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Properties

Label 36T2333
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, 2333);
 

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 $t$:  $2333$
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:  $(1,8,25,14,19)(2,7,26,13,20)(3,11,27,17,21,5,10,30,16,24)(4,12,28,18,22,6,9,29,15,23)(33,36)(34,35)$, $(1,27,2,28)(3,26,4,25)(5,29)(6,30)(7,34,8,33)(9,31,10,32)(11,35)(12,36)(13,15,14,16)(17,18)(19,21,20,22)(23,24)$
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: $S_4$, $\PSL(2,5)$

Degree 9: None

Degree 12: None

Degree 18: 18T145

Low degree siblings

20T203, 24T2963, 30T258, 30T262, 36T2334, 36T2335, 36T2336, 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 $2^{12},1^{12}$ $3$ $2$ $12$ $( 1, 2)( 3, 4)( 7, 8)( 9,10)(13,14)(15,16)(19,20)(21,22)(25,26)(27,28)(31,32)(33,34)$
2B $2^{12},1^{12}$ $6$ $2$ $12$ $( 1, 6)( 2, 5)( 7,11)( 8,12)(13,17)(14,18)(19,23)(20,24)(25,29)(26,30)(31,35)(32,36)$
2C $2^{12},1^{12}$ $15$ $2$ $12$ $( 7,13)( 8,14)( 9,15)(10,16)(11,17)(12,18)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)$
2D $2^{16},1^{4}$ $45$ $2$ $16$ $( 3, 4)( 5, 6)( 7,20)( 8,19)( 9,21)(10,22)(11,23)(12,24)(13,26)(14,25)(15,27)(16,28)(17,29)(18,30)(33,34)(35,36)$
2E $2^{16},1^{4}$ $90$ $2$ $16$ $( 1,25)( 2,26)( 3,29)( 4,30)( 5,28)( 6,27)( 9,11)(10,12)(13,20)(14,19)(15,24)(16,23)(17,22)(18,21)(33,35)(34,36)$
3A $3^{12}$ $8$ $3$ $24$ $( 1, 4, 6)( 2, 3, 5)( 7,10,11)( 8, 9,12)(13,16,17)(14,15,18)(19,22,23)(20,21,24)(25,28,29)(26,27,30)(31,34,35)(32,33,36)$
3B $3^{12}$ $20$ $3$ $24$ $( 1,19,31)( 2,20,32)( 3,21,33)( 4,22,34)( 5,24,36)( 6,23,35)( 7,26,13)( 8,25,14)( 9,28,15)(10,27,16)(11,30,17)(12,29,18)$
3C $3^{12}$ $160$ $3$ $24$ $( 1,17,22)( 2,18,21)( 3,13,23)( 4,14,24)( 5,15,19)( 6,16,20)( 7,35,27)( 8,36,28)( 9,31,30)(10,32,29)(11,34,25)(12,33,26)$
4A $4^{6},2^{6}$ $6$ $4$ $24$ $( 1, 4, 2, 3)( 5, 6)( 7,10, 8, 9)(11,12)(13,16,14,15)(17,18)(19,22,20,21)(23,24)(25,28,26,27)(29,30)(31,34,32,33)(35,36)$
4B $4^{6},2^{6}$ $90$ $4$ $24$ $( 1,34, 2,33)( 3,31, 4,32)( 5,35)( 6,36)( 7,21, 8,22)( 9,20,10,19)(11,23)(12,24)(13,16,14,15)(17,18)(25,28,26,27)(29,30)$
5A1 $5^{6},1^{6}$ $12$ $5$ $24$ $( 7,26,32,13,20)( 8,25,31,14,19)( 9,28,34,15,22)(10,27,33,16,21)(11,30,36,17,24)(12,29,35,18,23)$
5A2 $5^{6},1^{6}$ $12$ $5$ $24$ $( 7,32,20,26,13)( 8,31,19,25,14)( 9,34,22,28,15)(10,33,21,27,16)(11,36,24,30,17)(12,35,23,29,18)$
6A $6^{4},3^{4}$ $60$ $6$ $28$ $( 1,32,19, 2,31,20)( 3,34,21, 4,33,22)( 5,36,24)( 6,35,23)( 7,14,26, 8,13,25)( 9,16,28,10,15,27)(11,17,30)(12,18,29)$
6B $6^{4},3^{4}$ $120$ $6$ $28$ $( 1,29,14, 6,25,18)( 2,30,13, 5,26,17)( 3,27,16)( 4,28,15)( 7,24,32,11,20,36)( 8,23,31,12,19,35)( 9,22,34)(10,21,33)$
6C $6^{4},3^{4}$ $120$ $6$ $28$ $( 1, 4, 6)( 2, 3, 5)( 7,16,11,13,10,17)( 8,15,12,14, 9,18)(19,22,23)(20,21,24)(25,34,29,31,28,35)(26,33,30,32,27,36)$
10A1 $10^{2},5^{2},2^{2},1^{2}$ $36$ $10$ $28$ $( 1,26,14,32,19, 2,25,13,31,20)( 3,27,16,33,21)( 4,28,15,34,22)( 5,29,17,35,24, 6,30,18,36,23)( 7, 8)(11,12)$
10A3 $10^{2},5^{2},2^{2},1^{2}$ $36$ $10$ $28$ $( 1,32,25,20,14, 2,31,26,19,13)( 3,33,27,21,16)( 4,34,28,22,15)( 5,35,30,23,17, 6,36,29,24,18)( 7, 8)(11,12)$
10B1 $10^{2},5^{2},2^{2},1^{2}$ $72$ $10$ $28$ $( 1, 8,25,14,19)( 2, 7,26,13,20)( 3,11,27,17,21, 5,10,30,16,24)( 4,12,28,18,22, 6, 9,29,15,23)(33,36)(34,35)$
10B3 $10^{2},5^{2},2^{2},1^{2}$ $72$ $10$ $28$ $( 1,14, 8,19,25)( 2,13, 7,20,26)( 3,17,10,24,27, 5,16,11,21,30)( 4,18, 9,23,28, 6,15,12,22,29)(33,36)(34,35)$
12A $12^{2},6^{2}$ $120$ $12$ $32$ $( 1,21,32, 4,19,33, 2,22,31, 3,20,34)( 5,23,36, 6,24,35)( 7,28,14,10,26,15, 8,27,13, 9,25,16)(11,29,17,12,30,18)$
15A1 $15^{2},3^{2}$ $96$ $15$ $32$ $( 1, 6, 4)( 2, 5, 3)( 7,36,21,26,17,10,32,24,27,13,11,33,20,30,16)( 8,35,22,25,18, 9,31,23,28,14,12,34,19,29,15)$
15A2 $15^{2},3^{2}$ $96$ $15$ $32$ $( 1, 4, 6)( 2, 3, 5)( 7,21,17,32,27,11,20,16,36,26,10,24,13,33,30)( 8,22,18,31,28,12,19,15,35,25, 9,23,14,34,29)$
20A1 $20,10,4,2$ $72$ $20$ $32$ $( 1,36,26,23,14, 5,32,29,19,17, 2,35,25,24,13, 6,31,30,20,18)( 3,34,27,22,16, 4,33,28,21,15)( 7,12, 8,11)( 9,10)$
20A3 $20,10,4,2$ $72$ $20$ $32$ $( 1,23,32,17,25, 6,20,36,14,29, 2,24,31,18,26, 5,19,35,13,30)( 3,22,33,15,27, 4,21,34,16,28)( 7,11, 8,12)( 9,10)$

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