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Properties

Label 36T1195
Degree $36$
Order $648$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^3:D_{12}$

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Copy content magma:G := TransitiveGroup(36, 1195);
 

Group invariants

Abstract group:  $C_3^3:D_{12}$
Copy content magma:IdentifyGroup(G);
 
Order:  $648=2^{3} \cdot 3^{4}$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  yes
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 $t$:  $1195$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,11,8,17,13,22,21,30,25,36,31,5)(2,12,9,18,15,23,19,28,26,34,33,6)(3,10,7,16,14,24,20,29,27,35,32,4)$, $(1,20,2,19,3,21)(4,6)(7,15,9,14,8,13)(10,35)(11,34)(12,36)(16,28)(17,30)(18,29)(22,23)(25,32,26,33,27,31)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$ x 2
$8$:  $D_{4}$
$12$:  $D_{6}$ x 2
$24$:  $D_{12}$, $(C_6\times C_2):C_2$
$36$:  $S_3^2$
$72$:  $C_3^2:D_4$, 12T38
$216$:  12T116, 12T118

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: $D_{4}$

Degree 6: $S_3^2$

Degree 9: None

Degree 12: 12T38, 12T118, 12T120, 12T169

Degree 18: None

Low degree siblings

12T169 x 2, 18T210 x 2, 24T1521 x 2, 24T1533, 36T1105 x 2, 36T1106 x 2, 36T1107 x 2, 36T1153, 36T1154, 36T1160, 36T1167, 36T1173, 36T1190 x 2, 36T1195, 36T1225 x 2

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 $2^{18}$ $9$ $2$ $18$ $( 1,20)( 2,21)( 3,19)( 4,22)( 5,23)( 6,24)( 7,25)( 8,26)( 9,27)(10,28)(11,29)(12,30)(13,32)(14,33)(15,31)(16,36)(17,34)(18,35)$
2B $2^{18}$ $18$ $2$ $18$ $( 1,23)( 2,22)( 3,24)( 4,21)( 5,20)( 6,19)( 7,17)( 8,16)( 9,18)(10,14)(11,15)(12,13)(25,34)(26,36)(27,35)(28,33)(29,31)(30,32)$
2C $2^{17},1^{2}$ $54$ $2$ $17$ $( 1,27)( 2,26)( 3,25)( 4,24)( 5,23)( 6,22)( 7,21)( 8,20)( 9,19)(10,16)(11,18)(12,17)(13,14)(28,36)(29,35)(30,34)(31,32)$
3A $3^{12}$ $2$ $3$ $24$ $( 1,27,15)( 2,25,14)( 3,26,13)( 4,17,28)( 5,18,29)( 6,16,30)( 7,33,21)( 8,32,19)( 9,31,20)(10,22,34)(11,23,35)(12,24,36)$
3B $3^{12}$ $2$ $3$ $24$ $( 1,13,25)( 2,15,26)( 3,14,27)( 4,16,29)( 5,17,30)( 6,18,28)( 7,20,32)( 8,21,31)( 9,19,33)(10,24,35)(11,22,36)(12,23,34)$
3C $3^{6},1^{18}$ $4$ $3$ $12$ $( 4, 6, 5)(10,11,12)(16,18,17)(22,23,24)(28,30,29)(34,35,36)$
3D $3^{12}$ $4$ $3$ $24$ $( 1, 3, 2)( 4, 5, 6)( 7, 8, 9)(10,12,11)(13,14,15)(16,17,18)(19,20,21)(22,24,23)(25,27,26)(28,29,30)(31,33,32)(34,36,35)$
3E $3^{12}$ $4$ $3$ $24$ $( 1,26,14)( 2,27,13)( 3,25,15)( 4, 6, 5)( 7,31,19)( 8,33,20)( 9,32,21)(10,12,11)(16,18,17)(22,24,23)(28,30,29)(34,36,35)$
3F1 $3^{12}$ $4$ $3$ $24$ $( 1,14,26)( 2,13,27)( 3,15,25)( 4,30,18)( 5,28,16)( 6,29,17)( 7,20,32)( 8,21,31)( 9,19,33)(10,35,24)(11,36,22)(12,34,23)$
3F-1 $3^{12}$ $4$ $3$ $24$ $( 1,26,14)( 2,27,13)( 3,25,15)( 4,18,30)( 5,16,28)( 6,17,29)( 7,32,20)( 8,31,21)( 9,33,19)(10,24,35)(11,22,36)(12,23,34)$
3G $3^{12}$ $8$ $3$ $24$ $( 1,27,15)( 2,25,14)( 3,26,13)( 4, 6, 5)( 7,32,20)( 8,31,21)( 9,33,19)(10,12,11)(16,18,17)(22,24,23)(28,30,29)(34,36,35)$
3H $3^{12}$ $8$ $3$ $24$ $( 1,26,14)( 2,27,13)( 3,25,15)( 4,28,17)( 5,29,18)( 6,30,16)( 7,33,21)( 8,32,19)( 9,31,20)(10,36,23)(11,34,24)(12,35,22)$
3I $3^{12}$ $8$ $3$ $24$ $( 1,15,27)( 2,14,25)( 3,13,26)( 4,30,18)( 5,28,16)( 6,29,17)( 7,21,33)( 8,19,32)( 9,20,31)(10,35,24)(11,36,22)(12,34,23)$
3J $3^{12}$ $8$ $3$ $24$ $( 1,25,13)( 2,26,15)( 3,27,14)( 4,28,17)( 5,29,18)( 6,30,16)( 7,32,20)( 8,31,21)( 9,33,19)(10,36,23)(11,34,24)(12,35,22)$
3K $3^{9},1^{9}$ $8$ $3$ $18$ $( 1, 3, 2)( 4,18,30)( 5,16,28)( 6,17,29)(10,23,36)(11,24,34)(12,22,35)(13,14,15)(25,27,26)$
3L1 $3^{9},1^{9}$ $8$ $3$ $18$ $( 4,29,16)( 5,30,17)( 6,28,18)( 7, 8, 9)(10,34,22)(11,35,23)(12,36,24)(19,20,21)(31,33,32)$
3L-1 $3^{9},1^{9}$ $8$ $3$ $18$ $( 1, 2, 3)( 4,29,16)( 5,30,17)( 6,28,18)(10,34,22)(11,35,23)(12,36,24)(13,15,14)(25,26,27)$
4A $4^{9}$ $54$ $4$ $27$ $( 1,29,19,12)( 2,30,20,10)( 3,28,21,11)( 4,33,23,13)( 5,32,24,15)( 6,31,22,14)( 7,35,26,17)( 8,36,27,18)( 9,34,25,16)$
6A $6^{6}$ $18$ $6$ $30$ $( 1,31,27,20,15, 9)( 2,33,25,21,14, 7)( 3,32,26,19,13, 8)( 4,10,17,22,28,34)( 5,11,18,23,29,35)( 6,12,16,24,30,36)$
6B $6^{6}$ $18$ $6$ $30$ $( 1, 9,13,19,25,33)( 2, 7,15,20,26,32)( 3, 8,14,21,27,31)( 4,12,16,23,29,34)( 5,10,17,24,30,35)( 6,11,18,22,28,36)$
6C1 $6^{6}$ $18$ $6$ $30$ $( 1,11,27,23,15,35)( 2,10,25,22,14,34)( 3,12,26,24,13,36)( 4,33,17,21,28, 7)( 5,31,18,20,29, 9)( 6,32,16,19,30, 8)$
6C-1 $6^{6}$ $18$ $6$ $30$ $( 1,24,15,12,27,36)( 2,23,14,11,25,35)( 3,22,13,10,26,34)( 4,32,28, 8,17,19)( 5,33,29, 7,18,21)( 6,31,30, 9,16,20)$
6D $6^{6}$ $36$ $6$ $30$ $( 1, 5, 3, 6, 2, 4)( 7,34, 8,36, 9,35)(10,32,12,31,11,33)(13,30,14,28,15,29)(16,25,17,27,18,26)(19,24,20,23,21,22)$
6E $6^{6}$ $36$ $6$ $30$ $( 1,32,26,21,14, 9)( 2,31,27,19,13, 7)( 3,33,25,20,15, 8)( 4,23, 6,22, 5,24)(10,29,12,28,11,30)(16,34,18,36,17,35)$
6F1 $6^{6}$ $36$ $6$ $30$ $( 1,18,14, 4,26,30)( 2,17,13, 6,27,29)( 3,16,15, 5,25,28)( 7,10,20,35,32,24)( 8,12,21,34,31,23)( 9,11,19,36,33,22)$
6F-1 $6^{6}$ $36$ $6$ $30$ $( 1,30,26, 4,14,18)( 2,29,27, 6,13,17)( 3,28,25, 5,15,16)( 7,24,32,35,20,10)( 8,23,31,34,21,12)( 9,22,33,36,19,11)$
6G $6^{3},2^{8},1^{2}$ $108$ $6$ $23$ $( 1,27)( 2,26)( 3,25)( 4,23, 6,24, 5,22)( 7,21)( 8,20)( 9,19)(10,17,11,16,12,18)(13,14)(28,35,30,36,29,34)(31,32)$
12A1 $12^{3}$ $54$ $12$ $33$ $( 1,23, 9,29,13,34,19, 4,25,12,33,16)( 2,24, 7,30,15,35,20, 5,26,10,32,17)( 3,22, 8,28,14,36,21, 6,27,11,31,18)$
12A5 $12^{3}$ $54$ $12$ $33$ $( 1,34,33,29,25,23,19,16,13,12, 9, 4)( 2,35,32,30,26,24,20,17,15,10, 7, 5)( 3,36,31,28,27,22,21,18,14,11, 8, 6)$

Malle's constant $a(G)$:     $1/12$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 3A 3B 3C 3D 3E 3F1 3F-1 3G 3H 3I 3J 3K 3L1 3L-1 4A 6A 6B 6C1 6C-1 6D 6E 6F1 6F-1 6G 12A1 12A5
Size 1 9 18 54 2 2 4 4 4 4 4 8 8 8 8 8 8 8 54 18 18 18 18 36 36 36 36 108 54 54
2 P 1A 1A 1A 1A 3A 3B 3C 3D 3E 3F-1 3F1 3G 3H 3I 3J 3K 3L-1 3L1 2A 3A 3B 3A 3A 3D 3E 3F1 3F-1 3C 6B 6B
3 P 1A 2A 2B 2C 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 4A 2A 2A 2B 2B 2B 2A 2B 2B 2C 4A 4A
Type
648.723.1a 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 1 1 1 1 1
648.723.1b 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 −1 −1 −1 1 1
648.723.1c 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 −1 −1 1 −1 −1
648.723.1d 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 1 1 −1 −1 −1
648.723.2a R 2 2 0 0 2 −1 2 2 −1 2 2 −1 2 −1 −1 −1 −1 −1 2 0 2 0 −1 0 −1 0 0 0 −1 −1
648.723.2b R 2 2 2 0 −1 2 2 2 −1 −1 −1 −1 −1 −1 2 −1 2 −1 0 −1 −1 −1 2 2 −1 −1 −1 0 0 0
648.723.2c R 2 −2 0 0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 0 0 −2 0 −2 0 −2 0 0 0 0 0
648.723.2d R 2 2 −2 0 −1 2 2 2 −1 −1 −1 −1 −1 −1 2 −1 2 −1 0 1 −1 1 2 −2 −1 1 1 0 0 0
648.723.2e R 2 2 0 0 2 −1 2 2 −1 2 2 −1 2 −1 −1 −1 −1 −1 −2 0 2 0 −1 0 −1 0 0 0 1 1
648.723.2f1 R 2 −2 0 0 2 −1 2 2 −1 2 2 −1 2 −1 −1 −1 −1 −1 0 0 −2 0 1 0 1 0 0 0 −ζ12−1−ζ12 ζ12−1+ζ12
648.723.2f2 R 2 −2 0 0 2 −1 2 2 −1 2 2 −1 2 −1 −1 −1 −1 −1 0 0 −2 0 1 0 1 0 0 0 ζ12−1+ζ12 −ζ12−1−ζ12
648.723.2g1 C 2 −2 0 0 −1 2 2 2 −1 −1 −1 −1 −1 −1 2 −1 2 −1 0 −1−2ζ3 1 1+2ζ3 −2 0 1 1+2ζ3 −1−2ζ3 0 0 0
648.723.2g2 C 2 −2 0 0 −1 2 2 2 −1 −1 −1 −1 −1 −1 2 −1 2 −1 0 1+2ζ3 1 −1−2ζ3 −2 0 1 −1−2ζ3 1+2ζ3 0 0 0
648.723.4a R 4 4 0 0 −2 −2 4 4 1 −2 −2 1 −2 1 −2 1 −2 1 0 0 −2 0 −2 0 1 0 0 0 0 0
648.723.4b R 4 0 0 2 4 4 1 −2 4 −2 −2 1 1 −2 1 −2 −2 1 0 0 0 0 0 0 0 0 0 −1 0 0
648.723.4c R 4 0 2 0 4 4 −2 1 4 1 1 −2 −2 1 −2 1 1 −2 0 2 0 2 0 −1 0 −1 −1 0 0 0
648.723.4d R 4 0 −2 0 4 4 −2 1 4 1 1 −2 −2 1 −2 1 1 −2 0 −2 0 −2 0 1 0 1 1 0 0 0
648.723.4e R 4 0 0 −2 4 4 1 −2 4 −2 −2 1 1 −2 1 −2 −2 1 0 0 0 0 0 0 0 0 0 1 0 0
648.723.4f R 4 −4 0 0 −2 −2 4 4 1 −2 −2 1 −2 1 −2 1 −2 1 0 0 2 0 2 0 −1 0 0 0 0 0
648.723.4g1 C 4 0 2 0 −2 4 −2 1 −2 −2−3ζ3 1+3ζ3 1 1 1+3ζ3 −2 −2−3ζ3 1 1 0 2ζ3 0 2ζ3−1 0 −1 0 −ζ3−1 −ζ3 0 0 0
648.723.4g2 C 4 0 2 0 −2 4 −2 1 −2 1+3ζ3 −2−3ζ3 1 1 −2−3ζ3 −2 1+3ζ3 1 1 0 2ζ3−1 0 2ζ3 0 −1 0 −ζ3 −ζ3−1 0 0 0
648.723.4h1 C 4 0 −2 0 −2 4 −2 1 −2 −2−3ζ3 1+3ζ3 1 1 1+3ζ3 −2 −2−3ζ3 1 1 0 −2ζ3 0 −2ζ3−1 0 1 0 ζ3−1 ζ3 0 0 0
648.723.4h2 C 4 0 −2 0 −2 4 −2 1 −2 1+3ζ3 −2−3ζ3 1 1 −2−3ζ3 −2 1+3ζ3 1 1 0 −2ζ3−1 0 −2ζ3 0 1 0 ζ3 ζ3−1 0 0 0
648.723.8a R 8 0 0 0 8 −4 −4 2 −4 2 2 2 −4 −1 2 −1 −1 2 0 0 0 0 0 0 0 0 0 0 0 0
648.723.8b R 8 0 0 0 8 −4 2 −4 −4 −4 −4 −1 2 2 −1 2 2 −1 0 0 0 0 0 0 0 0 0 0 0 0
648.723.8c R 8 0 0 0 −4 8 2 −4 −4 2 2 −1 −1 2 2 2 −4 −1 0 0 0 0 0 0 0 0 0 0 0 0
648.723.8d R 8 0 0 0 −4 −4 2 −4 2 2 2 −4 −1 −1 −1 −1 2 5 0 0 0 0 0 0 0 0 0 0 0 0
648.723.8e R 8 0 0 0 −4 −4 2 −4 2 2 2 5 −1 −1 −1 −1 2 −4 0 0 0 0 0 0 0 0 0 0 0 0
648.723.8f1 C 8 0 0 0 −4 −4 −4 2 2 −4−6ζ3 2+6ζ3 −1 2 −1−3ζ3 2 2+3ζ3 −1 −1 0 0 0 0 0 0 0 0 0 0 0 0
648.723.8f2 C 8 0 0 0 −4 −4 −4 2 2 2+6ζ3 −4−6ζ3 −1 2 2+3ζ3 2 −1−3ζ3 −1 −1 0 0 0 0 0 0 0 0 0 0 0 0

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Regular extensions

Data not computed