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Properties

Label 36T2336
Degree $36$
Order $1440$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $S_4\times A_5$

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Copy content magma:G := TransitiveGroup(36, 2336);
 

Group invariants

Abstract group:  $S_4\times A_5$
Copy content magma:IdentifyGroup(G);
 
Order:  $1440=2^{5} \cdot 3^{2} \cdot 5$
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 $t$:  $2336$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $2$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,6,2,5)(3,4)(7,23,13,35,25,11,20,17,32,29,8,24,14,36,26,12,19,18,31,30)(9,22,16,33,27,10,21,15,34,28)$, $(1,20,7,32,13)(2,19,8,31,14)(3,23,9,35,15,5,22,12,34,18)(4,24,10,36,16,6,21,11,33,17)(27,30)(28,29)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$6$:  $S_3$
$24$:  $S_4$
$60$:  $A_5$
$120$:  $A_5\times C_2$
$360$:  $A_5 \times S_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 4: None

Degree 6: $\PSL(2,5)$

Degree 9: None

Degree 12: None

Degree 18: 18T145

Low degree siblings

20T203, 24T2963, 30T258, 30T262, 36T2333, 36T2334, 36T2335, 40T1200, 40T1206

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 $2^{12},1^{12}$ $3$ $2$ $12$ $( 1, 2)( 5, 6)( 7, 8)(11,12)(13,14)(17,18)(19,20)(23,24)(25,26)(29,30)(31,32)(35,36)$
2B $2^{12},1^{12}$ $6$ $2$ $12$ $( 1, 3)( 2, 4)( 7, 9)( 8,10)(13,15)(14,16)(19,21)(20,22)(25,27)(26,28)(31,33)(32,34)$
2C $2^{18}$ $15$ $2$ $18$ $( 1, 8)( 2, 7)( 3,10)( 4, 9)( 5,11)( 6,12)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)$
2D $2^{14},1^{8}$ $45$ $2$ $14$ $( 1,25)( 2,26)( 3,28)( 4,27)( 5,29)( 6,30)( 7,32)( 8,31)( 9,33)(10,34)(11,35)(12,36)(13,14)(19,20)$
2E $2^{18}$ $90$ $2$ $18$ $( 1, 2)( 3, 5)( 4, 6)( 7,26)( 8,25)( 9,30)(10,29)(11,28)(12,27)(13,14)(15,18)(16,17)(19,31)(20,32)(21,35)(22,36)(23,33)(24,34)$
3A $3^{12}$ $8$ $3$ $24$ $( 1, 5, 3)( 2, 6, 4)( 7,12, 9)( 8,11,10)(13,18,15)(14,17,16)(19,24,21)(20,23,22)(25,30,27)(26,29,28)(31,36,33)(32,35,34)$
3B $3^{12}$ $20$ $3$ $24$ $( 1,31, 8)( 2,32, 7)( 3,33,10)( 4,34, 9)( 5,36,11)( 6,35,12)(13,20,25)(14,19,26)(15,22,27)(16,21,28)(17,24,29)(18,23,30)$
3C $3^{12}$ $160$ $3$ $24$ $( 1,15,35)( 2,16,36)( 3,18,31)( 4,17,32)( 5,14,33)( 6,13,34)( 7,27,23)( 8,28,24)( 9,30,19)(10,29,20)(11,25,22)(12,26,21)$
4A $4^{6},2^{6}$ $6$ $4$ $24$ $( 1, 5, 2, 6)( 3, 4)( 7,12, 8,11)( 9,10)(13,18,14,17)(15,16)(19,24,20,23)(21,22)(25,30,26,29)(27,28)(31,36,32,35)(33,34)$
4B $4^{6},2^{4},1^{4}$ $90$ $4$ $22$ $( 1,29, 2,30)( 3,27)( 4,28)( 5,25, 6,26)( 7,11, 8,12)(13,23,14,24)(15,21)(16,22)(17,20,18,19)(31,35,32,36)$
5A1 $5^{6},1^{6}$ $12$ $5$ $24$ $( 1,31,14, 7,26)( 2,32,13, 8,25)( 3,33,16, 9,28)( 4,34,15,10,27)( 5,36,17,12,29)( 6,35,18,11,30)$
5A2 $5^{6},1^{6}$ $12$ $5$ $24$ $( 1,14,26,31, 7)( 2,13,25,32, 8)( 3,16,28,33, 9)( 4,15,27,34,10)( 5,17,29,36,12)( 6,18,30,35,11)$
6A $6^{4},3^{4}$ $60$ $6$ $28$ $( 1, 7,31, 2, 8,32)( 3,10,33)( 4, 9,34)( 5,12,36, 6,11,35)(13,26,20,14,25,19)(15,27,22)(16,28,21)(17,30,24,18,29,23)$
6B $6^{4},3^{4}$ $120$ $6$ $28$ $( 1,28,13, 3,26,15)( 2,27,14, 4,25,16)( 5,29,18)( 6,30,17)( 7,21,31, 9,19,33)( 8,22,32,10,20,34)(11,23,35)(12,24,36)$
6C $6^{6}$ $120$ $6$ $30$ $( 1,11, 3, 8, 5,10)( 2,12, 4, 7, 6, 9)(13,36,15,31,18,33)(14,35,16,32,17,34)(19,23,21,20,24,22)(25,29,27,26,30,28)$
10A1 $10^{2},5^{2},2^{2},1^{2}$ $36$ $10$ $28$ $( 1, 2)( 3, 4)( 7,13,25,20,32, 8,14,26,19,31)( 9,15,27,22,34,10,16,28,21,33)(11,18,29,23,36)(12,17,30,24,35)$
10A3 $10^{2},5^{2},2^{2},1^{2}$ $36$ $10$ $28$ $( 1, 2)( 3, 4)( 7,20,14,31,25, 8,19,13,32,26)( 9,22,16,33,27,10,21,15,34,28)(11,23,18,36,29)(12,24,17,35,30)$
10B1 $10^{2},5^{2},2^{2},1^{2}$ $72$ $10$ $28$ $( 1,32, 8,26,19)( 2,31, 7,25,20)( 3,35,10,29,21, 5,34,11,28,24)( 4,36, 9,30,22, 6,33,12,27,23)(15,18)(16,17)$
10B3 $10^{2},5^{2},2^{2},1^{2}$ $72$ $10$ $28$ $( 1,26,32,19, 8)( 2,25,31,20, 7)( 3,29,34,24,10, 5,28,35,21,11)( 4,30,33,23, 9, 6,27,36,22,12)(15,18)(16,17)$
12A $12^{2},6^{2}$ $120$ $12$ $32$ $( 1,35, 7, 5,31,12, 2,36, 8, 6,32,11)( 3,34,10, 4,33, 9)(13,24,26,18,20,29,14,23,25,17,19,30)(15,21,27,16,22,28)$
15A1 $15^{2},3^{2}$ $96$ $15$ $32$ $( 1,16,29,31, 9, 5,14,28,36, 7, 3,17,26,33,12)( 2,15,30,32,10, 6,13,27,35, 8, 4,18,25,34,11)(19,21,24)(20,22,23)$
15A2 $15^{2},3^{2}$ $96$ $15$ $32$ $( 1,29, 9,14,36, 3,26,12,16,31, 5,28, 7,17,33)( 2,30,10,13,35, 4,25,11,15,32, 6,27, 8,18,34)(19,24,21)(20,23,22)$
20A1 $20,10,4,2$ $72$ $20$ $32$ $( 1, 4, 2, 3)( 5, 6)( 7,22,13,34,25,10,20,16,32,28, 8,21,14,33,26, 9,19,15,31,27)(11,24,18,35,29,12,23,17,36,30)$
20A3 $20,10,4,2$ $72$ $20$ $32$ $( 1, 3, 2, 4)( 5, 6)( 7,34,20,28,14, 9,31,22,25,16, 8,33,19,27,13,10,32,21,26,15)(11,35,23,30,18,12,36,24,29,17)$

Malle's constant $a(G)$:     $1/12$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 2E 3A 3B 3C 4A 4B 5A1 5A2 6A 6B 6C 10A1 10A3 10B1 10B3 12A 15A1 15A2 20A1 20A3
Size 1 3 6 15 45 90 8 20 160 6 90 12 12 60 120 120 36 36 72 72 120 96 96 72 72
2 P 1A 1A 1A 1A 1A 1A 3A 3B 3C 2A 2A 5A2 5A1 3B 3B 3A 5A2 5A1 5A1 5A2 6A 15A2 15A1 10A1 10A3
3 P 1A 2A 2B 2C 2D 2E 1A 1A 1A 4A 4B 5A2 5A1 2A 2B 2C 10A3 10A1 10B3 10B1 4A 5A1 5A2 20A3 20A1
5 P 1A 2A 2B 2C 2D 2E 3A 3B 3C 4A 4B 1A 1A 6A 6B 6C 2A 2A 2B 2B 12A 3A 3A 4A 4A
Type
1440.5848.1a 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
1440.5848.1b 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
1440.5848.2a R 2 2 0 2 2 0 −1 2 −1 0 0 2 2 2 0 −1 2 2 0 0 0 −1 −1 0 0
1440.5848.3a R 3 −1 1 3 −1 1 0 3 0 −1 −1 3 3 −1 1 0 −1 −1 1 1 −1 0 0 −1 −1
1440.5848.3b R 3 −1 −1 3 −1 −1 0 3 0 1 1 3 3 −1 −1 0 −1 −1 −1 −1 1 0 0 1 1
1440.5848.3c1 R 3 3 3 −1 −1 −1 3 0 0 3 −1 −ζ5−1−ζ5 −ζ5−2−ζ52 0 0 −1 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 0 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5
1440.5848.3c2 R 3 3 3 −1 −1 −1 3 0 0 3 −1 −ζ5−2−ζ52 −ζ5−1−ζ5 0 0 −1 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 0 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52
1440.5848.3d1 R 3 3 −3 −1 −1 1 3 0 0 −3 1 −ζ5−1−ζ5 −ζ5−2−ζ52 0 0 −1 −ζ5−1−ζ5 −ζ5−2−ζ52 ζ5−2+ζ52 ζ5−1+ζ5 0 −ζ5−2−ζ52 −ζ5−1−ζ5 ζ5−2+ζ52 ζ5−1+ζ5
1440.5848.3d2 R 3 3 −3 −1 −1 1 3 0 0 −3 1 −ζ5−2−ζ52 −ζ5−1−ζ5 0 0 −1 −ζ5−2−ζ52 −ζ5−1−ζ5 ζ5−1+ζ5 ζ5−2+ζ52 0 −ζ5−1−ζ5 −ζ5−2−ζ52 ζ5−1+ζ5 ζ5−2+ζ52
1440.5848.4a R 4 4 4 0 0 0 4 1 1 4 0 −1 −1 1 1 0 −1 −1 −1 −1 1 −1 −1 −1 −1
1440.5848.4b R 4 4 −4 0 0 0 4 1 1 −4 0 −1 −1 1 −1 0 −1 −1 1 1 −1 −1 −1 1 1
1440.5848.5a R 5 5 5 1 1 1 5 −1 −1 5 1 0 0 −1 −1 1 0 0 0 0 −1 0 0 0 0
1440.5848.5b R 5 5 −5 1 1 −1 5 −1 −1 −5 −1 0 0 −1 1 1 0 0 0 0 1 0 0 0 0
1440.5848.6a1 R 6 6 0 −2 −2 0 −3 0 0 0 0 −2ζ5−1−2ζ5 −2ζ5−2−2ζ52 0 0 1 −2ζ5−1−2ζ5 −2ζ5−2−2ζ52 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5 0 0
1440.5848.6a2 R 6 6 0 −2 −2 0 −3 0 0 0 0 −2ζ5−2−2ζ52 −2ζ5−1−2ζ5 0 0 1 −2ζ5−2−2ζ52 −2ζ5−1−2ζ5 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52 0 0
1440.5848.8a R 8 8 0 0 0 0 −4 2 −1 0 0 −2 −2 2 0 0 −2 −2 0 0 0 1 1 0 0
1440.5848.9a1 R 9 −3 −3 −3 1 1 0 0 0 3 −1 −3ζ5−1−3ζ5 −3ζ5−2−3ζ52 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 0 0 0 −ζ5−2−ζ52 −ζ5−1−ζ5
1440.5848.9a2 R 9 −3 −3 −3 1 1 0 0 0 3 −1 −3ζ5−2−3ζ52 −3ζ5−1−3ζ5 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 0 0 0 −ζ5−1−ζ5 −ζ5−2−ζ52
1440.5848.9b1 R 9 −3 3 −3 1 −1 0 0 0 −3 1 −3ζ5−1−3ζ5 −3ζ5−2−3ζ52 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5
1440.5848.9b2 R 9 −3 3 −3 1 −1 0 0 0 −3 1 −3ζ5−2−3ζ52 −3ζ5−1−3ζ5 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52
1440.5848.10a R 10 10 0 2 2 0 −5 −2 1 0 0 0 0 −2 0 −1 0 0 0 0 0 0 0 0 0
1440.5848.12a R 12 −4 4 0 0 0 0 3 0 −4 0 −3 −3 −1 1 0 1 1 −1 −1 −1 0 0 1 1
1440.5848.12b R 12 −4 −4 0 0 0 0 3 0 4 0 −3 −3 −1 −1 0 1 1 1 1 1 0 0 −1 −1
1440.5848.15a R 15 −5 5 3 −1 1 0 −3 0 −5 −1 0 0 1 −1 0 0 0 0 0 1 0 0 0 0
1440.5848.15b R 15 −5 −5 3 −1 −1 0 −3 0 5 1 0 0 1 1 0 0 0 0 0 −1 0 0 0 0

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Regular extensions

Data not computed