LMFDB - Galois group: 36T2046

Show commands: Magma

magma:G := TransitiveGroup(36, 2046);

Group invariants

Abstract group:	$C_3^4:(C_2^2\times C_4)$	magma:IdentifyGroup(G);
Order:	$1296=2^{4} \cdot 3^{4}$	magma:Order(G);
Cyclic:	no	magma:IsCyclic(G);
Abelian:	no	magma:IsAbelian(G);
Solvable:	yes	magma:IsSolvable(G);
Nilpotency class:	not nilpotent	magma:NilpotencyClass(G);

Group action invariants

Degree $n$:	$36$	magma:t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$2046$	magma:t, n := TransitiveGroupIdentification(G); t;
Parity:	$-1$	magma:IsEven(G);
Primitive:	no	magma:IsPrimitive(G);
$\card{\Aut(F/K)}$:	$2$	magma:Order(Centralizer(SymmetricGroup(n), G));
Generators:	$(1,9,14,4,8,15)(2,10,13,3,7,16)(5,11,18)(6,12,17)(19,30,32,24,25,36)(20,29,31,23,26,35)(21,28,33)(22,27,34)$, $(1,36,6,31)(2,35,5,32)(3,34,4,33)(7,29,18,19)(8,30,17,20)(9,28,16,22)(10,27,15,21)(11,25,13,23)(12,26,14,24)$, $(1,17,9,2,18,10)(3,14,12,4,13,11)(5,16,8,6,15,7)(19,26,32,20,25,31)(21,27,33,22,28,34)(23,30,35,24,29,36)$	magma:Generators(G);

Low degree resolvents

$\card{(G/N)}$ Galois groups for stem field(s)
$2$: $C_2$ x 7
$4$: $C_4$ x 4, $C_2^2$ x 7
$8$: $C_4\times C_2$ x 6, $C_2^3$
$16$: $C_4\times C_2^2$
$36$: $C_3^2:C_4$ x 2
$72$: 12T40 x 6
$144$: 24T241 x 2
$648$: 12T171

Resolvents shown for degrees $\leq 47$

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 7
$4$:	$C_4$ x 4, $C_2^2$ x 7
$8$:	$C_4\times C_2$ x 6, $C_2^3$
$16$:	$C_4\times C_2^2$
$36$:	$C_3^2:C_4$ x 2
$72$:	12T40 x 6
$144$:	24T241 x 2
$648$:	12T171

Subfields

Degree 2: $C_2$
Degree 3: None
Degree 4: $C_4$
Degree 6: $C_3^2:C_4$ x 2
Degree 9: None
Degree 12: 12T41 x 2
Degree 18: 18T190

Low degree siblings

24T2857 x 8, 24T2860 x 8, 36T2046 x 15, 36T2047 x 16, 36T2059 x 16
Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

Label	Cycle Type	Size	Order	Index	Representative
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	$2^{12},1^{12}$	$9$	$2$	$12$	$( 1, 5)( 2, 6)( 7,12)( 8,11)(13,17)(14,18)(21,24)(22,23)(27,29)(28,30)(33,36)(34,35)$
2C	$2^{18}$	$9$	$2$	$18$	$( 1, 6)( 2, 5)( 3, 4)( 7,11)( 8,12)( 9,10)(13,18)(14,17)(15,16)(19,20)(21,23)(22,24)(25,26)(27,30)(28,29)(31,32)(33,35)(34,36)$
2D	$2^{18}$	$9$	$2$	$18$	$( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,12)( 6,11)(13,14)(15,16)(17,18)(19,26)(20,25)(21,27)(22,28)(23,30)(24,29)(31,32)(33,34)(35,36)$
2E	$2^{12},1^{12}$	$9$	$2$	$12$	$( 1, 8)( 2, 7)( 3,10)( 4, 9)( 5,11)( 6,12)(19,25)(20,26)(21,28)(22,27)(23,29)(24,30)$
2F	$2^{16},1^{4}$	$81$	$2$	$16$	$( 1, 9)( 2,10)( 3, 7)( 4, 8)( 5,11)( 6,12)(13,16)(14,15)(19,25)(20,26)(21,30)(22,29)(23,27)(24,28)(33,36)(34,35)$
2G	$2^{18}$	$81$	$2$	$18$	$( 1, 7)( 2, 8)( 3,11)( 4,12)( 5,10)( 6, 9)(13,14)(15,17)(16,18)(19,23)(20,24)(21,22)(25,35)(26,36)(27,33)(28,34)(29,32)(30,31)$
3A	$3^{6},1^{18}$	$4$	$3$	$12$	$(19,32,25)(20,31,26)(21,33,28)(22,34,27)(23,35,29)(24,36,30)$
3B	$3^{12}$	$4$	$3$	$24$	$( 1, 8,14)( 2, 7,13)( 3,10,16)( 4, 9,15)( 5,11,18)( 6,12,17)(19,32,25)(20,31,26)(21,33,28)(22,34,27)(23,35,29)(24,36,30)$
3C	$3^{12}$	$4$	$3$	$24$	$( 1, 4, 5)( 2, 3, 6)( 7,10,12)( 8, 9,11)(13,16,17)(14,15,18)(19,24,21)(20,23,22)(25,30,28)(26,29,27)(31,35,34)(32,36,33)$
3D	$3^{6},1^{18}$	$4$	$3$	$12$	$( 1, 5, 4)( 2, 6, 3)( 7,12,10)( 8,11, 9)(13,17,16)(14,18,15)$
3E	$3^{12}$	$8$	$3$	$24$	$( 1,18, 9)( 2,17,10)( 3,13,12)( 4,14,11)( 5,15, 8)( 6,16, 7)(19,25,32)(20,26,31)(21,28,33)(22,27,34)(23,29,35)(24,30,36)$
3F	$3^{6},1^{18}$	$8$	$3$	$12$	$( 1,18, 9)( 2,17,10)( 3,13,12)( 4,14,11)( 5,15, 8)( 6,16, 7)$
3G	$3^{12}$	$8$	$3$	$24$	$( 1,18, 9)( 2,17,10)( 3,13,12)( 4,14,11)( 5,15, 8)( 6,16, 7)(19,32,25)(20,31,26)(21,33,28)(22,34,27)(23,35,29)(24,36,30)$
3H	$3^{12}$	$8$	$3$	$24$	$( 1, 4, 5)( 2, 3, 6)( 7,10,12)( 8, 9,11)(13,16,17)(14,15,18)(19,30,33)(20,29,34)(21,25,36)(22,26,35)(23,27,31)(24,28,32)$
3I	$3^{12}$	$8$	$3$	$24$	$( 1,15,11)( 2,16,12)( 3,17, 7)( 4,18, 8)( 5,14, 9)( 6,13,10)(19,33,30)(20,34,29)(21,36,25)(22,35,26)(23,31,27)(24,32,28)$
3J	$3^{12}$	$8$	$3$	$24$	$( 1,14, 8)( 2,13, 7)( 3,16,10)( 4,15, 9)( 5,18,11)( 6,17,12)(19,24,21)(20,23,22)(25,30,28)(26,29,27)(31,35,34)(32,36,33)$
3K	$3^{12}$	$8$	$3$	$24$	$( 1,15,11)( 2,16,12)( 3,17, 7)( 4,18, 8)( 5,14, 9)( 6,13,10)(19,28,36)(20,27,35)(21,30,32)(22,29,31)(23,26,34)(24,25,33)$
3L	$3^{12}$	$8$	$3$	$24$	$( 1, 9,18)( 2,10,17)( 3,12,13)( 4,11,14)( 5, 8,15)( 6, 7,16)(19,24,21)(20,23,22)(25,30,28)(26,29,27)(31,35,34)(32,36,33)$
4A1	$4^{9}$	$81$	$4$	$27$	$( 1,28, 7,34)( 2,27, 8,33)( 3,29,11,32)( 4,30,12,31)( 5,25,10,35)( 6,26, 9,36)(13,22,14,21)(15,24,17,20)(16,23,18,19)$
4A-1	$4^{9}$	$81$	$4$	$27$	$( 1,34, 7,28)( 2,33, 8,27)( 3,32,11,29)( 4,31,12,30)( 5,35,10,25)( 6,36, 9,26)(13,21,14,22)(15,20,17,24)(16,19,18,23)$
4B1	$4^{9}$	$81$	$4$	$27$	$( 1,27,12,32)( 2,28,11,31)( 3,30, 9,35)( 4,29,10,36)( 5,26, 7,33)( 6,25, 8,34)(13,21,18,20)(14,22,17,19)(15,23,16,24)$
4B-1	$4^{9}$	$81$	$4$	$27$	$( 1,32,12,27)( 2,31,11,28)( 3,35, 9,30)( 4,36,10,29)( 5,33, 7,26)( 6,34, 8,25)(13,20,18,21)(14,19,17,22)(15,24,16,23)$
4C1	$4^{9}$	$81$	$4$	$27$	$( 1,33,17,29)( 2,34,18,30)( 3,31,15,25)( 4,32,16,26)( 5,36,13,27)( 6,35,14,28)( 7,22,11,24)( 8,21,12,23)( 9,19,10,20)$
4C-1	$4^{9}$	$81$	$4$	$27$	$( 1,29,17,33)( 2,30,18,34)( 3,25,15,31)( 4,26,16,32)( 5,27,13,36)( 6,28,14,35)( 7,24,11,22)( 8,23,12,21)( 9,20,10,19)$
4D1	$4^{9}$	$81$	$4$	$27$	$( 1,19,13,26)( 2,20,14,25)( 3,22,18,30)( 4,21,17,29)( 5,24,16,27)( 6,23,15,28)( 7,31, 8,32)( 9,33,12,35)(10,34,11,36)$
4D-1	$4^{9}$	$81$	$4$	$27$	$( 1,26,13,19)( 2,25,14,20)( 3,30,18,22)( 4,29,17,21)( 5,27,16,24)( 6,28,15,23)( 7,32, 8,31)( 9,35,12,33)(10,36,11,34)$
6A	$6^{3},2^{9}$	$4$	$6$	$24$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,26,32,20,25,31)(21,27,33,22,28,34)(23,30,35,24,29,36)$
6B	$6^{6}$	$4$	$6$	$30$	$( 1,13, 8, 2,14, 7)( 3,15,10, 4,16, 9)( 5,17,11, 6,18,12)(19,26,32,20,25,31)(21,27,33,22,28,34)(23,30,35,24,29,36)$
6C	$6^{6}$	$4$	$6$	$30$	$( 1, 6, 4, 2, 5, 3)( 7,11,10, 8,12, 9)(13,18,16,14,17,15)(19,22,24,20,21,23)(25,27,30,26,28,29)(31,33,35,32,34,36)$
6D	$6^{3},2^{9}$	$4$	$6$	$24$	$( 1, 3, 5, 2, 4, 6)( 7, 9,12, 8,10,11)(13,15,17,14,16,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)$
6E	$6^{6}$	$8$	$6$	$30$	$( 1,10,18, 2, 9,17)( 3,11,13, 4,12,14)( 5, 7,15, 6, 8,16)(19,31,25,20,32,26)(21,34,28,22,33,27)(23,36,29,24,35,30)$
6F	$6^{3},2^{9}$	$8$	$6$	$24$	$( 1,10,18, 2, 9,17)( 3,11,13, 4,12,14)( 5, 7,15, 6, 8,16)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)$
6G	$6^{6}$	$8$	$6$	$30$	$( 1,10,18, 2, 9,17)( 3,11,13, 4,12,14)( 5, 7,15, 6, 8,16)(19,26,32,20,25,31)(21,27,33,22,28,34)(23,30,35,24,29,36)$
6H	$6^{6}$	$8$	$6$	$30$	$( 1, 6, 4, 2, 5, 3)( 7,11,10, 8,12, 9)(13,18,16,14,17,15)(19,34,30,20,33,29)(21,35,25,22,36,26)(23,32,27,24,31,28)$
6I	$6^{6}$	$8$	$6$	$30$	$( 1,12,15, 2,11,16)( 3, 8,17, 4, 7,18)( 5,10,14, 6, 9,13)(19,29,33,20,30,34)(21,26,36,22,25,35)(23,28,31,24,27,32)$
6J	$6^{6}$	$8$	$6$	$30$	$( 1, 7,14, 2, 8,13)( 3, 9,16, 4,10,15)( 5,12,18, 6,11,17)(19,22,24,20,21,23)(25,27,30,26,28,29)(31,33,35,32,34,36)$
6K	$6^{6}$	$8$	$6$	$30$	$( 1,12,15, 2,11,16)( 3, 8,17, 4, 7,18)( 5,10,14, 6, 9,13)(19,35,28,20,36,27)(21,31,30,22,32,29)(23,33,26,24,34,25)$
6L	$6^{6}$	$8$	$6$	$30$	$( 1,17, 9, 2,18,10)( 3,14,12, 4,13,11)( 5,16, 8, 6,15, 7)(19,22,24,20,21,23)(25,27,30,26,28,29)(31,33,35,32,34,36)$
6M	$6^{4},3^{4}$	$36$	$6$	$28$	$( 1,11,14, 5, 8,18)( 2,12,13, 6, 7,17)( 3,10,16)( 4, 9,15)(19,32,25)(20,31,26)(21,36,28,24,33,30)(22,35,27,23,34,29)$
6N	$6^{6}$	$36$	$6$	$30$	$( 1,12,14, 6, 8,17)( 2,11,13, 5, 7,18)( 3, 9,16, 4,10,15)(19,31,25,20,32,26)(21,35,28,23,33,29)(22,36,27,24,34,30)$
6O	$6^{2},3^{2},2^{6},1^{6}$	$36$	$6$	$20$	$( 1,11,14, 5, 8,18)( 2,12,13, 6, 7,17)( 3,10,16)( 4, 9,15)(21,24)(22,23)(27,29)(28,30)(33,36)(34,35)$
6P	$6^{3},2^{9}$	$36$	$6$	$24$	$( 1,10, 5, 7, 4,12)( 2, 9, 6, 8, 3,11)(13,15,17,14,16,18)(19,26)(20,25)(21,27)(22,28)(23,30)(24,29)(31,32)(33,34)(35,36)$
6Q	$6^{2},3^{2},2^{6},1^{6}$	$36$	$6$	$20$	$( 1, 9, 5, 8, 4,11)( 2,10, 6, 7, 3,12)(13,16,17)(14,15,18)(19,25)(20,26)(21,28)(22,27)(23,29)(24,30)$
6R	$6^{3},2^{9}$	$36$	$6$	$24$	$( 1,12,14, 6, 8,17)( 2,11,13, 5, 7,18)( 3, 9,16, 4,10,15)(19,20)(21,23)(22,24)(25,26)(27,30)(28,29)(31,32)(33,35)(34,36)$
6S	$6^{6}$	$36$	$6$	$30$	$( 1, 6, 4, 2, 5, 3)( 7,18,10,14,12,15)( 8,17, 9,13,11,16)(19,27,24,26,21,29)(20,28,23,25,22,30)(31,33,35,32,34,36)$
6T	$6^{4},3^{4}$	$36$	$6$	$28$	$( 1, 4, 5)( 2, 3, 6)( 7,16,12,13,10,17)( 8,15,11,14, 9,18)(19,24,21)(20,23,22)(25,36,28,32,30,33)(26,35,27,31,29,34)$

Malle's constant $a(G)$: $1/12$

magma:ConjugacyClasses(G);

Character table

48 x 48 character table

magma:CharacterTable(G);

Regular extensions

Data not computed

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This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	36T2046

Degree	$36$
Order	$1296$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	no
Group:	$C_3^4:(C_2^2\times C_4)$