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Properties

Label 36T1173
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, 1173);
 

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 $t$:  $1173$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $3$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,32,3,33,2,31)(4,24,6,22,5,23)(7,13,9,15,8,14)(10,29,11,30,12,28)(16,34,17,36,18,35)(19,25,21,26,20,27)$, $(1,35,15,11,26,23,3,34,13,10,27,24)(2,36,14,12,25,22)(4,7,30,32,16,21,6,9,29,31,17,20)(5,8,28,33,18,19)$
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: $C_2$ x 3

Degree 3: None

Degree 4: $C_2^2$

Degree 6: $S_3^2$, $C_3^2:D_4$

Degree 9: None

Degree 12: $S_3^2$, 12T34

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

Malle's constant $a(G)$:     $1/12$

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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