LMFDB - Galois group: 24T27

Show commands: Magma

magma: G := TransitiveGroup(24, 27);

Group action invariants

Degree $n$:	$24$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$27$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:	$C_4\times D_6$
Parity:	$1$	magma: IsEven(G);
Primitive:	no	magma: IsPrimitive(G);
magma: NilpotencyClass(G);
$\card{\Aut(F/K)}$:	$8$	magma: Order(Centralizer(SymmetricGroup(n), G));
Generators:	(3,11)(4,12)(5,22)(6,21)(9,17)(10,18)(15,24)(16,23), (1,23,2,24)(3,21,4,22)(5,19,6,20)(7,17,8,18)(9,16,10,15)(11,13,12,14), (1,20,2,19)(3,9,4,10)(5,24,6,23)(7,13,8,14)(11,17,12,18)(15,21,16,22)	magma: Generators(G);

Low degree resolvents

|G/N| Galois groups for stem field(s)
$2$: $C_2$ x 7
$4$: $C_4$ x 4, $C_2^2$ x 7
$6$: $S_3$
$8$: $C_4\times C_2$ x 6, $C_2^3$
$12$: $D_{6}$ x 3
$16$: $C_4\times C_2^2$
$24$: $S_3 \times C_2^2$, $S_3 \times C_4$ x 2

Resolvents shown for degrees $\leq 47$

\|G/N\|	Galois groups for stem field(s)
$2$:	$C_2$ x 7
$4$:	$C_4$ x 4, $C_2^2$ x 7
$6$:	$S_3$
$8$:	$C_4\times C_2$ x 6, $C_2^3$
$12$:	$D_{6}$ x 3
$16$:	$C_4\times C_2^2$
$24$:	$S_3 \times C_2^2$, $S_3 \times C_4$ x 2

Subfields

Degree 2: $C_2$ x 3
Degree 3: $S_3$
Degree 4: $C_4$ x 2, $C_2^2$
Degree 6: $D_{6}$ x 3
Degree 8: $C_4\times C_2$
Degree 12: $S_3 \times C_2^2$, $S_3 \times C_4$ x 2

Low degree siblings

24T27 x 3
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	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, 1, 1, 1, 1, 1, 1, 1, 1 $	$3$	$2$	$( 3,11)( 4,12)( 5,22)( 6,21)( 9,17)(10,18)(15,24)(16,23)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$3$	$2$	$( 1, 2)( 3,12)( 4,11)( 5,21)( 6,22)( 7, 8)( 9,18)(10,17)(13,14)(15,23)(16,24) (19,20)$
	$ 4, 4, 4, 4, 4, 4 $	$3$	$4$	$( 1, 3, 2, 4)( 5,23, 6,24)( 7,22, 8,21)( 9,19,10,20)(11,18,12,17)(13,15,14,16)$
	$ 12, 12 $	$2$	$12$	$( 1, 3,18,20, 9,11, 2, 4,17,19,10,12)( 5, 8,21,24,14,16, 6, 7,22,23,13,15)$
	$ 4, 4, 4, 4, 4, 4 $	$3$	$4$	$( 1, 4, 2, 3)( 5,24, 6,23)( 7,21, 8,22)( 9,20,10,19)(11,17,12,18)(13,16,14,15)$
	$ 12, 12 $	$2$	$12$	$( 1, 4,18,19, 9,12, 2, 3,17,20,10,11)( 5, 7,21,23,14,15, 6, 8,22,24,13,16)$
	$ 6, 6, 6, 6 $	$2$	$6$	$( 1, 5, 9,14,17,22)( 2, 6,10,13,18,21)( 3, 8,11,16,19,23)( 4, 7,12,15,20,24)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$3$	$2$	$( 1, 5)( 2, 6)( 3,16)( 4,15)( 7,12)( 8,11)( 9,22)(10,21)(13,18)(14,17)(19,23) (20,24)$
	$ 6, 6, 6, 6 $	$2$	$6$	$( 1, 6, 9,13,17,21)( 2, 5,10,14,18,22)( 3, 7,11,15,19,24)( 4, 8,12,16,20,23)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$3$	$2$	$( 1, 6)( 2, 5)( 3,15)( 4,16)( 7,11)( 8,12)( 9,21)(10,22)(13,17)(14,18)(19,24) (20,23)$
	$ 4, 4, 4, 4, 4, 4 $	$3$	$4$	$( 1, 7, 2, 8)( 3, 5, 4, 6)( 9,24,10,23)(11,22,12,21)(13,19,14,20)(15,18,16,17)$
	$ 4, 4, 4, 4, 4, 4 $	$1$	$4$	$( 1, 7, 2, 8)( 3,22, 4,21)( 5,12, 6,11)( 9,15,10,16)(13,19,14,20)(17,24,18,23)$
	$ 4, 4, 4, 4, 4, 4 $	$3$	$4$	$( 1, 8, 2, 7)( 3, 6, 4, 5)( 9,23,10,24)(11,21,12,22)(13,20,14,19)(15,17,16,18)$
	$ 4, 4, 4, 4, 4, 4 $	$1$	$4$	$( 1, 8, 2, 7)( 3,21, 4,22)( 5,11, 6,12)( 9,16,10,15)(13,20,14,19)(17,23,18,24)$
	$ 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 $	$2$	$6$	$( 1,10,17, 2, 9,18)( 3,12,19, 4,11,20)( 5,13,22, 6,14,21)( 7,16,24, 8,15,23)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1,13)( 2,14)( 3,15)( 4,16)( 5,18)( 6,17)( 7,19)( 8,20)( 9,21)(10,22)(11,24) (12,23)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1,14)( 2,13)( 3,16)( 4,15)( 5,17)( 6,18)( 7,20)( 8,19)( 9,22)(10,21)(11,23) (12,24)$
	$ 12, 12 $	$2$	$12$	$( 1,15,18, 8, 9,24, 2,16,17, 7,10,23)( 3, 5,20,21,11,14, 4, 6,19,22,12,13)$
	$ 12, 12 $	$2$	$12$	$( 1,16,18, 7, 9,23, 2,15,17, 8,10,24)( 3, 6,20,22,11,13, 4, 5,19,21,12,14)$
	$ 4, 4, 4, 4, 4, 4 $	$1$	$4$	$( 1,19, 2,20)( 3,10, 4, 9)( 5,23, 6,24)( 7,14, 8,13)(11,18,12,17)(15,22,16,21)$
	$ 4, 4, 4, 4, 4, 4 $	$1$	$4$	$( 1,20, 2,19)( 3, 9, 4,10)( 5,24, 6,23)( 7,13, 8,14)(11,17,12,18)(15,21,16,22)$

magma: ConjugacyClasses(G);

Group invariants

Order:	$48=2^{4} \cdot 3$	magma: Order(G);
Cyclic:	no	magma: IsCyclic(G);
Abelian:	no	magma: IsAbelian(G);
Solvable:	yes	magma: IsSolvable(G);
Nilpotency class:	not nilpotent
Label:	48.35	magma: IdentifyGroup(G);
Character table:

		1A	2A	2B	2C	2D	2E	2F	2G	3A	4A1	4A-1	4B1	4B-1	4C1	4C-1	4D1	4D-1	6A	6B	6C	12A1	12A-1	12B1	12B-1
	Size	1	1	1	1	3	3	3	3	2	1	1	1	1	3	3	3	3	2	2	2	2	2	2	2
	2 P	1A	1A	1A	1A	1A	1A	1A	1A	3A	2A	2A	2A	2A	2A	2A	2A	2A	3A	3A	3A	6A	6A	6A	6A
	3 P	1A	2A	2B	2C	2D	2G	2E	2F	1A	4A1	4A-1	4B1	4B-1	4D-1	4C-1	4C1	4D1	2B	2C	2A	4A1	4A-1	4B1	4B-1
	Type
48.35.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
48.35.1b	R	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$
48.35.1c	R	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$
48.35.1d	R	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$
48.35.1e	R	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$
48.35.1f	R	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$
48.35.1g	R	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
48.35.1h	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$
48.35.1i1	C	$1$	$- 1$	$- 1$	$1$	$- 1$	$1$	$- 1$	$1$	$1$	$- i$	$i$	$- i$	$i$	$i$	$- i$	$i$	$- i$	$- 1$	$- 1$	$1$	$i$	$- i$	$i$	$- i$
48.35.1i2	C	$1$	$- 1$	$- 1$	$1$	$- 1$	$1$	$- 1$	$1$	$1$	$i$	$- i$	$i$	$- i$	$- i$	$i$	$- i$	$i$	$- 1$	$- 1$	$1$	$- i$	$i$	$- i$	$i$
48.35.1j1	C	$1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$1$	$- 1$	$1$	$- i$	$i$	$- i$	$i$	$- i$	$i$	$- i$	$i$	$- 1$	$- 1$	$1$	$i$	$- i$	$i$	$- i$
48.35.1j2	C	$1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$1$	$- 1$	$1$	$i$	$- i$	$i$	$- i$	$i$	$- i$	$i$	$- i$	$- 1$	$- 1$	$1$	$- i$	$i$	$- i$	$i$
48.35.1k1	C	$1$	$- 1$	$1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$1$	$- i$	$i$	$i$	$- i$	$i$	$- i$	$- i$	$i$	$- 1$	$1$	$- 1$	$i$	$- i$	$- i$	$i$
48.35.1k2	C	$1$	$- 1$	$1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$1$	$i$	$- i$	$- i$	$i$	$- i$	$i$	$i$	$- i$	$- 1$	$1$	$- 1$	$- i$	$i$	$i$	$- i$
48.35.1l1	C	$1$	$- 1$	$1$	$- 1$	$1$	$- 1$	$- 1$	$1$	$1$	$- i$	$i$	$i$	$- i$	$- i$	$i$	$i$	$- i$	$- 1$	$1$	$- 1$	$i$	$- i$	$- i$	$i$
48.35.1l2	C	$1$	$- 1$	$1$	$- 1$	$1$	$- 1$	$- 1$	$1$	$1$	$i$	$- i$	$- i$	$i$	$i$	$- i$	$- i$	$i$	$- 1$	$1$	$- 1$	$- i$	$i$	$i$	$- i$
48.35.2a	R	$2$	$2$	$2$	$2$	$0$	$0$	$0$	$0$	$- 1$	$2$	$2$	$2$	$2$	$0$	$0$	$0$	$0$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$
48.35.2b	R	$2$	$2$	$- 2$	$- 2$	$0$	$0$	$0$	$0$	$- 1$	$- 2$	$- 2$	$2$	$2$	$0$	$0$	$0$	$0$	$- 1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$
48.35.2c	R	$2$	$2$	$- 2$	$- 2$	$0$	$0$	$0$	$0$	$- 1$	$2$	$2$	$- 2$	$- 2$	$0$	$0$	$0$	$0$	$- 1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$
48.35.2d	R	$2$	$2$	$2$	$2$	$0$	$0$	$0$	$0$	$- 1$	$- 2$	$- 2$	$- 2$	$- 2$	$0$	$0$	$0$	$0$	$- 1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$
48.35.2e1	C	$2$	$- 2$	$- 2$	$2$	$0$	$0$	$0$	$0$	$- 1$	$- 2 i$	$2 i$	$- 2 i$	$2 i$	$0$	$0$	$0$	$0$	$1$	$1$	$- 1$	$- i$	$i$	$- i$	$i$
48.35.2e2	C	$2$	$- 2$	$- 2$	$2$	$0$	$0$	$0$	$0$	$- 1$	$2 i$	$- 2 i$	$2 i$	$- 2 i$	$0$	$0$	$0$	$0$	$1$	$1$	$- 1$	$i$	$- i$	$i$	$- i$
48.35.2f1	C	$2$	$- 2$	$2$	$- 2$	$0$	$0$	$0$	$0$	$- 1$	$- 2 i$	$2 i$	$2 i$	$- 2 i$	$0$	$0$	$0$	$0$	$1$	$- 1$	$1$	$- i$	$i$	$i$	$- i$
48.35.2f2	C	$2$	$- 2$	$2$	$- 2$	$0$	$0$	$0$	$0$	$- 1$	$2 i$	$- 2 i$	$- 2 i$	$2 i$	$0$	$0$	$0$	$0$	$1$	$- 1$	$1$	$i$	$- i$	$- i$	$i$

magma: CharacterTable(G);

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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	24T27
Degree	$24$
Order	$48$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	no
Group:	$C_4\times D_6$