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Properties

Label 36T1154
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, 1154);
 

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 $t$:  $1154$
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:  $(1,22,7,29,14,36,20,6,27,12,31,18)(2,23,9,28,15,34,19,5,25,11,32,16)(3,24,8,30,13,35,21,4,26,10,33,17)$, $(1,22,2,24,3,23)(4,21)(5,20)(6,19)(7,16)(8,17)(9,18)(10,13,11,14,12,15)(25,35,26,34,27,36)(28,31)(29,32)(30,33)$
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
$24$:  $D_{12}$, $(C_6\times C_2):C_2$
$36$:  $S_3^2$
$72$:  $C_3^2:D_4$, $C_3:D_{12}$
$216$:  12T116, $C_3^2:D_{12}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: $D_{4}$

Degree 6: $D_{6}$

Degree 9: None

Degree 12: $D_{12}$, 12T120, $C_3^3:D_{12}$ x 2

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, 36T1160, 36T1167, 36T1173, 36T1190 x 2, 36T1195 x 2, 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 $2^{18}$ $9$ $2$ $18$ $( 1,21)( 2,20)( 3,19)( 4,22)( 5,24)( 6,23)( 7,25)( 8,27)( 9,26)(10,28)(11,29)(12,30)(13,32)(14,33)(15,31)(16,35)(17,36)(18,34)$
2B $2^{18}$ $18$ $2$ $18$ $( 1, 6)( 2, 4)( 3, 5)( 7,36)( 8,34)( 9,35)(10,32)(11,33)(12,31)(13,28)(14,29)(15,30)(16,26)(17,25)(18,27)(19,24)(20,22)(21,23)$
2C $2^{17},1^{2}$ $54$ $2$ $17$ $( 1,19)( 2,20)( 3,21)( 4,18)( 5,16)( 6,17)( 7,15)( 8,13)( 9,14)(10,12)(22,35)(23,34)(24,36)(25,31)(26,33)(27,32)(29,30)$
3A $3^{12}$ $2$ $3$ $24$ $( 1, 2, 3)( 4, 5, 6)( 7, 9, 8)(10,11,12)(13,14,15)(16,18,17)(19,21,20)(22,24,23)(25,26,27)(28,29,30)(31,32,33)(34,36,35)$
3B $3^{12}$ $2$ $3$ $24$ $( 1,27,14)( 2,25,15)( 3,26,13)( 4,30,17)( 5,28,16)( 6,29,18)( 7,31,20)( 8,33,21)( 9,32,19)(10,35,24)(11,34,23)(12,36,22)$
3C $3^{6},1^{18}$ $4$ $3$ $12$ $( 1, 3, 2)( 7, 9, 8)(13,15,14)(19,21,20)(25,27,26)(31,32,33)$
3D $3^{12}$ $4$ $3$ $24$ $( 1, 2, 3)( 4, 6, 5)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,24,23)(25,26,27)(28,30,29)(31,33,32)(34,36,35)$
3E $3^{12}$ $4$ $3$ $24$ $( 1,26,15)( 2,27,13)( 3,25,14)( 4,29,16)( 5,30,18)( 6,28,17)( 7,33,19)( 8,32,20)( 9,31,21)(10,36,23)(11,35,22)(12,34,24)$
3F1 $3^{6},1^{18}$ $4$ $3$ $12$ $( 1, 2, 3)(10,11,12)(13,14,15)(22,24,23)(25,26,27)(34,36,35)$
3F-1 $3^{6},1^{18}$ $4$ $3$ $12$ $( 1, 3, 2)(10,12,11)(13,15,14)(22,23,24)(25,27,26)(34,35,36)$
3G $3^{12}$ $8$ $3$ $24$ $( 1,26,15)( 2,27,13)( 3,25,14)( 4,30,17)( 5,28,16)( 6,29,18)( 7,33,19)( 8,32,20)( 9,31,21)(10,34,22)(11,36,24)(12,35,23)$
3H $3^{12}$ $8$ $3$ $24$ $( 1,13,25)( 2,14,26)( 3,15,27)( 4,16,29)( 5,18,30)( 6,17,28)( 7,19,33)( 8,20,32)( 9,21,31)(10,22,34)(11,24,36)(12,23,35)$
3I $3^{9},1^{9}$ $8$ $3$ $18$ $( 1, 2, 3)( 4, 6, 5)(10,12,11)(13,14,15)(16,17,18)(22,23,24)(25,26,27)(28,30,29)(34,35,36)$
3J $3^{12}$ $8$ $3$ $24$ $( 1,13,25)( 2,14,26)( 3,15,27)( 4,17,30)( 5,16,28)( 6,18,29)( 7,19,33)( 8,20,32)( 9,21,31)(10,24,35)(11,23,34)(12,22,36)$
3K $3^{12}$ $8$ $3$ $24$ $( 1,13,25)( 2,14,26)( 3,15,27)( 4,17,30)( 5,16,28)( 6,18,29)( 7,21,32)( 8,19,31)( 9,20,33)(10,23,36)(11,22,35)(12,24,34)$
3L1 $3^{12}$ $8$ $3$ $24$ $( 1,27,14)( 2,25,15)( 3,26,13)( 4,29,16)( 5,30,18)( 6,28,17)( 7,33,19)( 8,32,20)( 9,31,21)(10,35,24)(11,34,23)(12,36,22)$
3L-1 $3^{12}$ $8$ $3$ $24$ $( 1,27,14)( 2,25,15)( 3,26,13)( 4,30,17)( 5,28,16)( 6,29,18)( 7,33,19)( 8,32,20)( 9,31,21)(10,36,23)(11,35,22)(12,34,24)$
4A $4^{9}$ $54$ $4$ $27$ $( 1,28,20,12)( 2,30,19,11)( 3,29,21,10)( 4,32,23,15)( 5,31,22,14)( 6,33,24,13)( 7,36,27,16)( 8,35,26,18)( 9,34,25,17)$
6A $6^{6}$ $18$ $6$ $30$ $( 1,19, 2,21, 3,20)( 4,23, 5,22, 6,24)( 7,27, 9,25, 8,26)(10,30,11,28,12,29)(13,31,14,32,15,33)(16,36,18,35,17,34)$
6B $6^{6}$ $18$ $6$ $30$ $( 1,31,27,20,14, 7)( 2,32,25,19,15, 9)( 3,33,26,21,13, 8)( 4,34,30,23,17,11)( 5,36,28,22,16,12)( 6,35,29,24,18,10)$
6C1 $6^{6}$ $18$ $6$ $30$ $( 1, 5, 2, 6, 3, 4)( 7,34, 9,36, 8,35)(10,31,11,32,12,33)(13,30,14,28,15,29)(16,25,18,26,17,27)(19,22,21,24,20,23)$
6C-1 $6^{6}$ $18$ $6$ $30$ $( 1,17, 3,18, 2,16)( 4,13, 6,15, 5,14)( 7,10, 8,12, 9,11)(19,34,20,35,21,36)(22,32,23,31,24,33)(25,28,27,30,26,29)$
6D $6^{6}$ $36$ $6$ $30$ $( 1,22, 2,24, 3,23)( 4,20, 6,21, 5,19)( 7,18, 8,16, 9,17)(10,13,11,14,12,15)(25,35,26,34,27,36)(28,32,30,31,29,33)$
6E $6^{6}$ $36$ $6$ $30$ $( 1,31,26,21,15, 9)( 2,32,27,20,13, 8)( 3,33,25,19,14, 7)( 4,34,29,24,16,12)( 5,36,30,23,18,10)( 6,35,28,22,17,11)$
6F1 $6^{3},2^{9}$ $36$ $6$ $24$ $( 1,23, 2,22, 3,24)( 4,19)( 5,21)( 6,20)( 7,18)( 8,16)( 9,17)(10,14,11,15,12,13)(25,36,26,35,27,34)(28,33)(29,31)(30,32)$
6F-1 $6^{3},2^{9}$ $36$ $6$ $24$ $( 1,24, 3,22, 2,23)( 4,19)( 5,21)( 6,20)( 7,18)( 8,16)( 9,17)(10,13,12,15,11,14)(25,34,27,35,26,36)(28,33)(29,31)(30,32)$
6G $6^{3},2^{8},1^{2}$ $108$ $6$ $23$ $( 1,20, 3,19, 2,21)( 4,18)( 5,16)( 6,17)( 7,13, 9,15, 8,14)(10,12)(22,35)(23,34)(24,36)(25,33,27,31,26,32)(29,30)$
12A1 $12^{3}$ $54$ $12$ $33$ $( 1,36,31,28,27,22,20,16,14,12, 7, 5)( 2,34,32,30,25,23,19,17,15,11, 9, 4)( 3,35,33,29,26,24,21,18,13,10, 8, 6)$
12A5 $12^{3}$ $54$ $12$ $33$ $( 1,22, 7,28,14,36,20, 5,27,12,31,16)( 2,23, 9,30,15,34,19, 4,25,11,32,17)( 3,24, 8,29,13,35,21, 6,26,10,33,18)$

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

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed