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Properties

Label 36T12781
Degree $36$
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(36, 12781);
 

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 $n$:  $36$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $12781$
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:  $(1,2,4,8,16)(3,6,7,14,25)(5,11,21,32,35)(9,17,18,28,29)(10,19,30,22,33)(12,13,20,24,31)(15,27,34,36,23)$, $(2,3,5,10,18)(4,8,15,26,30)(6,12,17,25,22)(7,13,23,24,29)(9,14,19,27,31)(11,20,16,28,34)(21,32,35,33,36)$
Copy content magma:Generators(G);
 

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: None

Degree 4: None

Degree 6: None

Degree 9: None

Degree 12: None

Degree 18: None

Low degree siblings

27T993, 40T14344, 40T14345, 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 $1^{36}$ $1$ $1$ $0$ $()$
2A $2^{12},1^{12}$ $45$ $2$ $12$ $( 1,26)( 2,27)( 5,36)( 6,10)(11,33)(13,34)(14,29)(15,18)(16,19)(17,31)(24,25)(28,30)$
2B $2^{14},1^{8}$ $270$ $2$ $14$ $( 1,25)( 2,26)( 3,31)( 4,23)( 5, 8)( 9,30)(10,29)(11,34)(12,28)(15,24)(17,35)(18,27)(20,36)(21,22)$
3A1 $3^{12}$ $40$ $3$ $24$ $( 1,12,24)( 2, 3,14)( 4, 7,23)( 5,28,21)( 6,20,26)( 8,18,29)( 9,27,15)(10,25,35)(11,19,34)(13,16,33)(17,22,30)(31,36,32)$
3A-1 $3^{12}$ $40$ $3$ $24$ $( 1,24,12)( 2,14, 3)( 4,23, 7)( 5,21,28)( 6,26,20)( 8,29,18)( 9,15,27)(10,35,25)(11,34,19)(13,33,16)(17,30,22)(31,32,36)$
3B $3^{10},1^{6}$ $240$ $3$ $20$ $( 1, 5,11)( 2,28,21)( 3,19,34)( 6,23,30)( 7,31,25)(10,27,20)(12,24,14)(13,15,22)(16,35,18)(17,32,29)$
3C $3^{11},1^{3}$ $480$ $3$ $22$ $( 1,19,14)( 2,34, 5)( 4,22,35)( 6,17,15)( 7, 9,32)( 8,23,20)(10,31,18)(11,24,28)(13,36,27)(16,29,26)(25,30,33)$
4A $4^{6},2^{6}$ $540$ $4$ $24$ $( 1,26)( 2,33,17,19)( 3,13,22,34)( 4,35)( 5,31,18,27)( 6,12)( 7,10)( 8, 9,21,32)(11,14,16,30)(15,28,36,29)(20,24)(23,25)$
4B $4^{7},2^{3},1^{2}$ $3240$ $4$ $24$ $( 1, 9,25,30)( 2,15,26,24)( 3,35,31,17)( 4,29,23,10)( 5,20, 8,36)( 6,13)( 7,19)(11,22,34,21)(12,18,28,27)(14,33)$
5A $5^{7},1$ $5184$ $5$ $28$ $( 1,11,31,26,24)( 2,22,19,13,34)( 3,21,28,15, 8)( 4,23,35,16,27)( 5,25,14, 7,12)( 6,18,20,30,10)(17,29,32,36,33)$
6A1 $6^{4},3^{4}$ $360$ $6$ $28$ $( 1,24,12)( 2,30, 3,17,14,22)( 4,23, 7)( 5, 8,28,18,21,29)( 6,26,20)( 9,36,27,32,15,31)(10,35,25)(11,13,19,16,34,33)$
6A-1 $6^{4},3^{4}$ $360$ $6$ $28$ $( 1,12,24)( 2,22,14,17, 3,30)( 4, 7,23)( 5,29,21,18,28, 8)( 6,20,26)( 9,31,15,32,27,36)(10,25,35)(11,33,34,16,19,13)$
6B1 $6^{3},3^{4},2^{3}$ $720$ $6$ $26$ $( 1,13,29)( 2,25)( 3,10)( 4, 7,31,36,28,21)( 5,32,23)( 6,15,11)( 8,18,16,33,12,24)( 9,27,20,26,19,34)(14,35)(17,22,30)$
6B-1 $6^{3},3^{4},2^{3}$ $720$ $6$ $26$ $( 1,29,13)( 2,25)( 3,10)( 4,21,28,36,31, 7)( 5,23,32)( 6,11,15)( 8,24,12,33,16,18)( 9,34,19,26,20,27)(14,35)(17,30,22)$
6C $6^{4},3^{3},1^{3}$ $1440$ $6$ $26$ $( 1,29,19,26,14,16)( 2,36,34,27, 5,13)( 4,35,22)( 6,18,17,10,15,31)( 7,32, 9)( 8,20,23)(11,30,24,33,28,25)$
6D $6^{4},3^{2},2^{2},1^{2}$ $2160$ $6$ $26$ $( 1,11, 5)( 2,15,28,22,21,13)( 3, 6,19,23,34,30)( 4, 8)( 7,25,31)(10,18,27,16,20,35)(12,32,24,29,14,17)(26,36)$
9A1 $9^{4}$ $2880$ $9$ $32$ $( 1, 7,21,12,23, 5,24, 4,28)( 2,25,22, 3,35,30,14,10,17)( 6,31,34,20,36,11,26,32,19)( 8, 9,16,18,27,33,29,15,13)$
9A-1 $9^{4}$ $2880$ $9$ $32$ $( 1,28, 4,24, 5,23,12,21, 7)( 2,17,10,14,30,35, 3,22,25)( 6,19,32,26,11,36,20,34,31)( 8,13,15,29,33,27,18,16, 9)$
12A1 $12^{2},6^{2}$ $2160$ $12$ $32$ $( 1, 6,24,26,12,20)( 2,34,30,33, 3,11,17,13,14,19,22,16)( 4,10,23,35, 7,25)( 5,15, 8,31,28, 9,18,36,21,27,29,32)$
12A-1 $12^{2},6^{2}$ $2160$ $12$ $32$ $( 1,20,12,26,24, 6)( 2,16,22,19,14,13,17,11, 3,33,30,34)( 4,25, 7,35,23,10)( 5,32,29,27,21,36,18, 9,28,31, 8,15)$

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