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Properties

Label	24T8
Degree	$24$
Order	$24$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	no
Group:	$C_3:C_8$

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magma: G := TransitiveGroup(24, 8);

Group action invariants

Degree $n$:		$24$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:		$8$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:		$C_3:C_8$
Parity:		$-1$	magma: IsEven(G);
Primitive:		no	magma: IsPrimitive(G);
magma: NilpotencyClass(G);
$\card{\Aut(F/K)}$:		$24$	magma: Order(Centralizer(SymmetricGroup(n), G));
Generators:		(1,8,13,20,2,7,14,19)(3,17,16,6,4,18,15,5)(9,23,21,12,10,24,22,11), (1,17,9)(2,18,10)(3,19,11)(4,20,12)(5,22,14)(6,21,13)(7,24,15)(8,23,16)	magma: Generators(G);

Low degree resolvents

|G/N| Galois groups for stem field(s)
$2$: $C_2$
$4$: $C_4$
$6$: $S_3$
$8$: $C_8$
$12$: $C_3 : C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$
Degree 3: $S_3$
Degree 4: $C_4$
Degree 6: $S_3$
Degree 8: $C_8$
Degree 12: $C_3 : C_4$

Low degree siblings

There are no siblings with degree $\leq 47$

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

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, 1, 1, 1, 1, 1, 1, 1 $	$1$	$1$	$()$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)$
	$ 8, 8, 8 $	$3$	$8$	$( 1, 3,14,15, 2, 4,13,16)( 5,24,18,12, 6,23,17,11)( 7,10,20,21, 8, 9,19,22)$
	$ 8, 8, 8 $	$3$	$8$	$( 1, 4,14,16, 2, 3,13,15)( 5,23,18,11, 6,24,17,12)( 7, 9,20,22, 8,10,19,21)$
	$ 12, 12 $	$2$	$12$	$( 1, 5,10,13,17,22, 2, 6, 9,14,18,21)( 3, 7,12,16,19,24, 4, 8,11,15,20,23)$
	$ 12, 12 $	$2$	$12$	$( 1, 6,10,14,17,21, 2, 5, 9,13,18,22)( 3, 8,12,15,19,23, 4, 7,11,16,20,24)$
	$ 8, 8, 8 $	$3$	$8$	$( 1, 7,13,19, 2, 8,14,20)( 3,18,16, 5, 4,17,15, 6)( 9,24,21,11,10,23,22,12)$
	$ 8, 8, 8 $	$3$	$8$	$( 1, 8,13,20, 2, 7,14,19)( 3,17,16, 6, 4,18,15, 5)( 9,23,21,12,10,24,22,11)$
	$ 3, 3, 3, 3, 3, 3, 3, 3 $	$2$	$3$	$( 1, 9,17)( 2,10,18)( 3,11,19)( 4,12,20)( 5,14,22)( 6,13,21)( 7,15,24) ( 8,16,23)$
	$ 6, 6, 6, 6 $	$2$	$6$	$( 1,10,17, 2, 9,18)( 3,12,19, 4,11,20)( 5,13,22, 6,14,21)( 7,16,24, 8,15,23)$
	$ 4, 4, 4, 4, 4, 4 $	$1$	$4$	$( 1,13, 2,14)( 3,16, 4,15)( 5,17, 6,18)( 7,19, 8,20)( 9,21,10,22)(11,23,12,24)$
	$ 4, 4, 4, 4, 4, 4 $	$1$	$4$	$( 1,14, 2,13)( 3,15, 4,16)( 5,18, 6,17)( 7,20, 8,19)( 9,22,10,21)(11,24,12,23)$

magma: ConjugacyClasses(G);

Group invariants

Order:		$24=2^{3} \cdot 3$	magma: Order(G);
Cyclic:		no	magma: IsCyclic(G);
Abelian:		no	magma: IsAbelian(G);
Solvable:		yes	magma: IsSolvable(G);
Nilpotency class:		not nilpotent
Label:		24.1	magma: IdentifyGroup(G);
Character table:

		1A	2A	3A	4A1	4A-1	6A	8A1	8A-1	8A3	8A-3	12A1	12A-1
	Size	1	1	2	1	1	2	3	3	3	3	2	2
	2 P	1A	1A	3A	2A	2A	3A	4A1	4A-1	4A-1	4A1	6A	6A
	3 P	1A	2A	1A	4A-1	4A1	2A	8A3	8A-3	8A1	8A-1	4A1	4A-1
	Type
24.1.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
24.1.1b	R	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$
24.1.1c1	C	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$- i$	$i$	$i$	$- i$	$- 1$	$- 1$
24.1.1c2	C	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$i$	$- i$	$- i$	$i$	$- 1$	$- 1$
24.1.1d1	C	$1$	$- 1$	$1$	$- ζ_{8}^{2}$	$ζ_{8}^{2}$	$- 1$	$ζ_{8}^{3}$	$- ζ_{8}$	$ζ_{8}$	$- ζ_{8}^{3}$	$ζ_{8}^{2}$	$- ζ_{8}^{2}$
24.1.1d2	C	$1$	$- 1$	$1$	$ζ_{8}^{2}$	$- ζ_{8}^{2}$	$- 1$	$- ζ_{8}$	$ζ_{8}^{3}$	$- ζ_{8}^{3}$	$ζ_{8}$	$- ζ_{8}^{2}$	$ζ_{8}^{2}$
24.1.1d3	C	$1$	$- 1$	$1$	$- ζ_{8}^{2}$	$ζ_{8}^{2}$	$- 1$	$- ζ_{8}^{3}$	$ζ_{8}$	$- ζ_{8}$	$ζ_{8}^{3}$	$ζ_{8}^{2}$	$- ζ_{8}^{2}$
24.1.1d4	C	$1$	$- 1$	$1$	$ζ_{8}^{2}$	$- ζ_{8}^{2}$	$- 1$	$ζ_{8}$	$- ζ_{8}^{3}$	$ζ_{8}^{3}$	$- ζ_{8}$	$- ζ_{8}^{2}$	$ζ_{8}^{2}$
24.1.2a	R	$2$	$2$	$- 1$	$2$	$2$	$- 1$	$0$	$0$	$0$	$0$	$- 1$	$- 1$
24.1.2b	S	$2$	$2$	$- 1$	$- 2$	$- 2$	$- 1$	$0$	$0$	$0$	$0$	$1$	$1$
24.1.2c1	C	$2$	$- 2$	$- 1$	$- 2 i$	$2 i$	$1$	$0$	$0$	$0$	$0$	$- i$	$i$
24.1.2c2	C	$2$	$- 2$	$- 1$	$2 i$	$- 2 i$	$1$	$0$	$0$	$0$	$0$	$i$	$- i$

magma: CharacterTable(G);