LMFDB - Galois group: 36T2285

Show commands: Magma

magma:G := TransitiveGroup(36, 2285);

Group invariants

Abstract group:	$C_6^2:S_3^2$	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 $t$:	$2285$	magma:t, n := TransitiveGroupIdentification(G); t;
Parity:	$-1$	magma:IsEven(G);
Primitive:	no	magma:IsPrimitive(G);
$\card{\Aut(F/K)}$:	$1$	magma:Order(Centralizer(SymmetricGroup(n), G));
Generators:	$(1,36,5,20,25,22,30,9,13,10,16,33)(2,34,6,21,27,23,29,7,15,11,17,31)(3,35,4,19,26,24,28,8,14,12,18,32)$, $(1,18,19,36,2,17,20,35,3,16,21,34)(4,33,24,13)(5,32,22,14)(6,31,23,15)(7,10,26,28,9,11,27,30,8,12,25,29)$	magma:Generators(G);

Low degree resolvents

$\card{(G/N)}$ Galois groups for stem field(s)
$2$: $C_2$ x 3
$3$: $C_3$
$4$: $C_2^2$
$6$: $S_3$ x 2, $C_6$ x 3
$12$: $D_{6}$ x 2, $C_6\times C_2$
$18$: $S_3\times C_3$ x 2
$24$: $S_4$
$36$: $S_3^2$, $C_6\times S_3$ x 2
$48$: $S_4\times C_2$
$54$: $C_3^2 : C_6$
$72$: 12T45
$108$: 12T70, 18T41
$144$: 12T83, 18T61
$324$: 18T121
$432$: 24T1328

Resolvents shown for degrees $\leq 47$

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$3$:	$C_3$
$4$:	$C_2^2$
$6$:	$S_3$ x 2, $C_6$ x 3
$12$:	$D_{6}$ x 2, $C_6\times C_2$
$18$:	$S_3\times C_3$ x 2
$24$:	$S_4$
$36$:	$S_3^2$, $C_6\times S_3$ x 2
$48$:	$S_4\times C_2$
$54$:	$C_3^2 : C_6$
$72$:	12T45
$108$:	12T70, 18T41
$144$:	12T83, 18T61
$324$:	18T121
$432$:	24T1328

Subfields

Degree 2: None
Degree 3: $S_3$
Degree 4: $S_4$
Degree 6: None
Degree 9: $C_3^2 : C_6$
Degree 12: 12T83
Degree 18: None

Low degree siblings

36T2272
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	$2^{18}$	$3$	$2$	$18$	$( 1,10)( 2,11)( 3,12)( 4,32)( 5,33)( 6,31)( 7,17)( 8,18)( 9,16)(13,22)(14,24)(15,23)(19,28)(20,30)(21,29)(25,36)(26,35)(27,34)$
2B	$2^{9},1^{18}$	$6$	$2$	$9$	$( 7,34)( 8,35)( 9,36)(10,20)(11,21)(12,19)(22,33)(23,31)(24,32)$
2C	$2^{16},1^{4}$	$9$	$2$	$16$	$( 2, 3)( 4,18)( 5,17)( 6,16)( 7,33)( 8,32)( 9,31)(11,12)(13,27)(14,26)(15,25)(19,21)(22,34)(23,36)(24,35)(28,29)$
2D	$2^{18}$	$27$	$2$	$18$	$( 1,34)( 2,36)( 3,35)( 4,31)( 5,33)( 6,32)( 7,30)( 8,28)( 9,29)(10,27)(11,25)(12,26)(13,22)(14,23)(15,24)(16,21)(17,20)(18,19)$
2E	$2^{17},1^{2}$	$54$	$2$	$17$	$( 1, 5)( 2, 4)( 3, 6)( 8, 9)(10,22)(11,24)(12,23)(13,30)(14,29)(15,28)(16,26)(17,27)(18,25)(19,31)(20,33)(21,32)(35,36)$
3A	$3^{12}$	$2$	$3$	$24$	$( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,14,15)(16,18,17)(19,21,20)(22,24,23)(25,26,27)(28,29,30)(31,33,32)(34,36,35)$
3B1	$3^{8},1^{12}$	$3$	$3$	$16$	$( 4, 5, 6)( 7, 9, 8)(13,15,14)(16,18,17)(22,23,24)(25,26,27)(31,32,33)(34,36,35)$
3B-1	$3^{8},1^{12}$	$3$	$3$	$16$	$( 4, 6, 5)( 7, 8, 9)(13,14,15)(16,17,18)(22,24,23)(25,27,26)(31,33,32)(34,35,36)$
3C	$3^{12}$	$6$	$3$	$24$	$( 1,25,15)( 2,27,14)( 3,26,13)( 4,29,17)( 5,28,18)( 6,30,16)( 7,32,21)( 8,33,19)( 9,31,20)(10,36,23)(11,34,24)(12,35,22)$
3D1	$3^{12}$	$6$	$3$	$24$	$( 1,27,15)( 2,26,14)( 3,25,13)( 4,29,18)( 5,28,16)( 6,30,17)( 7,31,20)( 8,32,21)( 9,33,19)(10,34,23)(11,35,24)(12,36,22)$
3D-1	$3^{12}$	$6$	$3$	$24$	$( 1,15,27)( 2,14,26)( 3,13,25)( 4,18,29)( 5,16,28)( 6,17,30)( 7,20,31)( 8,21,32)( 9,19,33)(10,23,34)(11,24,35)(12,22,36)$
3E	$3^{9},1^{9}$	$8$	$3$	$18$	$( 1,10,20)( 2,11,21)( 3,12,19)( 7,27,34)( 8,26,35)( 9,25,36)(13,22,33)(14,24,32)(15,23,31)$
3F	$3^{12}$	$16$	$3$	$24$	$( 1,11,19)( 2,12,20)( 3,10,21)( 4, 5, 6)( 7,26,36)( 8,25,34)( 9,27,35)(13,23,32)(14,22,31)(15,24,33)(16,17,18)(28,30,29)$
3G1	$3^{11},1^{3}$	$24$	$3$	$22$	$( 4,22,31)( 5,23,32)( 6,24,33)( 7,16,35)( 8,17,36)( 9,18,34)(10,20,30)(11,21,29)(12,19,28)(13,15,14)(25,26,27)$
3G-1	$3^{11},1^{3}$	$24$	$3$	$22$	$( 4,31,22)( 5,32,23)( 6,33,24)( 7,35,16)( 8,36,17)( 9,34,18)(10,30,20)(11,29,21)(12,28,19)(13,14,15)(25,27,26)$
3H	$3^{12}$	$48$	$3$	$24$	$( 1,24, 8)( 2,22, 9)( 3,23, 7)( 4,18,30)( 5,16,29)( 6,17,28)(10,32,26)(11,33,25)(12,31,27)(13,36,21)(14,35,20)(15,34,19)$
3I1	$3^{12}$	$48$	$3$	$24$	$( 1,24,16)( 2,22,17)( 3,23,18)( 4,25,10)( 5,27,11)( 6,26,12)( 7,21,33)( 8,19,31)( 9,20,32)(13,34,29)(14,36,30)(15,35,28)$
3I-1	$3^{12}$	$48$	$3$	$24$	$( 1,16,24)( 2,17,22)( 3,18,23)( 4,10,25)( 5,11,27)( 6,12,26)( 7,33,21)( 8,31,19)( 9,32,20)(13,29,34)(14,30,36)(15,28,35)$
4A	$4^{9}$	$6$	$4$	$27$	$( 1,30,10,20)( 2,29,11,21)( 3,28,12,19)( 4,24,32,14)( 5,22,33,13)( 6,23,31,15)( 7,27,17,34)( 8,26,18,35)( 9,25,16,36)$
4B	$4^{9}$	$54$	$4$	$27$	$( 1,20,30,10)( 2,19,29,12)( 3,21,28,11)( 4,35,14, 8)( 5,34,13, 7)( 6,36,15, 9)(16,23,25,31)(17,22,27,33)(18,24,26,32)$
6A	$6^{6}$	$6$	$6$	$30$	$( 1,11, 3,10, 2,12)( 4,33, 6,32, 5,31)( 7,18, 9,17, 8,16)(13,23,14,22,15,24)(19,30,21,28,20,29)(25,34,26,36,27,35)$
6B1	$6^{4},2^{4},1^{4}$	$9$	$6$	$24$	$( 2, 3)( 4,16, 5,18, 6,17)( 7,32, 9,33, 8,31)(11,12)(13,26,15,27,14,25)(19,21)(22,35,23,34,24,36)(28,29)$
6B-1	$6^{4},2^{4},1^{4}$	$9$	$6$	$24$	$( 2, 3)( 4,17, 6,18, 5,16)( 7,31, 8,33, 9,32)(11,12)(13,25,14,27,15,26)(19,21)(22,36,24,34,23,35)(28,29)$
6C1	$6^{4},2^{6}$	$9$	$6$	$26$	$( 1,30)( 2,29)( 3,28)( 4,15, 5,14, 6,13)( 7,35, 9,34, 8,36)(10,20)(11,21)(12,19)(16,27,18,25,17,26)(22,32,23,33,24,31)$
6C-1	$6^{4},2^{6}$	$9$	$6$	$26$	$( 1,30)( 2,29)( 3,28)( 4,13, 6,14, 5,15)( 7,36, 8,34, 9,35)(10,20)(11,21)(12,19)(16,26,17,25,18,27)(22,31,24,33,23,32)$
6D	$6^{3},3^{6}$	$12$	$6$	$27$	$( 1, 2, 3)( 4, 5, 6)( 7,35, 9,34, 8,36)(10,21,12,20,11,19)(13,15,14)(16,17,18)(22,31,24,33,23,32)(25,27,26)(28,30,29)$
6E	$6^{6}$	$18$	$6$	$30$	$( 1, 6,25,30,15,16)( 2, 4,27,29,14,17)( 3, 5,26,28,13,18)( 7,11,32,34,21,24)( 8,12,33,35,19,22)( 9,10,31,36,20,23)$
6F1	$6^{2},3^{4},2^{3},1^{6}$	$18$	$6$	$21$	$( 4, 5, 6)( 7,36, 8,34, 9,35)(10,20)(11,21)(12,19)(13,15,14)(16,18,17)(22,31,24,33,23,32)(25,26,27)$
6F-1	$6^{2},3^{4},2^{3},1^{6}$	$18$	$6$	$21$	$( 4, 6, 5)( 7,35, 9,34, 8,36)(10,20)(11,21)(12,19)(13,14,15)(16,17,18)(22,32,23,33,24,31)(25,27,26)$
6G1	$6^{6}$	$18$	$6$	$30$	$( 1, 5,25,30,13,16)( 2, 6,27,29,15,17)( 3, 4,26,28,14,18)( 7,11,31,34,21,23)( 8,12,32,35,19,24)( 9,10,33,36,20,22)$
6G-1	$6^{6}$	$18$	$6$	$30$	$( 1,16,13,30,25, 5)( 2,17,15,29,27, 6)( 3,18,14,28,26, 4)( 7,23,21,34,31,11)( 8,24,19,35,32,12)( 9,22,20,36,33,10)$
6H1	$6^{4},2^{6}$	$27$	$6$	$26$	$( 1,36, 3,34, 2,35)( 4,31)( 5,33)( 6,32)( 7,29, 8,30, 9,28)(10,25,12,27,11,26)(13,22)(14,23)(15,24)(16,19,17,21,18,20)$
6H-1	$6^{4},2^{6}$	$27$	$6$	$26$	$( 1,36, 2,35, 3,34)( 4,32)( 5,31)( 6,33)( 7,30, 9,29, 8,28)(10,25,11,26,12,27)(13,23)(14,24)(15,22)(16,21,18,19,17,20)$
6I	$6^{3},3^{6}$	$36$	$6$	$27$	$( 1,15,25)( 2,14,27)( 3,13,26)( 4,17,29)( 5,18,28)( 6,16,30)( 7,11,32,34,21,24)( 8,12,33,35,19,22)( 9,10,31,36,20,23)$
6J1	$6^{3},3^{6}$	$36$	$6$	$27$	$( 1,15,27)( 2,14,26)( 3,13,25)( 4,18,29)( 5,16,28)( 6,17,30)( 7,10,31,34,20,23)( 8,11,32,35,21,24)( 9,12,33,36,19,22)$
6J-1	$6^{3},3^{6}$	$36$	$6$	$27$	$( 1,15,26)( 2,14,25)( 3,13,27)( 4,16,29)( 5,17,28)( 6,18,30)( 7,12,33,34,19,22)( 8,10,31,35,20,23)( 9,11,32,36,21,24)$
6K1	$6^{4},2^{5},1^{2}$	$54$	$6$	$25$	$( 1, 4, 3, 5, 2, 6)( 8, 9)(10,24,12,22,11,23)(13,29,15,30,14,28)(16,26)(17,27)(18,25)(19,33,21,31,20,32)(35,36)$
6K-1	$6^{4},2^{5},1^{2}$	$54$	$6$	$25$	$( 1, 6, 2, 5, 3, 4)( 8, 9)(10,23,11,22,12,24)(13,28,14,30,15,29)(16,26)(17,27)(18,25)(19,32,20,31,21,33)(35,36)$
6L	$6^{4},3,2^{4},1$	$72$	$6$	$26$	$( 1,21,10, 2,20,11)( 3,19,12)( 4,16)( 5,18)( 6,17)( 7,23,27,31,34,15)( 8,22,26,33,35,13)( 9,24,25,32,36,14)(29,30)$
6M1	$6^{5},3,2,1$	$72$	$6$	$28$	$( 1, 2)( 4, 7,22,16,31,35)( 5, 9,23,18,32,34)( 6, 8,24,17,33,36)(10,29,20,11,30,21)(12,28,19)(13,25,15,26,14,27)$
6M-1	$6^{5},3,2,1$	$72$	$6$	$28$	$( 1, 2)( 4,35,31,16,22, 7)( 5,34,32,18,23, 9)( 6,36,33,17,24, 8)(10,21,30,11,20,29)(12,19,28)(13,27,14,26,15,25)$
12A	$12^{3}$	$12$	$12$	$33$	$( 1,19,11,30, 3,21,10,28, 2,20,12,29)( 4,15,33,24, 6,13,32,23, 5,14,31,22)( 7,36,18,27, 9,35,17,25, 8,34,16,26)$
12B1	$12^{2},4^{3}$	$18$	$12$	$31$	$( 1,10,30,20)( 2,11,29,21)( 3,12,28,19)( 4,33,15,24, 5,31,14,22, 6,32,13,23)( 7,25,35,17, 9,26,34,16, 8,27,36,18)$
12B-1	$12^{2},4^{3}$	$18$	$12$	$31$	$( 1,10,30,20)( 2,11,29,21)( 3,12,28,19)( 4,31,13,24, 6,33,14,23, 5,32,15,22)( 7,26,36,17, 8,25,34,18, 9,27,35,16)$
12C	$12^{3}$	$36$	$12$	$33$	$( 1,36, 6,20,25,23,30, 9,15,10,16,31)( 2,34, 4,21,27,24,29, 7,14,11,17,32)( 3,35, 5,19,26,22,28, 8,13,12,18,33)$
12D1	$12^{3}$	$36$	$12$	$33$	$( 1,36, 5,20,25,22,30, 9,13,10,16,33)( 2,34, 6,21,27,23,29, 7,15,11,17,31)( 3,35, 4,19,26,24,28, 8,14,12,18,32)$
12D-1	$12^{3}$	$36$	$12$	$33$	$( 1,36, 4,20,25,24,30, 9,14,10,16,32)( 2,34, 5,21,27,22,29, 7,13,11,17,33)( 3,35, 6,19,26,23,28, 8,15,12,18,31)$
12E1	$12^{2},4^{3}$	$54$	$12$	$31$	$( 1,10,30,20)( 2,12,29,19)( 3,11,28,21)( 4, 9,13,35, 6, 7,14,36, 5, 8,15,34)(16,33,26,23,17,32,25,22,18,31,27,24)$
12E-1	$12^{2},4^{3}$	$54$	$12$	$31$	$( 1,20,30,10)( 2,19,29,12)( 3,21,28,11)( 4,34,15, 8, 5,36,14, 7, 6,35,13, 9)(16,24,27,31,18,22,25,32,17,23,26,33)$

Malle's constant $a(G)$: $1/9$

magma:ConjugacyClasses(G);

Character table

50 x 50 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	36T2285

Degree	$36$
Order	$1296$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	no
Group:	$C_6^2:S_3^2$