LMFDB - Galois group: 36T1364

Show commands: Magma

magma:G := TransitiveGroup(36, 1364);

Group invariants

Abstract group:	$S_4\times S_3^2$	magma:IdentifyGroup(G);
Order:	$864=2^{5} \cdot 3^{3}$	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 $t$:	$1364$	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:	$(1,14,25)(2,13,26)(3,17,28,6,15,29,4,18,27,5,16,30)(7,32,20)(8,31,19)(9,35,21,12,33,24,10,36,22,11,34,23)$, $(1,11,16,19,30,33)(2,12,15,20,29,34)(3,7,17,21,26,36)(4,8,18,22,25,35)(5,10,13,23,27,32)(6,9,14,24,28,31)$, $(1,33,29,7,15,24)(2,34,30,8,16,23)(3,35,25,9,17,20)(4,36,26,10,18,19)(5,32,27,11,14,22)(6,31,28,12,13,21)$	magma:Generators(G);

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 3
$8$: $C_2^3$
$12$: $D_{6}$ x 9
$24$: $S_4$, $S_3 \times C_2^2$ x 3
$36$: $S_3^2$ x 3
$48$: $S_4\times C_2$ x 3
$72$: 12T37 x 3
$96$: 12T48
$144$: 12T83 x 2
$216$: 12T117
$288$: 18T111 x 2

Resolvents shown for degrees $\leq 47$

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 7
$4$:	$C_2^2$ x 7
$6$:	$S_3$ x 3
$8$:	$C_2^3$
$12$:	$D_{6}$ x 9
$24$:	$S_4$, $S_3 \times C_2^2$ x 3
$36$:	$S_3^2$ x 3
$48$:	$S_4\times C_2$ x 3
$72$:	12T37 x 3
$96$:	12T48
$144$:	12T83 x 2
$216$:	12T117
$288$:	18T111 x 2

Subfields

Degree 2: $C_2$
Degree 3: $S_3$
Degree 4: None
Degree 6: $D_{6}$, $S_3^2$, $S_4\times C_2$ x 2
Degree 9: None
Degree 12: 12T48
Degree 18: 18T96

Low degree siblings

24T2661, 36T1363, 36T1365, 36T1366, 36T1432
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	$2^{18}$	$3$	$2$	$18$	$( 1,19)( 2,20)( 3,21)( 4,22)( 5,23)( 6,24)( 7,26)( 8,25)( 9,28)(10,27)(11,30)(12,29)(13,32)(14,31)(15,34)(16,33)(17,36)(18,35)$
2B	$2^{18}$	$3$	$2$	$18$	$( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,26)(20,25)(21,28)(22,27)(23,30)(24,29)$
2C	$2^{12},1^{12}$	$3$	$2$	$12$	$( 1, 2)( 3, 4)( 7, 8)( 9,10)(13,14)(15,16)(19,20)(21,22)(25,26)(27,28)(31,32)(33,34)$
2D	$2^{18}$	$6$	$2$	$18$	$( 1, 5)( 2, 6)( 3, 4)( 7,11)( 8,12)( 9,10)(13,18)(14,17)(15,16)(19,23)(20,24)(21,22)(25,29)(26,30)(27,28)(31,36)(32,35)(33,34)$
2E	$2^{18}$	$9$	$2$	$18$	$( 1,20)( 2,19)( 3,21)( 4,22)( 5,24)( 6,23)( 7,25)( 8,26)( 9,28)(10,27)(11,29)(12,30)(13,31)(14,32)(15,34)(16,33)(17,35)(18,36)$
2F	$2^{18}$	$9$	$2$	$18$	$( 1, 8)( 2, 7)( 3, 9)( 4,10)( 5,12)( 6,11)(13,32)(14,31)(15,33)(16,34)(17,36)(18,35)(19,25)(20,26)(21,28)(22,27)(23,29)(24,30)$
2G	$2^{18}$	$9$	$2$	$18$	$( 1, 2)( 3, 4)( 5, 6)( 7,19)( 8,20)( 9,21)(10,22)(11,23)(12,24)(13,25)(14,26)(15,28)(16,27)(17,30)(18,29)(31,32)(33,34)(35,36)$
2H	$2^{18}$	$18$	$2$	$18$	$( 1,10)( 2, 9)( 3, 8)( 4, 7)( 5,12)( 6,11)(13,33)(14,34)(15,31)(16,32)(17,36)(18,35)(19,27)(20,28)(21,25)(22,26)(23,29)(24,30)$
2I	$2^{18}$	$18$	$2$	$18$	$( 1,20)( 2,19)( 3,23)( 4,24)( 5,21)( 6,22)( 7,25)( 8,26)( 9,30)(10,29)(11,28)(12,27)(13,31)(14,32)(15,36)(16,35)(17,34)(18,33)$
2J	$2^{14},1^{8}$	$27$	$2$	$14$	$( 3, 4)( 7,20)( 8,19)( 9,21)(10,22)(11,24)(12,23)(13,26)(14,25)(15,28)(16,27)(17,29)(18,30)(33,34)$
2K	$2^{16},1^{4}$	$54$	$2$	$16$	$( 1, 3)( 2, 4)( 7, 9)( 8,10)(13,28)(14,27)(15,25)(16,26)(17,29)(18,30)(19,34)(20,33)(21,31)(22,32)(23,36)(24,35)$
3A	$3^{12}$	$2$	$3$	$24$	$( 1,25,14)( 2,26,13)( 3,27,15)( 4,28,16)( 5,29,17)( 6,30,18)( 7,32,20)( 8,31,19)( 9,33,22)(10,34,21)(11,35,24)(12,36,23)$
3B	$3^{12}$	$2$	$3$	$24$	$( 1,25,14)( 2,26,13)( 3,27,15)( 4,28,16)( 5,29,17)( 6,30,18)( 7,20,32)( 8,19,31)( 9,22,33)(10,21,34)(11,24,35)(12,23,36)$
3C	$3^{6},1^{18}$	$4$	$3$	$12$	$( 7,20,32)( 8,19,31)( 9,22,33)(10,21,34)(11,24,35)(12,23,36)$
3D	$3^{12}$	$8$	$3$	$24$	$( 1, 6, 4)( 2, 5, 3)( 7,12,10)( 8,11, 9)(13,17,15)(14,18,16)(19,24,22)(20,23,21)(25,30,28)(26,29,27)(31,35,33)(32,36,34)$
3E	$3^{12}$	$16$	$3$	$24$	$( 1,27,18)( 2,28,17)( 3,30,14)( 4,29,13)( 5,26,16)( 6,25,15)( 7,33,23)( 8,34,24)( 9,36,20)(10,35,19)(11,31,21)(12,32,22)$
3F	$3^{12}$	$16$	$3$	$24$	$( 1,15,30)( 2,16,29)( 3,18,25)( 4,17,26)( 5,13,28)( 6,14,27)( 7,33,23)( 8,34,24)( 9,36,20)(10,35,19)(11,31,21)(12,32,22)$
3G	$3^{12}$	$32$	$3$	$24$	$( 1,30,15)( 2,29,16)( 3,25,18)( 4,26,17)( 5,28,13)( 6,27,14)( 7,12, 9)( 8,11,10)(19,24,21)(20,23,22)(31,35,34)(32,36,33)$
4A	$4^{6},1^{12}$	$6$	$4$	$18$	$( 1, 4, 2, 3)( 7,10, 8, 9)(13,15,14,16)(19,22,20,21)(25,28,26,27)(31,33,32,34)$
4B	$4^{6},2^{6}$	$18$	$4$	$24$	$( 1,10, 2, 9)( 3, 7, 4, 8)( 5,11)( 6,12)(13,33,14,34)(15,32,16,31)(17,35)(18,36)(19,27,20,28)(21,26,22,25)(23,30)(24,29)$
4C	$4^{6},2^{6}$	$18$	$4$	$24$	$( 1,23, 2,24)( 3,21)( 4,22)( 5,20, 6,19)( 7,30, 8,29)( 9,28)(10,27)(11,25,12,26)(13,35,14,36)(15,34)(16,33)(17,32,18,31)$
4D	$4^{6},2^{6}$	$54$	$4$	$24$	$( 1, 2)( 3, 6, 4, 5)( 7, 8)( 9,12,10,11)(13,25)(14,26)(15,30,16,29)(17,27,18,28)(19,32)(20,31)(21,35,22,36)(23,34,24,33)$
6A	$6^{6}$	$6$	$6$	$30$	$( 1,31,25,19,14, 8)( 2,32,26,20,13, 7)( 3,34,27,21,15,10)( 4,33,28,22,16, 9)( 5,36,29,23,17,12)( 6,35,30,24,18,11)$
6B	$6^{6}$	$6$	$6$	$30$	$( 1,32,25, 7,14,20)( 2,31,26, 8,13,19)( 3,33,27, 9,15,22)( 4,34,28,10,16,21)( 5,35,29,11,17,24)( 6,36,30,12,18,23)$
6C	$6^{4},3^{4}$	$6$	$6$	$28$	$( 1,13,25, 2,14,26)( 3,16,27, 4,15,28)( 5,17,29)( 6,18,30)( 7,19,32, 8,20,31)( 9,21,33,10,22,34)(11,24,35)(12,23,36)$
6D	$6^{4},3^{4}$	$6$	$6$	$28$	$( 1,13,25, 2,14,26)( 3,16,27, 4,15,28)( 5,17,29)( 6,18,30)( 7,31,20, 8,32,19)( 9,34,22,10,33,21)(11,35,24)(12,36,23)$
6E	$6^{6}$	$12$	$6$	$30$	$( 1,29,14, 5,25,17)( 2,30,13, 6,26,18)( 3,28,15, 4,27,16)( 7,35,20,11,32,24)( 8,36,19,12,31,23)( 9,34,22,10,33,21)$
6F	$6^{6}$	$12$	$6$	$30$	$( 1,29,14, 5,25,17)( 2,30,13, 6,26,18)( 3,28,15, 4,27,16)( 7,24,32,11,20,35)( 8,23,31,12,19,36)( 9,21,33,10,22,34)$
6G	$6^{2},3^{2},2^{6},1^{6}$	$12$	$6$	$20$	$( 1, 2)( 3, 4)( 7,19,32, 8,20,31)( 9,21,33,10,22,34)(11,24,35)(12,23,36)(13,14)(15,16)(25,26)(27,28)$
6H	$6^{6}$	$18$	$6$	$30$	$( 1,32,25,20,14, 7)( 2,31,26,19,13, 8)( 3,34,27,21,15,10)( 4,33,28,22,16, 9)( 5,35,29,24,17,11)( 6,36,30,23,18,12)$
6I	$6^{6}$	$18$	$6$	$30$	$( 1,31,25, 8,14,19)( 2,32,26, 7,13,20)( 3,33,27, 9,15,22)( 4,34,28,10,16,21)( 5,36,29,12,17,23)( 6,35,30,11,18,24)$
6J	$6^{3},2^{9}$	$24$	$6$	$24$	$( 1, 5)( 2, 6)( 3, 4)( 7,35,20,11,32,24)( 8,36,19,12,31,23)( 9,34,22,10,33,21)(13,18)(14,17)(15,16)(25,29)(26,30)(27,28)$
6K	$6^{6}$	$24$	$6$	$30$	$( 1,33, 6,31, 4,35)( 2,34, 5,32, 3,36)( 7,15,12,13,10,17)( 8,16,11,14, 9,18)(19,28,24,25,22,30)(20,27,23,26,21,29)$
6L	$6^{6}$	$24$	$6$	$30$	$( 1,34, 6,32, 4,36)( 2,33, 5,31, 3,35)( 7,28,12,25,10,30)( 8,27,11,26, 9,29)(13,22,17,19,15,24)(14,21,18,20,16,23)$
6M	$6^{6}$	$36$	$6$	$30$	$( 1,34,25,10,14,21)( 2,33,26, 9,13,22)( 3,31,27, 8,15,19)( 4,32,28, 7,16,20)( 5,36,29,12,17,23)( 6,35,30,11,18,24)$
6N	$6^{6}$	$36$	$6$	$30$	$( 1, 7,14,20,25,32)( 2, 8,13,19,26,31)( 3,12,15,23,27,36)( 4,11,16,24,28,35)( 5,10,17,21,29,34)( 6, 9,18,22,30,33)$
6O	$6^{6}$	$48$	$6$	$30$	$( 1,24,27, 8,18,34)( 2,23,28, 7,17,33)( 3,19,30,10,14,35)( 4,20,29, 9,13,36)( 5,22,26,12,16,32)( 6,21,25,11,15,31)$
6P	$6^{6}$	$48$	$6$	$30$	$( 1,12,15,32,30,22)( 2,11,16,31,29,21)( 3, 7,18,33,25,23)( 4, 8,17,34,26,24)( 5,10,13,35,28,19)( 6, 9,14,36,27,20)$
6Q	$6^{6}$	$72$	$6$	$30$	$( 1, 5, 4, 2, 6, 3)( 7,24,10,19,12,22)( 8,23, 9,20,11,21)(13,30,15,25,17,28)(14,29,16,26,18,27)(31,36,33,32,35,34)$
12A	$12^{2},3^{4}$	$12$	$12$	$30$	$( 1,27,13, 4,25,15, 2,28,14, 3,26,16)( 5,29,17)( 6,30,18)( 7,33,19,10,32,22, 8,34,20, 9,31,21)(11,35,24)(12,36,23)$
12B	$12^{2},3^{4}$	$12$	$12$	$30$	$( 1,27,13, 4,25,15, 2,28,14, 3,26,16)( 5,29,17)( 6,30,18)( 7,22,31,10,20,33, 8,21,32, 9,19,34)(11,24,35)(12,23,36)$
12C	$12,4^{3},3^{2},1^{6}$	$24$	$12$	$24$	$( 1, 3, 2, 4)( 7,33,19,10,32,22, 8,34,20, 9,31,21)(11,35,24)(12,36,23)(13,16,14,15)(25,27,26,28)$
12D	$12^{2},6^{2}$	$36$	$12$	$32$	$( 1,22,13,10,25,33, 2,21,14, 9,26,34)( 3,19,16, 7,27,31, 4,20,15, 8,28,32)( 5,24,17,11,29,35)( 6,23,18,12,30,36)$
12E	$12^{2},6^{2}$	$36$	$12$	$32$	$( 1,35,26,23,14,11, 2,36,25,24,13,12)( 3,34,27,21,15,10)( 4,33,28,22,16, 9)( 5,31,30,20,17, 8, 6,32,29,19,18, 7)$

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

magma:ConjugacyClasses(G);

Character table

45 x 45 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	36T1364

Degree	$36$
Order	$864$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	no
Group:	$S_4\times S_3^2$