 3 0 3 −1 0 2 2 −1 −1 2 −1 0 0 0 0
25920.a.20a R 20 4 4 2 2 5 −1 0 0 0 −2 −2 1 1 1 1 −1 −1 0 0
25920.a.24a R 24 8 0 6 6 0 3 0 0 −1 2 2 2 2 −1 0 0 0 0 0
25920.a.30a R 30 −10 2 3 3 3 3 −2 0 0 −1 −1 −1 −1 −1 −1 0 0 1 1
25920.a.30b1 C 30 6 2 −6−9ζ3 3+9ζ3 −3 0 2 0 0 −1+ζ3 −2−ζ3 −1−2ζ3 1+2ζ3 0 −1 0 0 −ζ3 −ζ3−1
25920.a.30b2 C 30 6 2 3+9ζ3 −6−9ζ3 −3 0 2 0 0 −2−ζ3 −1+ζ3 1+2ζ3 −1−2ζ3 0 −1 0 0 −ζ3−1 −ζ3
25920.a.40a1 C 40 −8 0 −8−6ζ3 −2+6ζ3 −2 1 0 0 0 −2ζ3−1 −2ζ3 −2ζ3−1 −2ζ3 1 0 ζ3−1 ζ3 0 0
25920.a.40a2 C 40 −8 0 −2+6ζ3 −8−6ζ3 −2 1 0 0 0 −2ζ3 −2ζ3−1 −2ζ3 −2ζ3−1 1 0 ζ3 ζ3−1 0 0
25920.a.45a1 C 45 −3 −3 −9ζ3 −9ζ3−1 0 0 1 1 0 3ζ3−1 3ζ3 0 0 0 0 0 0 ζ3 ζ3−1
25920.a.45a2 C 45 −3 −3 −9ζ3−1 −9ζ3 0 0 1 1 0 3ζ3 3ζ3−1 0 0 0 0 0 0 ζ3−1 ζ3
25920.a.60a R 60 −4 4 6 6 −3 −3 0 0 0 2 2 −1 −1 −1 1 0 0 0 0
25920.a.64a R 64 0 0 −8 −8 4 −2 0 0 −1 0 0 0 0 0 0 1 1 0 0
25920.a.81a R 81 9 −3 0 0 0 0 −3 −1 1 0 0 0 0 0 0 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed