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Properties

Label	36T49
Degree	$36$
Order	$72$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	no
Group:	$F_9$

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magma: G := TransitiveGroup(36, 49);

Group action invariants

Degree $n$:		$36$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:		$49$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:		$F_9$
Parity:		$-1$	magma: IsEven(G);
Primitive:		no	magma: IsPrimitive(G);
magma: NilpotencyClass(G);
$\card{\Aut(F/K)}$:		$4$	magma: Order(Centralizer(SymmetricGroup(n), G));
Generators:		(1,3,2,4)(5,13,18,29,34,25,22,10)(6,14,17,30,33,26,21,9)(7,16,20,31,36,28,24,12)(8,15,19,32,35,27,23,11), (1,22,27,6)(2,21,28,5)(3,24,25,8)(4,23,26,7)(9,35,19,29)(10,36,20,30)(11,34,18,31)(12,33,17,32)(13,14)(15,16)	magma: Generators(G);

Low degree resolvents

$\card{(G/N)}$ Galois groups for stem field(s)
$2$: $C_2$
$4$: $C_4$
$8$: $C_8$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$
Degree 3: None
Degree 4: $C_4$
Degree 6: None
Degree 9: $C_3^2:C_8$
Degree 12: 12T46
Degree 18: 18T28

Low degree siblings

9T15, 12T46, 18T28, 24T81
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	$2^{16},1^{4}$	$9$	$2$	$16$	$( 5,34)( 6,33)( 7,36)( 8,35)( 9,30)(10,29)(11,32)(12,31)(13,25)(14,26)(15,27)(16,28)(17,21)(18,22)(19,23)(20,24)$
3A	$3^{12}$	$8$	$3$	$24$	$( 1,31,12)( 2,32,11)( 3,30, 9)( 4,29,10)( 5,21,15)( 6,22,16)( 7,23,13)( 8,24,14)(17,34,27)(18,33,28)(19,36,25)(20,35,26)$
4A1	$4^{8},2^{2}$	$9$	$4$	$26$	$( 1, 2)( 3, 4)( 5,18,34,22)( 6,17,33,21)( 7,20,36,24)( 8,19,35,23)( 9,14,30,26)(10,13,29,25)(11,15,32,27)(12,16,31,28)$
4A-1	$4^{8},2^{2}$	$9$	$4$	$26$	$( 1, 2)( 3, 4)( 5,22,34,18)( 6,21,33,17)( 7,24,36,20)( 8,23,35,19)( 9,26,30,14)(10,25,29,13)(11,27,32,15)(12,28,31,16)$
8A1	$8^{4},4$	$9$	$8$	$31$	$( 1, 4, 2, 3)( 5,29,22,13,34,10,18,25)( 6,30,21,14,33, 9,17,26)( 7,31,24,16,36,12,20,28)( 8,32,23,15,35,11,19,27)$
8A-1	$8^{4},4$	$9$	$8$	$31$	$( 1, 3, 2, 4)( 5,25,18,10,34,13,22,29)( 6,26,17, 9,33,14,21,30)( 7,28,20,12,36,16,24,31)( 8,27,19,11,35,15,23,32)$
8A3	$8^{4},4$	$9$	$8$	$31$	$( 1, 3, 2, 4)( 5,13,18,29,34,25,22,10)( 6,14,17,30,33,26,21, 9)( 7,16,20,31,36,28,24,12)( 8,15,19,32,35,27,23,11)$
8A-3	$8^{4},4$	$9$	$8$	$31$	$( 1, 4, 2, 3)( 5,10,22,25,34,29,18,13)( 6, 9,21,26,33,30,17,14)( 7,12,24,28,36,31,20,16)( 8,11,23,27,35,32,19,15)$

Malle's constant $a(G)$: $1/16$

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:		not nilpotent
Label:		72.39	magma: IdentifyGroup(G);
Character table:

		1A	2A	3A	4A1	4A-1	8A1	8A-1	8A3	8A-3
	Size	1	9	8	9	9	9	9	9	9
	2 P	1A	1A	3A	2A	2A	4A-1	4A1	4A1	4A-1
	3 P	1A	2A	1A	4A-1	4A1	8A1	8A-1	8A3	8A-3
	Type
72.39.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
72.39.1b	R	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$
72.39.1c1	C	$1$	$1$	$1$	$- 1$	$- 1$	$- i$	$i$	$i$	$- i$
72.39.1c2	C	$1$	$1$	$1$	$- 1$	$- 1$	$i$	$- i$	$- i$	$i$
72.39.1d1	C	$1$	$- 1$	$1$	$- ζ_{8}^{2}$	$ζ_{8}^{2}$	$ζ_{8}^{3}$	$- ζ_{8}$	$ζ_{8}$	$- ζ_{8}^{3}$
72.39.1d2	C	$1$	$- 1$	$1$	$ζ_{8}^{2}$	$- ζ_{8}^{2}$	$- ζ_{8}$	$ζ_{8}^{3}$	$- ζ_{8}^{3}$	$ζ_{8}$
72.39.1d3	C	$1$	$- 1$	$1$	$- ζ_{8}^{2}$	$ζ_{8}^{2}$	$- ζ_{8}^{3}$	$ζ_{8}$	$- ζ_{8}$	$ζ_{8}^{3}$
72.39.1d4	C	$1$	$- 1$	$1$	$ζ_{8}^{2}$	$- ζ_{8}^{2}$	$ζ_{8}$	$- ζ_{8}^{3}$	$ζ_{8}^{3}$	$- ζ_{8}$
72.39.8a	R	$8$	$0$	$- 1$	$0$	$0$	$0$	$0$	$0$	$0$

magma: CharacterTable(G);