LMFDB - Galois group: 36T1365

Show commands: Magma

magma:G := TransitiveGroup(36, 1365);

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 $t$:	$1365$	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:	$(1,9,29,32,15,24)(2,10,30,31,16,23)(3,11,25,33,17,20)(4,12,26,34,18,19)(5,7,27,35,14,22)(6,8,28,36,13,21)$, $(1,31,25,19,14,8)(2,32,26,20,13,7)(3,35,27,24,15,11)(4,36,28,23,16,12)(5,33,29,22,17,9)(6,34,30,21,18,10)$, $(1,14)(2,13)(3,17,4,18)(5,16,6,15)(9,11,10,12)(19,31)(20,32)(21,36,22,35)(23,33,24,34)(27,29,28,30)$	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: $D_{6}$, $S_4$, $S_3^2$, $S_4\times C_2$
Degree 9: None
Degree 12: $C_2 \times S_4$
Degree 18: 18T96

Low degree siblings

24T2661, 36T1363, 36T1364, 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	$2^{18}$	$3$	$2$	$18$	$( 1,20)( 2,19)( 3,22)( 4,21)( 5,24)( 6,23)( 7,14)( 8,13)( 9,15)(10,16)(11,17)(12,18)(25,32)(26,31)(27,33)(28,34)(29,35)(30,36)$
2B	$2^{18}$	$3$	$2$	$18$	$( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)(13,19)(14,20)(15,22)(16,21)(17,24)(18,23)(25,32)(26,31)(27,33)(28,34)(29,35)(30,36)$
2C	$2^{12},1^{12}$	$3$	$2$	$12$	$( 1, 2)( 5, 6)( 7, 8)(11,12)(13,14)(17,18)(19,20)(23,24)(25,26)(29,30)(31,32)(35,36)$
2D	$2^{18}$	$6$	$2$	$18$	$( 1, 4)( 2, 3)( 5, 6)( 7,10)( 8, 9)(11,12)(13,15)(14,16)(17,18)(19,22)(20,21)(23,24)(25,28)(26,27)(29,30)(31,33)(32,34)(35,36)$
2E	$2^{18}$	$9$	$2$	$18$	$( 1,19)( 2,20)( 3,21)( 4,22)( 5,24)( 6,23)( 7,13)( 8,14)( 9,16)(10,15)(11,17)(12,18)(25,31)(26,32)(27,34)(28,33)(29,35)(30,36)$
2F	$2^{18}$	$9$	$2$	$18$	$( 1, 8)( 2, 7)( 3,10)( 4, 9)( 5,11)( 6,12)(13,20)(14,19)(15,21)(16,22)(17,24)(18,23)(25,31)(26,32)(27,34)(28,33)(29,35)(30,36)$
2G	$2^{12},1^{12}$	$9$	$2$	$12$	$( 1,14)( 2,13)( 3,15)( 4,16)( 5,17)( 6,18)(19,31)(20,32)(21,34)(22,33)(23,36)(24,35)$
2H	$2^{18}$	$18$	$2$	$18$	$( 1,12)( 2,11)( 3,10)( 4, 9)( 5, 8)( 6, 7)(13,24)(14,23)(15,21)(16,22)(17,19)(18,20)(25,36)(26,35)(27,34)(28,33)(29,31)(30,32)$
2I	$2^{18}$	$18$	$2$	$18$	$( 1,19)( 2,20)( 3,23)( 4,24)( 5,21)( 6,22)( 7,13)( 8,14)( 9,18)(10,17)(11,16)(12,15)(25,31)(26,32)(27,36)(28,35)(29,34)(30,33)$
2J	$2^{16},1^{4}$	$27$	$2$	$16$	$( 1,13)( 2,14)( 3,16)( 4,15)( 5,17)( 6,18)( 7, 8)( 9,10)(19,32)(20,31)(21,33)(22,34)(23,36)(24,35)(25,26)(27,28)$
2K	$2^{18}$	$54$	$2$	$18$	$( 1, 6)( 2, 5)( 3, 4)( 7,12)( 8,11)( 9,10)(13,29)(14,30)(15,28)(16,27)(17,26)(18,25)(19,35)(20,36)(21,33)(22,34)(23,32)(24,31)$
3A	$3^{12}$	$2$	$3$	$24$	$( 1,25,14)( 2,26,13)( 3,27,15)( 4,28,16)( 5,29,17)( 6,30,18)( 7,20,32)( 8,19,31)( 9,22,33)(10,21,34)(11,24,35)(12,23,36)$
3B	$3^{12}$	$2$	$3$	$24$	$( 1,14,25)( 2,13,26)( 3,15,27)( 4,16,28)( 5,17,29)( 6,18,30)( 7,20,32)( 8,19,31)( 9,22,33)(10,21,34)(11,24,35)(12,23,36)$
3C	$3^{6},1^{18}$	$4$	$3$	$12$	$( 1,14,25)( 2,13,26)( 3,15,27)( 4,16,28)( 5,17,29)( 6,18,30)$
3D	$3^{12}$	$8$	$3$	$24$	$( 1, 3, 6)( 2, 4, 5)( 7, 9,12)( 8,10,11)(13,16,17)(14,15,18)(19,21,24)(20,22,23)(25,27,30)(26,28,29)(31,34,35)(32,33,36)$
3E	$3^{12}$	$16$	$3$	$24$	$( 1,29,15)( 2,30,16)( 3,25,17)( 4,26,18)( 5,27,14)( 6,28,13)( 7,24,33)( 8,23,34)( 9,20,35)(10,19,36)(11,22,32)(12,21,31)$
3F	$3^{12}$	$16$	$3$	$24$	$( 1,29,15)( 2,30,16)( 3,25,17)( 4,26,18)( 5,27,14)( 6,28,13)( 7,35,22)( 8,36,21)( 9,32,24)(10,31,23)(11,33,20)(12,34,19)$
3G	$3^{12}$	$32$	$3$	$24$	$( 1, 3, 5)( 2, 4, 6)( 7,22,35)( 8,21,36)( 9,24,32)(10,23,31)(11,20,33)(12,19,34)(13,16,18)(14,15,17)(25,27,29)(26,28,30)$
4A	$4^{6},1^{12}$	$6$	$4$	$18$	$( 1, 6, 2, 5)( 7,12, 8,11)(13,17,14,18)(19,24,20,23)(25,30,26,29)(31,35,32,36)$
4B	$4^{6},2^{6}$	$18$	$4$	$24$	$( 1,12, 2,11)( 3, 9)( 4,10)( 5, 7, 6, 8)(13,24,14,23)(15,22)(16,21)(17,20,18,19)(25,36,26,35)(27,33)(28,34)(29,32,30,31)$
4C	$4^{6},2^{6}$	$18$	$4$	$24$	$( 1,21, 2,22)( 3,20, 4,19)( 5,24)( 6,23)( 7,16, 8,15)( 9,14,10,13)(11,17)(12,18)(25,34,26,33)(27,32,28,31)(29,35)(30,36)$
4D	$4^{6},2^{4},1^{4}$	$54$	$4$	$22$	$( 3, 5, 4, 6)( 9,11,10,12)(13,26)(14,25)(15,29,16,30)(17,28,18,27)(19,31)(20,32)(21,36,22,35)(23,33,24,34)$
6A	$6^{6}$	$6$	$6$	$30$	$( 1, 7,25,20,14,32)( 2, 8,26,19,13,31)( 3, 9,27,22,15,33)( 4,10,28,21,16,34)( 5,11,29,24,17,35)( 6,12,30,23,18,36)$
6B	$6^{6}$	$6$	$6$	$30$	$( 1,32,14, 7,25,20)( 2,31,13, 8,26,19)( 3,33,15, 9,27,22)( 4,34,16,10,28,21)( 5,35,17,11,29,24)( 6,36,18,12,30,23)$
6C	$6^{4},3^{4}$	$6$	$6$	$28$	$( 1,13,25, 2,14,26)( 3,15,27)( 4,16,28)( 5,18,29, 6,17,30)( 7,31,20, 8,32,19)( 9,33,22)(10,34,21)(11,36,24,12,35,23)$
6D	$6^{4},3^{4}$	$6$	$6$	$28$	$( 1,26,14, 2,25,13)( 3,27,15)( 4,28,16)( 5,30,17, 6,29,18)( 7,31,20, 8,32,19)( 9,33,22)(10,34,21)(11,36,24,12,35,23)$
6E	$6^{6}$	$12$	$6$	$30$	$( 1,28,14, 4,25,16)( 2,27,13, 3,26,15)( 5,30,17, 6,29,18)( 7,21,32,10,20,34)( 8,22,31, 9,19,33)(11,23,35,12,24,36)$
6F	$6^{6}$	$12$	$6$	$30$	$( 1,16,25, 4,14,28)( 2,15,26, 3,13,27)( 5,18,29, 6,17,30)( 7,21,32,10,20,34)( 8,22,31, 9,19,33)(11,23,35,12,24,36)$
6G	$6^{2},3^{2},2^{6},1^{6}$	$12$	$6$	$20$	$( 1,13,25, 2,14,26)( 3,15,27)( 4,16,28)( 5,18,29, 6,17,30)( 7, 8)(11,12)(19,20)(23,24)(31,32)(35,36)$
6H	$6^{6}$	$18$	$6$	$30$	$( 1, 8,25,19,14,31)( 2, 7,26,20,13,32)( 3,10,27,21,15,34)( 4, 9,28,22,16,33)( 5,11,29,24,17,35)( 6,12,30,23,18,36)$
6I	$6^{6}$	$18$	$6$	$30$	$( 1,31,14, 8,25,19)( 2,32,13, 7,26,20)( 3,34,15,10,27,21)( 4,33,16, 9,28,22)( 5,35,17,11,29,24)( 6,36,18,12,30,23)$
6J	$6^{3},2^{9}$	$24$	$6$	$24$	$( 1,28,14, 4,25,16)( 2,27,13, 3,26,15)( 5,30,17, 6,29,18)( 7,10)( 8, 9)(11,12)(19,22)(20,21)(23,24)(31,33)(32,34)(35,36)$
6K	$6^{6}$	$24$	$6$	$30$	$( 1,36, 3,32, 6,33)( 2,35, 4,31, 5,34)( 7,30, 9,25,12,27)( 8,29,10,26,11,28)(13,24,16,19,17,21)(14,23,15,20,18,22)$
6L	$6^{6}$	$24$	$6$	$30$	$( 1,23, 3,20, 6,22)( 2,24, 4,19, 5,21)( 7,30, 9,25,12,27)( 8,29,10,26,11,28)(13,35,16,31,17,34)(14,36,15,32,18,33)$
6M	$6^{6}$	$36$	$6$	$30$	$( 1,36,14,12,25,23)( 2,35,13,11,26,24)( 3,34,15,10,27,21)( 4,33,16, 9,28,22)( 5,31,17, 8,29,19)( 6,32,18, 7,30,20)$
6N	$6^{6}$	$36$	$6$	$30$	$( 1,31,14,19,25, 8)( 2,32,13,20,26, 7)( 3,36,15,23,27,12)( 4,35,16,24,28,11)( 5,34,17,21,29,10)( 6,33,18,22,30, 9)$
6O	$6^{6}$	$48$	$6$	$30$	$( 1,33,29, 7,15,24)( 2,34,30, 8,16,23)( 3,35,25, 9,17,20)( 4,36,26,10,18,19)( 5,32,27,11,14,22)( 6,31,28,12,13,21)$
6P	$6^{6}$	$48$	$6$	$30$	$( 1,33,29,20,15,11)( 2,34,30,19,16,12)( 3,35,25,22,17, 7)( 4,36,26,21,18, 8)( 5,32,27,24,14, 9)( 6,31,28,23,13,10)$
6Q	$6^{4},3^{4}$	$72$	$6$	$28$	$( 1,15, 6,14, 3,18)( 2,16, 5,13, 4,17)( 7, 9,12)( 8,10,11)(19,34,24,31,21,35)(20,33,23,32,22,36)(25,27,30)(26,28,29)$
12A	$12^{2},3^{4}$	$12$	$12$	$30$	$( 1,29,13, 6,25,17, 2,30,14, 5,26,18)( 3,27,15)( 4,28,16)( 7,24,31,12,20,35, 8,23,32,11,19,36)( 9,22,33)(10,21,34)$
12B	$12^{2},3^{4}$	$12$	$12$	$30$	$( 1,17,26, 6,14,29, 2,18,25, 5,13,30)( 3,15,27)( 4,16,28)( 7,24,31,12,20,35, 8,23,32,11,19,36)( 9,22,33)(10,21,34)$
12C	$12,4^{3},3^{2},1^{6}$	$24$	$12$	$24$	$( 1,29,13, 6,25,17, 2,30,14, 5,26,18)( 3,27,15)( 4,28,16)( 7,11, 8,12)(19,23,20,24)(31,36,32,35)$
12D	$12^{2},6^{2}$	$36$	$12$	$32$	$( 1,24,26,12,14,35, 2,23,25,11,13,36)( 3,22,27, 9,15,33)( 4,21,28,10,16,34)( 5,19,30, 7,17,31, 6,20,29, 8,18,32)$
12E	$12^{2},6^{2}$	$36$	$12$	$32$	$( 1, 9,26,21,14,33, 2,10,25,22,13,34)( 3, 8,28,20,15,31, 4, 7,27,19,16,32)( 5,11,29,24,17,35)( 6,12,30,23,18,36)$

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	36T1365

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