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Properties

Label	35T28
Degree	$35$
Order	$2520$
Cyclic	no
Abelian	no
Solvable	no
Primitive	yes
$p$-group	no
Group:	$A_7$

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magma: G := TransitiveGroup(35, 28);

Group invariants

Abstract group:		$A_7$	magma: IdentifyGroup(G);
Order:		$2520=2^{3} \cdot 3^{2} \cdot 5 \cdot 7$	magma: Order(G);
Cyclic:		no	magma: IsCyclic(G);
Abelian:		no	magma: IsAbelian(G);
Solvable:		no	magma: IsSolvable(G);
Nilpotency class:		not nilpotent	magma: NilpotencyClass(G);

Group action invariants

Degree $n$:		$35$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:		$28$	magma: t, n := TransitiveGroupIdentification(G); t;
Parity:		$1$	magma: IsEven(G);
Primitive:		yes	magma: IsPrimitive(G);
$\card{\Aut(F/K)}$:		$1$	magma: Order(Centralizer(SymmetricGroup(n), G));
Generators:		$(1,2,4,8,3,6,12)(5,10,15,19,25,29,22)(7,14,18,24,28,34,9)(11,16,20,27,23,13,17)(21,26,32,35,30,31,33)$, $(2,5,11)(3,7,15)(4,9,10)(6,13,14)(16,21,23)(17,18,22)(19,26,20)(25,30,32)(27,33,34)(28,31,35)$	magma: Generators(G);

Low degree resolvents

none
Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: None
Degree 7: None

Low degree siblings

7T6, 15T47 x 2, 21T33, 42T294, 42T299
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	$1^{35}$	$1$	$1$	$0$	$()$
2A	$2^{14},1^{7}$	$105$	$2$	$14$	$( 1,30)( 3,19)( 5,34)( 7,35)( 8, 9)(11,33)(13,29)(15,31)(16,28)(17,32)(18,25)(20,21)(22,24)(23,26)$
3A	$3^{10},1^{5}$	$70$	$3$	$20$	$( 1,12,24)( 2,22,30)( 3, 9,19)( 4,20,15)( 5,17,32)( 6,23,35)( 7,10,26)(11,18,25)(13,16,28)(14,21,31)$
3B	$3^{11},1^{2}$	$280$	$3$	$22$	$( 1,33, 9)( 2,27,35)( 3,29,30)( 4,17,26)( 5,18,10)( 6,16,32)( 7,21,12)( 8,31,22)(13,15,14)(19,28,34)(23,24,25)$
4A	$4^{7},2^{3},1$	$630$	$4$	$24$	$( 1,22,30,24)( 2,12)( 3, 9,19, 8)( 4, 6)( 5,18,34,25)( 7,21,35,20)(10,14)(11,17,33,32)(13,16,29,28)(15,23,31,26)$
5A	$5^{7}$	$504$	$5$	$28$	$( 1,28,26,19,17)( 2,10, 8,27, 5)( 3,32,24,16, 7)( 4,33, 9,34,11)( 6,14,12,29,13)(15,25,21,30,23)(18,31,22,35,20)$
6A	$6^{4},3^{2},2^{2},1$	$210$	$6$	$26$	$( 1,16,12,28,24,13)( 2,30,22)( 3,19, 9)( 4,25,20,11,15,18)( 5, 7,17,10,32,26)( 6,31,23,14,35,21)( 8,27)(33,34)$
7A1	$7^{5}$	$360$	$7$	$30$	$( 1,20,33, 6,16, 9,32)( 2,19,30,35,15,11,14)( 3,24,31,34,18,13,10)( 4,27,12,28, 8,25,29)( 5,17,22,26, 7,23,21)$
7A-1	$7^{5}$	$360$	$7$	$30$	$( 1,32, 9,16, 6,33,20)( 2,14,11,15,35,30,19)( 3,10,13,18,34,31,24)( 4,29,25, 8,28,12,27)( 5,21,23, 7,26,22,17)$

Malle's constant $a(G)$: $1/14$

magma: ConjugacyClasses(G);

Character table

		1A	2A	3A	3B	4A	5A	6A	7A1	7A-1
	Size	1	105	70	280	630	504	210	360	360
	2 P	1A	1A	3A	3B	2A	5A	3A	7A1	7A-1
	3 P	1A	2A	1A	1A	4A	5A	2A	7A-1	7A1
	5 P	1A	2A	3A	3B	4A	1A	6A	7A-1	7A1
	7 P	1A	2A	3A	3B	4A	5A	6A	1A	1A
	Type
2520.a.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
2520.a.6a	R	$6$	$2$	$3$	$0$	$0$	$1$	$- 1$	$- 1$	$- 1$
2520.a.10a1	C	$10$	$- 2$	$1$	$1$	$0$	$0$	$1$	$- ζ_{7}^{- 3} - 1 - ζ_{7} - ζ_{7}^{2}$	$ζ_{7}^{- 3} + ζ_{7} + ζ_{7}^{2}$
2520.a.10a2	C	$10$	$- 2$	$1$	$1$	$0$	$0$	$1$	$ζ_{7}^{- 3} + ζ_{7} + ζ_{7}^{2}$	$- ζ_{7}^{- 3} - 1 - ζ_{7} - ζ_{7}^{2}$
2520.a.14a	R	$14$	$2$	$- 1$	$2$	$0$	$- 1$	$- 1$	$0$	$0$
2520.a.14b	R	$14$	$2$	$2$	$- 1$	$0$	$- 1$	$2$	$0$	$0$
2520.a.15a	R	$15$	$- 1$	$3$	$0$	$- 1$	$0$	$- 1$	$1$	$1$
2520.a.21a	R	$21$	$1$	$- 3$	$0$	$- 1$	$1$	$1$	$0$	$0$
2520.a.35a	R	$35$	$- 1$	$- 1$	$- 1$	$1$	$0$	$- 1$	$0$	$0$

magma: CharacterTable(G);

Regular extensions

Data not computed