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Properties

Label	15T21
Degree	$15$
Order	$360$
Cyclic	no
Abelian	no
Solvable	no
Primitive	no
$p$-group	no
Group:	$\GL(2,4):C_2$

Related objects

Number fields with this Galois group

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Show commands: Magma

magma: G := TransitiveGroup(15, 21);

Group action invariants

Degree $n$:		$15$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:		$21$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:		$\GL(2,4):C_2$
CHM label:		$3S_{5}(15)$
Parity:		$1$	magma: IsEven(G);
Primitive:		no	magma: IsPrimitive(G);
magma: NilpotencyClass(G);
$\card{\Aut(F/K)}$:		$1$	magma: Order(Centralizer(SymmetricGroup(n), G));
Generators:		(1,9,10,3,14)(2,15,7,12,6)(4,5,11,13,8), (1,2,15)(4,5,6)(8,9,10)(12,13,14), (1,4,10)(2,5,8)(3,7,11)(6,9,15)(12,14,13), (1,4)(2,6)(3,7)(5,15)(8,9)(12,13)	magma: Generators(G);

Low degree resolvents

|G/N| Galois groups for stem field(s)
$2$: $C_2$
$6$: $S_3$
$120$: $S_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None
Degree 5: $S_5$

Low degree siblings

15T21, 15T22, 18T146, 30T89, 30T93 x 2, 30T101, 36T554, 45T45
Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

Conjugacy classes

Label	Cycle Type	Size	Order	Representative
	$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$1$	$1$	$()$
	$ 2, 2, 2, 2, 2, 2, 1, 1, 1 $	$30$	$2$	$( 3,15)( 4, 9)( 5,11)( 6, 8)( 7,10)(13,14)$
	$ 3, 3, 3, 3, 1, 1, 1 $	$40$	$3$	$( 2, 3,15)( 4, 7, 5)( 9,11,10)(12,13,14)$
	$ 4, 4, 4, 2, 1 $	$90$	$4$	$( 2, 5,12,11)( 3, 9,10,14)( 4,15,13, 7)( 6, 8)$
	$ 2, 2, 2, 2, 2, 2, 1, 1, 1 $	$15$	$2$	$( 2, 5)( 3,13)( 4,10)( 7,14)( 9,15)(11,12)$
	$ 15 $	$24$	$15$	$( 1, 2, 5, 3,12, 8, 9,15,11,14, 6, 4,10, 7,13)$
	$ 6, 6, 3 $	$60$	$6$	$( 1, 2, 6, 9, 8, 4)( 3,10,12,11,15,14)( 5,13, 7)$
	$ 6, 6, 3 $	$30$	$6$	$( 1, 2, 6, 4, 8, 9)( 3,14, 7,12,11,13)( 5,15,10)$
	$ 5, 5, 5 $	$24$	$5$	$( 1, 2, 7,14, 5)( 3,13,15, 8, 9)( 4,11,12,10, 6)$
	$ 3, 3, 3, 3, 3 $	$20$	$3$	$( 1, 2,12)( 3,11, 7)( 4,13, 6)( 5,10,15)( 8, 9,14)$
	$ 15 $	$24$	$15$	$( 1, 2,13, 5,11, 6, 4,14,10, 3, 8, 9,12,15, 7)$
	$ 3, 3, 3, 3, 3 $	$2$	$3$	$( 1, 6, 8)( 2, 4, 9)( 3, 7,11)( 5,10,15)(12,13,14)$

magma: ConjugacyClasses(G);

Group invariants

Order:		$360=2^{3} \cdot 3^{2} \cdot 5$	magma: Order(G);
Cyclic:		no	magma: IsCyclic(G);
Abelian:		no	magma: IsAbelian(G);
Solvable:		no	magma: IsSolvable(G);
Nilpotency class:		not nilpotent
Label:		360.120	magma: IdentifyGroup(G);
Character table:

		1A	2A	2B	3A	3B	3C	4A	5A	6A	6B	15A1	15A-1
	Size	1	15	30	2	20	40	90	24	30	60	24	24
	2 P	1A	1A	1A	3A	3B	3C	2A	5A	3A	3B	15A1	15A-1
	3 P	1A	2A	2B	1A	1A	1A	4A	5A	2A	2B	5A	5A
	5 P	1A	2A	2B	3A	3B	3C	4A	1A	6A	6B	3A	3A
	Type
360.120.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
360.120.1b	R	$1$	$1$	$- 1$	$1$	$1$	$1$	$- 1$	$1$	$1$	$- 1$	$1$	$1$
360.120.2a	R	$2$	$2$	$0$	$- 1$	$2$	$- 1$	$0$	$2$	$- 1$	$0$	$- 1$	$- 1$
360.120.4a	R	$4$	$0$	$2$	$4$	$1$	$1$	$0$	$- 1$	$0$	$- 1$	$- 1$	$- 1$
360.120.4b	R	$4$	$0$	$- 2$	$4$	$1$	$1$	$0$	$- 1$	$0$	$1$	$- 1$	$- 1$
360.120.5a	R	$5$	$1$	$- 1$	$5$	$- 1$	$- 1$	$1$	$0$	$1$	$- 1$	$0$	$0$
360.120.5b	R	$5$	$1$	$1$	$5$	$- 1$	$- 1$	$- 1$	$0$	$1$	$1$	$0$	$0$
360.120.6a	R	$6$	$- 2$	$0$	$6$	$0$	$0$	$0$	$1$	$- 2$	$0$	$1$	$1$
360.120.6b1	C	$6$	$- 2$	$0$	$- 3$	$0$	$0$	$0$	$1$	$1$	$0$	$1 - 2 ζ_{15} - ζ_{15}^{2} + ζ_{15}^{3} - 2 ζ_{15}^{4} + ζ_{15}^{5} - ζ_{15}^{7}$	$- 2 + 2 ζ_{15} + ζ_{15}^{2} - ζ_{15}^{3} + 2 ζ_{15}^{4} - ζ_{15}^{5} + ζ_{15}^{7}$
360.120.6b2	C	$6$	$- 2$	$0$	$- 3$	$0$	$0$	$0$	$1$	$1$	$0$	$- 2 + 2 ζ_{15} + ζ_{15}^{2} - ζ_{15}^{3} + 2 ζ_{15}^{4} - ζ_{15}^{5} + ζ_{15}^{7}$	$1 - 2 ζ_{15} - ζ_{15}^{2} + ζ_{15}^{3} - 2 ζ_{15}^{4} + ζ_{15}^{5} - ζ_{15}^{7}$
360.120.8a	R	$8$	$0$	$0$	$- 4$	$2$	$- 1$	$0$	$- 2$	$0$	$0$	$1$	$1$
360.120.10a	R	$10$	$2$	$0$	$- 5$	$- 2$	$1$	$0$	$0$	$- 1$	$0$	$0$	$0$

magma: CharacterTable(G);