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Properties

Label 36T1042
Degree $36$
Order $648$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2\times C_3^2:D_{18}$

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Copy content magma:G := TransitiveGroup(36, 1042);
 

Group invariants

Abstract group:  $C_2\times C_3^2:D_{18}$
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Order:  $648=2^{3} \cdot 3^{4}$
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Cyclic:  no
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Abelian:  no
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Solvable:  yes
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Nilpotency class:   not nilpotent
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Group action invariants

Degree $n$:  $36$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $1042$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
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$\card{\Aut(F/K)}$:  $2$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,19,6,23,4,22)(2,20,5,24,3,21)(7,31,9,36,11,34)(8,32,10,35,12,33)(13,27,17,25,15,29)(14,28,18,26,16,30)$, $(1,36,14,25,12,20,6,31,18,27,8,21,4,34,16,29,10,24)(2,35,13,26,11,19,5,32,17,28,7,22,3,33,15,30,9,23)$, $(1,9,18,3,8,13,6,11,16,2,10,17,4,7,14,5,12,15)(19,27,35,24,26,34,22,29,32,20,28,36,23,25,33,21,30,31)$
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Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$6$:  $S_3$ x 2
$8$:  $C_2^3$
$12$:  $D_{6}$ x 6
$18$:  $D_{9}$
$24$:  $S_3 \times C_2^2$ x 2
$36$:  $S_3^2$, $D_{18}$ x 3
$72$:  12T37, 36T48
$108$:  $C_3^2 : D_{6} $, 18T50
$216$:  18T94, 36T228
$324$:  18T132

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: $S_3$

Degree 4: $C_2^2$

Degree 6: $D_{6}$ x 3

Degree 9: None

Degree 12: $S_3 \times C_2^2$

Degree 18: 18T132

Low degree siblings

36T1042

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 $2^{18}$ $1$ $2$ $18$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)$
2B $2^{18}$ $9$ $2$ $18$ $( 1,29)( 2,30)( 3,28)( 4,27)( 5,26)( 6,25)( 7,35)( 8,36)( 9,32)(10,31)(11,33)(12,34)(13,19)(14,20)(15,23)(16,24)(17,22)(18,21)$
2C $2^{18}$ $9$ $2$ $18$ $( 1,28)( 2,27)( 3,25)( 4,26)( 5,29)( 6,30)( 7,34)( 8,33)( 9,36)(10,35)(11,31)(12,32)(13,24)(14,23)(15,21)(16,22)(17,20)(18,19)$
2D $2^{18}$ $27$ $2$ $18$ $( 1,25)( 2,26)( 3,28)( 4,27)( 5,30)( 6,29)( 7,19)( 8,20)( 9,23)(10,24)(11,22)(12,21)(13,32)(14,31)(15,33)(16,34)(17,35)(18,36)$
2E $2^{16},1^{4}$ $27$ $2$ $16$ $( 1, 4)( 2, 3)( 7,13)( 8,14)( 9,15)(10,16)(11,17)(12,18)(19,35)(20,36)(21,34)(22,33)(23,32)(24,31)(27,29)(28,30)$
2F $2^{18}$ $27$ $2$ $18$ $( 1,11)( 2,12)( 3, 8)( 4, 7)( 5,10)( 6, 9)(13,14)(15,18)(16,17)(19,24)(20,23)(21,22)(25,35)(26,36)(27,33)(28,34)(29,32)(30,31)$
2G $2^{18}$ $27$ $2$ $18$ $( 1,23)( 2,24)( 3,20)( 4,19)( 5,21)( 6,22)( 7,31)( 8,32)( 9,36)(10,35)(11,34)(12,33)(13,29)(14,30)(15,25)(16,26)(17,27)(18,28)$
3A $3^{12}$ $2$ $3$ $24$ $( 1, 4, 6)( 2, 3, 5)( 7,11, 9)( 8,12,10)(13,15,17)(14,16,18)(19,22,23)(20,21,24)(25,27,29)(26,28,30)(31,34,36)(32,33,35)$
3B $3^{12}$ $2$ $3$ $24$ $( 1, 6, 4)( 2, 5, 3)( 7, 9,11)( 8,10,12)(13,17,15)(14,18,16)(19,22,23)(20,21,24)(25,27,29)(26,28,30)(31,34,36)(32,33,35)$
3C $3^{6},1^{18}$ $4$ $3$ $12$ $(19,23,22)(20,24,21)(25,29,27)(26,30,28)(31,36,34)(32,35,33)$
3D $3^{8},1^{12}$ $6$ $3$ $16$ $( 1, 4, 6)( 2, 3, 5)( 7, 9,11)( 8,10,12)(25,27,29)(26,28,30)(31,36,34)(32,35,33)$
3E $3^{8},1^{12}$ $12$ $3$ $16$ $( 7,11, 9)( 8,12,10)(13,17,15)(14,18,16)(19,22,23)(20,21,24)(25,29,27)(26,30,28)$
6A $6^{6}$ $2$ $6$ $30$ $( 1, 5, 4, 2, 6, 3)( 7,10,11, 8, 9,12)(13,18,15,14,17,16)(19,24,22,20,23,21)(25,30,27,26,29,28)(31,35,34,32,36,33)$
6B $6^{6}$ $2$ $6$ $30$ $( 1, 3, 6, 2, 4, 5)( 7,12, 9, 8,11,10)(13,16,17,14,15,18)(19,24,22,20,23,21)(25,30,27,26,29,28)(31,35,34,32,36,33)$
6C $6^{3},2^{9}$ $4$ $6$ $24$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,21,23,20,22,24)(25,28,29,26,27,30)(31,33,36,32,34,35)$
6D $6^{4},2^{6}$ $6$ $6$ $26$ $( 1, 5, 4, 2, 6, 3)( 7,12, 9, 8,11,10)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,30,27,26,29,28)(31,33,36,32,34,35)$
6E $6^{4},2^{6}$ $12$ $6$ $26$ $( 1, 2)( 3, 4)( 5, 6)( 7,10,11, 8, 9,12)(13,16,17,14,15,18)(19,24,22,20,23,21)(25,28,29,26,27,30)(31,32)(33,34)(35,36)$
6F $6^{6}$ $18$ $6$ $30$ $( 1,27, 6,29, 4,25)( 2,28, 5,30, 3,26)( 7,33, 9,35,11,32)( 8,34,10,36,12,31)(13,23,17,19,15,22)(14,24,18,20,16,21)$
6G $6^{6}$ $18$ $6$ $30$ $( 1,26, 6,28, 4,30)( 2,25, 5,27, 3,29)( 7,31, 9,34,11,36)( 8,32,10,33,12,35)(13,21,17,24,15,20)(14,22,18,23,16,19)$
6H $6^{6}$ $54$ $6$ $30$ $( 1,27, 6,25, 4,29)( 2,28, 5,26, 3,30)( 7,22, 9,19,11,23)( 8,21,10,20,12,24)(13,33,17,32,15,35)(14,34,18,31,16,36)$
6I $6^{4},2^{4},1^{4}$ $54$ $6$ $24$ $( 1, 4)( 2, 3)( 7,15,11,13, 9,17)( 8,16,12,14,10,18)(19,33,23,35,22,32)(20,34,24,36,21,31)(27,29)(28,30)$
6J $6^{4},2^{6}$ $54$ $6$ $26$ $( 1, 7, 6,11, 4, 9)( 2, 8, 5,12, 3,10)(13,14)(15,18)(16,17)(19,24)(20,23)(21,22)(25,33,29,35,27,32)(26,34,30,36,28,31)$
6K $6^{6}$ $54$ $6$ $30$ $( 1,22, 4,23, 6,19)( 2,21, 3,24, 5,20)( 7,36,11,31, 9,34)( 8,35,12,32,10,33)(13,27,15,29,17,25)(14,28,16,30,18,26)$
9A1 $9^{4}$ $6$ $9$ $32$ $( 1,16, 8, 6,14,10, 4,18,12)( 2,15, 7, 5,13, 9, 3,17,11)(19,35,30,22,32,26,23,33,28)(20,36,29,21,31,25,24,34,27)$
9A2 $9^{4}$ $6$ $9$ $32$ $( 1, 8,14, 4,12,16, 6,10,18)( 2, 7,13, 3,11,15, 5, 9,17)(19,30,32,23,28,35,22,26,33)(20,29,31,24,27,36,21,25,34)$
9A4 $9^{4}$ $6$ $9$ $32$ $( 1,14,12, 6,18, 8, 4,16,10)( 2,13,11, 5,17, 7, 3,15, 9)(19,32,28,22,33,30,23,35,26)(20,31,27,21,34,29,24,36,25)$
9B1 $9^{4}$ $12$ $9$ $32$ $( 1,14, 8, 6,18,10, 4,16,12)( 2,13, 7, 5,17, 9, 3,15,11)(19,32,26,22,33,28,23,35,30)(20,31,25,21,34,27,24,36,29)$
9B2 $9^{4}$ $12$ $9$ $32$ $( 1, 8,18, 4,12,14, 6,10,16)( 2, 7,17, 3,11,13, 5, 9,15)(19,26,33,23,30,32,22,28,35)(20,25,34,24,29,31,21,27,36)$
9B4 $9^{4}$ $12$ $9$ $32$ $( 1,18,12, 6,16, 8, 4,14,10)( 2,17,11, 5,15, 7, 3,13, 9)(19,33,30,22,35,26,23,32,28)(20,34,29,21,36,25,24,31,27)$
18A1 $18^{2}$ $6$ $18$ $34$ $( 1, 9,16, 3, 8,17, 6,11,14, 2,10,15, 4, 7,18, 5,12,13)(19,25,35,24,30,34,22,27,32,20,26,36,23,29,33,21,28,31)$
18A5 $18^{2}$ $6$ $18$ $34$ $( 1, 7,14, 3,12,15, 6, 9,18, 2, 8,13, 4,11,16, 5,10,17)(19,29,32,24,28,36,22,25,33,20,30,31,23,27,35,21,26,34)$
18A7 $18^{2}$ $6$ $18$ $34$ $( 1,17,10, 5,16,11, 4,13, 8, 2,18, 9, 6,15,12, 3,14, 7)(19,31,28,21,33,29,23,36,26,20,32,27,22,34,30,24,35,25)$
18B1 $18^{2}$ $12$ $18$ $34$ $( 1, 9,14, 3, 8,15, 6,11,18, 2,10,13, 4, 7,16, 5,12,17)(19,27,32,24,26,36,22,29,33,20,28,31,23,25,35,21,30,34)$
18B5 $18^{2}$ $12$ $18$ $34$ $( 1, 7,18, 3,12,13, 6, 9,16, 2, 8,17, 4,11,14, 5,10,15)(19,25,33,24,30,31,22,27,35,20,26,34,23,29,32,21,28,36)$
18B7 $18^{2}$ $12$ $18$ $34$ $( 1,17, 8, 5,16, 9, 4,13,12, 2,18, 7, 6,15,10, 3,14,11)(19,34,28,21,35,29,23,31,26,20,33,27,22,36,30,24,32,25)$
18C1 $18^{2}$ $18$ $18$ $34$ $( 1,36,14,27,12,24, 6,31,18,29, 8,20, 4,34,16,25,10,21)( 2,35,13,28,11,23, 5,32,17,30, 7,19, 3,33,15,26, 9,22)$
18C5 $18^{2}$ $18$ $18$ $34$ $( 1,34,18,27,10,20, 6,36,16,29,12,21, 4,31,14,25, 8,24)( 2,33,17,28, 9,19, 5,35,15,30,11,22, 3,32,13,26, 7,23)$
18C7 $18^{2}$ $18$ $18$ $34$ $( 1,20,12,25,18,36, 4,24,10,29,14,34, 6,21, 8,27,16,31)( 2,19,11,26,17,35, 3,23, 9,30,13,33, 5,22, 7,28,15,32)$
18D1 $18^{2}$ $18$ $18$ $34$ $( 1,33,14,26,12,22, 6,35,18,28, 8,23, 4,32,16,30,10,19)( 2,34,13,25,11,21, 5,36,17,27, 7,24, 3,31,15,29, 9,20)$
18D5 $18^{2}$ $18$ $18$ $34$ $( 1,32,18,26,10,23, 6,33,16,28,12,19, 4,35,14,30, 8,22)( 2,31,17,25, 9,24, 5,34,15,27,11,20, 3,36,13,29, 7,21)$
18D7 $18^{2}$ $18$ $18$ $34$ $( 1,23,12,30,18,33, 4,22,10,28,14,32, 6,19, 8,26,16,35)( 2,24,11,29,17,34, 3,21, 9,27,13,31, 5,20, 7,25,15,36)$

Malle's constant $a(G)$:     $1/12$

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Character table

42 x 42 character table

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Regular extensions

Data not computed