LMFDB - Galois group: 36T17

Show commands: Magma

magma: G := TransitiveGroup(36, 17);

Group action invariants

Degree $n$:	$36$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$17$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:	$D_4\times C_3^2$
Parity:	$-1$	magma: IsEven(G);
Primitive:	no	magma: IsPrimitive(G);
magma: NilpotencyClass(G);
$\card{\Aut(F/K)}$:	$18$	magma: Order(Centralizer(SymmetricGroup(n), G));
Generators:	(1,27,15,4,26,13)(2,28,16,3,25,14)(5,32,20,8,29,17)(6,31,19,7,30,18)(9,35,23,12,33,21)(10,36,24,11,34,22), (1,9,29,2,10,30)(3,12,31)(4,11,32)(5,16,24,6,15,23)(7,14,21)(8,13,22)(17,27,36)(18,28,35)(19,26,33,20,25,34)	magma: Generators(G);

Low degree resolvents

|G/N| Galois groups for stem field(s)
$2$: $C_2$ x 3
$3$: $C_3$ x 4
$4$: $C_2^2$
$6$: $C_6$ x 12
$8$: $D_{4}$
$9$: $C_3^2$
$12$: $C_6\times C_2$ x 4
$18$: $C_6 \times C_3$ x 3
$24$: $D_4 \times C_3$ x 4
$36$: 36T4

Resolvents shown for degrees $\leq 47$

\|G/N\|	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$3$:	$C_3$ x 4
$4$:	$C_2^2$
$6$:	$C_6$ x 12
$8$:	$D_{4}$
$9$:	$C_3^2$
$12$:	$C_6\times C_2$ x 4
$18$:	$C_6 \times C_3$ x 3
$24$:	$D_4 \times C_3$ x 4
$36$:	36T4

Subfields

Degree 2: $C_2$
Degree 3: $C_3$ x 4
Degree 4: $D_{4}$
Degree 6: $C_6$ x 4
Degree 9: $C_3^2$
Degree 12: $D_4 \times C_3$ x 4
Degree 18: $C_6 \times C_3$

Low degree siblings

36T17
Siblings are shown with degree $\leq 47$

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

Conjugacy classes

Cycle Type	Size	Order	Representative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$1$	$1$	$()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$2$	$2$	$( 3, 4)( 7, 8)(11,12)(13,14)(17,18)(21,22)(27,28)(31,32)(35,36)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$2$	$2$	$( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,16)(14,15)(17,19)(18,20)(21,24) (22,23)(25,27)(26,28)(29,31)(30,32)(33,36)(34,35)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4 $	$2$	$4$	$( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,11,10,12)(13,15,14,16)(17,20,18,19)(21,23,22,24) (25,27,26,28)(29,31,30,32)(33,36,34,35)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$1$	$3$	$( 1, 5,34)( 2, 6,33)( 3, 7,35)( 4, 8,36)( 9,16,19)(10,15,20)(11,13,17) (12,14,18)(21,28,31)(22,27,32)(23,25,30)(24,26,29)$
$ 6, 6, 6, 3, 3, 3, 3, 3, 3 $	$2$	$6$	$( 1, 5,34)( 2, 6,33)( 3, 8,35, 4, 7,36)( 9,16,19)(10,15,20)(11,14,17,12,13,18) (21,27,31,22,28,32)(23,25,30)(24,26,29)$
$ 6, 6, 6, 6, 6, 6 $	$1$	$6$	$( 1, 6,34, 2, 5,33)( 3, 8,35, 4, 7,36)( 9,15,19,10,16,20)(11,14,17,12,13,18) (21,27,31,22,28,32)(23,26,30,24,25,29)$
$ 6, 6, 6, 6, 6, 6 $	$2$	$6$	$( 1, 7,34, 3, 5,35)( 2, 8,33, 4, 6,36)( 9,13,19,11,16,17)(10,14,20,12,15,18) (21,26,31,24,28,29)(22,25,32,23,27,30)$
$ 12, 12, 12 $	$2$	$12$	$( 1, 7,33, 4, 5,35, 2, 8,34, 3, 6,36)( 9,13,20,12,16,17,10,14,19,11,15,18) (21,25,32,24,28,30,22,26,31,23,27,29)$
$ 6, 6, 6, 6, 6, 6 $	$1$	$6$	$( 1, 9,29, 2,10,30)( 3,11,31, 4,12,32)( 5,16,24, 6,15,23)( 7,13,21, 8,14,22) (17,28,36,18,27,35)(19,26,33,20,25,34)$
$ 6, 6, 6, 3, 3, 3, 3, 3, 3 $	$2$	$6$	$( 1, 9,29, 2,10,30)( 3,12,31)( 4,11,32)( 5,16,24, 6,15,23)( 7,14,21)( 8,13,22) (17,27,36)(18,28,35)(19,26,33,20,25,34)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$1$	$3$	$( 1,10,29)( 2, 9,30)( 3,12,31)( 4,11,32)( 5,15,24)( 6,16,23)( 7,14,21) ( 8,13,22)(17,27,36)(18,28,35)(19,25,33)(20,26,34)$
$ 6, 6, 6, 6, 6, 6 $	$2$	$6$	$( 1,11,29, 4,10,32)( 2,12,30, 3, 9,31)( 5,13,24, 8,15,22)( 6,14,23, 7,16,21) (17,26,36,20,27,34)(18,25,35,19,28,33)$
$ 12, 12, 12 $	$2$	$12$	$( 1,11,30, 3,10,32, 2,12,29, 4, 9,31)( 5,13,23, 7,15,22, 6,14,24, 8,16,21) (17,25,35,20,27,33,18,26,36,19,28,34)$
$ 12, 12, 12 $	$2$	$12$	$( 1,13,25, 3,15,27, 2,14,26, 4,16,28)( 5,17,30, 7,20,32, 6,18,29, 8,19,31) ( 9,21,34,11,23,35,10,22,33,12,24,36)$
$ 6, 6, 6, 6, 6, 6 $	$2$	$6$	$( 1,13,26, 4,15,27)( 2,14,25, 3,16,28)( 5,17,29, 8,20,32)( 6,18,30, 7,19,31) ( 9,21,33,12,23,35)(10,22,34,11,24,36)$
$ 6, 6, 6, 3, 3, 3, 3, 3, 3 $	$2$	$6$	$( 1,15,26)( 2,16,25)( 3,13,28, 4,14,27)( 5,20,29)( 6,19,30)( 7,17,31, 8,18,32) ( 9,23,33)(10,24,34)(11,21,36,12,22,35)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$1$	$3$	$( 1,15,26)( 2,16,25)( 3,14,28)( 4,13,27)( 5,20,29)( 6,19,30)( 7,18,31) ( 8,17,32)( 9,23,33)(10,24,34)(11,22,36)(12,21,35)$
$ 6, 6, 6, 6, 6, 6 $	$1$	$6$	$( 1,16,26, 2,15,25)( 3,13,28, 4,14,27)( 5,19,29, 6,20,30)( 7,17,31, 8,18,32) ( 9,24,33,10,23,34)(11,21,36,12,22,35)$
$ 6, 6, 6, 6, 6, 6 $	$2$	$6$	$( 1,17,24, 4,20,22)( 2,18,23, 3,19,21)( 5,11,26, 8,10,27)( 6,12,25, 7, 9,28) (13,29,36,15,32,34)(14,30,35,16,31,33)$
$ 12, 12, 12 $	$2$	$12$	$( 1,17,23, 3,20,22, 2,18,24, 4,19,21)( 5,11,25, 7,10,27, 6,12,26, 8, 9,28) (13,30,35,15,32,33,14,29,36,16,31,34)$
$ 6, 6, 6, 6, 6, 6 $	$1$	$6$	$( 1,19,24, 2,20,23)( 3,17,21, 4,18,22)( 5, 9,26, 6,10,25)( 7,11,28, 8,12,27) (13,31,36,14,32,35)(15,30,34,16,29,33)$
$ 6, 6, 6, 3, 3, 3, 3, 3, 3 $	$2$	$6$	$( 1,19,24, 2,20,23)( 3,18,21)( 4,17,22)( 5, 9,26, 6,10,25)( 7,12,28)( 8,11,27) (13,32,36)(14,31,35)(15,30,34,16,29,33)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$1$	$3$	$( 1,20,24)( 2,19,23)( 3,18,21)( 4,17,22)( 5,10,26)( 6, 9,25)( 7,12,28) ( 8,11,27)(13,32,36)(14,31,35)(15,29,34)(16,30,33)$
$ 12, 12, 12 $	$2$	$12$	$( 1,21,19, 4,24,18, 2,22,20, 3,23,17)( 5,28, 9, 8,26,12, 6,27,10, 7,25,11) (13,34,31,16,36,29,14,33,32,15,35,30)$
$ 6, 6, 6, 6, 6, 6 $	$2$	$6$	$( 1,21,20, 3,24,18)( 2,22,19, 4,23,17)( 5,28,10, 7,26,12)( 6,27, 9, 8,25,11) (13,33,32,16,36,30)(14,34,31,15,35,29)$
$ 6, 6, 6, 3, 3, 3, 3, 3, 3 $	$2$	$6$	$( 1,23,20, 2,24,19)( 3,21,18)( 4,22,17)( 5,25,10, 6,26, 9)( 7,28,12)( 8,27,11) (13,36,32)(14,35,31)(15,33,29,16,34,30)$
$ 6, 6, 6, 6, 6, 6 $	$1$	$6$	$( 1,23,20, 2,24,19)( 3,22,18, 4,21,17)( 5,25,10, 6,26, 9)( 7,27,12, 8,28,11) (13,35,32,14,36,31)(15,33,29,16,34,30)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$1$	$3$	$( 1,24,20)( 2,23,19)( 3,21,18)( 4,22,17)( 5,26,10)( 6,25, 9)( 7,28,12) ( 8,27,11)(13,36,32)(14,35,31)(15,34,29)(16,33,30)$
$ 6, 6, 6, 6, 6, 6 $	$1$	$6$	$( 1,25,15, 2,26,16)( 3,27,14, 4,28,13)( 5,30,20, 6,29,19)( 7,32,18, 8,31,17) ( 9,34,23,10,33,24)(11,35,22,12,36,21)$
$ 6, 6, 6, 3, 3, 3, 3, 3, 3 $	$2$	$6$	$( 1,25,15, 2,26,16)( 3,28,14)( 4,27,13)( 5,30,20, 6,29,19)( 7,31,18)( 8,32,17) ( 9,34,23,10,33,24)(11,36,22)(12,35,21)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$1$	$3$	$( 1,26,15)( 2,25,16)( 3,28,14)( 4,27,13)( 5,29,20)( 6,30,19)( 7,31,18) ( 8,32,17)( 9,33,23)(10,34,24)(11,36,22)(12,35,21)$
$ 6, 6, 6, 6, 6, 6 $	$2$	$6$	$( 1,27,15, 4,26,13)( 2,28,16, 3,25,14)( 5,32,20, 8,29,17)( 6,31,19, 7,30,18) ( 9,35,23,12,33,21)(10,36,24,11,34,22)$
$ 12, 12, 12 $	$2$	$12$	$( 1,27,16, 3,26,13, 2,28,15, 4,25,14)( 5,32,19, 7,29,17, 6,31,20, 8,30,18) ( 9,35,24,11,33,21,10,36,23,12,34,22)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$1$	$3$	$( 1,29,10)( 2,30, 9)( 3,31,12)( 4,32,11)( 5,24,15)( 6,23,16)( 7,21,14) ( 8,22,13)(17,36,27)(18,35,28)(19,33,25)(20,34,26)$
$ 6, 6, 6, 3, 3, 3, 3, 3, 3 $	$2$	$6$	$( 1,29,10)( 2,30, 9)( 3,32,12, 4,31,11)( 5,24,15)( 6,23,16)( 7,22,14, 8,21,13) (17,35,27,18,36,28)(19,33,25)(20,34,26)$
$ 6, 6, 6, 6, 6, 6 $	$1$	$6$	$( 1,30,10, 2,29, 9)( 3,32,12, 4,31,11)( 5,23,15, 6,24,16)( 7,22,14, 8,21,13) (17,35,27,18,36,28)(19,34,25,20,33,26)$
$ 6, 6, 6, 6, 6, 6 $	$2$	$6$	$( 1,31,10, 3,29,12)( 2,32, 9, 4,30,11)( 5,21,15, 7,24,14)( 6,22,16, 8,23,13) (17,33,27,19,36,25)(18,34,28,20,35,26)$
$ 12, 12, 12 $	$2$	$12$	$( 1,31, 9, 4,29,12, 2,32,10, 3,30,11)( 5,21,16, 8,24,14, 6,22,15, 7,23,13) (17,34,28,19,36,26,18,33,27,20,35,25)$
$ 6, 6, 6, 3, 3, 3, 3, 3, 3 $	$2$	$6$	$( 1,33, 5, 2,34, 6)( 3,35, 7)( 4,36, 8)( 9,20,16,10,19,15)(11,17,13)(12,18,14) (21,31,28)(22,32,27)(23,29,25,24,30,26)$
$ 6, 6, 6, 6, 6, 6 $	$1$	$6$	$( 1,33, 5, 2,34, 6)( 3,36, 7, 4,35, 8)( 9,20,16,10,19,15)(11,18,13,12,17,14) (21,32,28,22,31,27)(23,29,25,24,30,26)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$1$	$3$	$( 1,34, 5)( 2,33, 6)( 3,35, 7)( 4,36, 8)( 9,19,16)(10,20,15)(11,17,13) (12,18,14)(21,31,28)(22,32,27)(23,30,25)(24,29,26)$
$ 12, 12, 12 $	$2$	$12$	$( 1,35, 6, 4,34, 7, 2,36, 5, 3,33, 8)( 9,17,15,12,19,13,10,18,16,11,20,14) (21,30,27,24,31,25,22,29,28,23,32,26)$
$ 6, 6, 6, 6, 6, 6 $	$2$	$6$	$( 1,35, 5, 3,34, 7)( 2,36, 6, 4,33, 8)( 9,17,16,11,19,13)(10,18,15,12,20,14) (21,29,28,24,31,26)(22,30,27,23,32,25)$

magma: ConjugacyClasses(G);

Group invariants

Order:	$72=2^{3} \cdot 3^{2}$	magma: Order(G);
Cyclic:	no	magma: IsCyclic(G);
Abelian:	no	magma: IsAbelian(G);
Solvable:	yes	magma: IsSolvable(G);
Nilpotency class:	$2$
Label:	72.37	magma: IdentifyGroup(G);

Character table: not available.

magma: CharacterTable(G);

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Group action invariants

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Group invariants

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	36T17
Degree	$36$
Order	$72$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	no
Group:	$D_4\times C_3^2$