LMFDB - Galois group: 24T400

Show commands: Magma

magma: G := TransitiveGroup(24, 400);

Group action invariants

Degree $n$:	$24$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$400$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:	$C_2^3\times S_4$
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:	(1,14)(2,13)(3,8,16,19)(4,7,15,20)(5,9,17,22)(6,10,18,21)(11,23)(12,24), (1,22,17,14,9,5)(2,21,18,13,10,6)(3,23,19,16,11,8)(4,24,20,15,12,7), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24), (1,24)(2,23)(3,6)(4,5)(7,9)(8,10)(11,13)(12,14)(15,17)(16,18)(19,21)(20,22)	magma: Generators(G);

Low degree resolvents

|G/N| Galois groups for stem field(s)
$2$: $C_2$ x 15
$4$: $C_2^2$ x 35
$6$: $S_3$
$8$: $C_2^3$ x 15
$12$: $D_{6}$ x 7
$16$: $C_2^4$
$24$: $S_4$, $S_3 \times C_2^2$ x 7
$48$: $S_4\times C_2$ x 7, 24T30
$96$: 12T48 x 7

Resolvents shown for degrees $\leq 47$

\|G/N\|	Galois groups for stem field(s)
$2$:	$C_2$ x 15
$4$:	$C_2^2$ x 35
$6$:	$S_3$
$8$:	$C_2^3$ x 15
$12$:	$D_{6}$ x 7
$16$:	$C_2^4$
$24$:	$S_4$, $S_3 \times C_2^2$ x 7
$48$:	$S_4\times C_2$ x 7, 24T30
$96$:	12T48 x 7

Subfields

Degree 2: $C_2$ x 3
Degree 3: $S_3$
Degree 4: $C_2^2$
Degree 6: $D_{6}$ x 3, $S_4\times C_2$ x 4
Degree 8: None
Degree 12: $S_3 \times C_2^2$, 12T48 x 6

Low degree siblings

24T400 x 55, 32T2168 x 8
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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$3$	$2$	$( 7,20)( 8,19)( 9,22)(10,21)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $	$6$	$2$	$( 3, 8)( 4, 7)( 5, 9)( 6,10)(15,20)(16,19)(17,22)(18,21)$
	$ 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $	$6$	$4$	$( 3, 8,16,19)( 4, 7,15,20)( 5, 9,17,22)( 6,10,18,21)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $	$3$	$2$	$( 3,16)( 4,15)( 5,17)( 6,18)( 7,20)( 8,19)( 9,22)(10,21)$
	$ 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 $	$3$	$2$	$( 1, 2)( 3, 4)( 5, 6)( 7,19)( 8,20)( 9,21)(10,22)(11,12)(13,14)(15,16)(17,18) (23,24)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$6$	$2$	$( 1, 2)( 3, 7)( 4, 8)( 5,10)( 6, 9)(11,12)(13,14)(15,19)(16,20)(17,21)(18,22) (23,24)$
	$ 4, 4, 4, 4, 2, 2, 2, 2 $	$6$	$4$	$( 1, 2)( 3, 7,16,20)( 4, 8,15,19)( 5,10,17,21)( 6, 9,18,22)(11,12)(13,14) (23,24)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$3$	$2$	$( 1, 2)( 3,15)( 4,16)( 5,18)( 6,17)( 7,19)( 8,20)( 9,21)(10,22)(11,12)(13,14) (23,24)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$6$	$2$	$( 1, 3)( 2, 4)( 5,23)( 6,24)( 7,10)( 8, 9)(11,17)(12,18)(13,15)(14,16)(19,22) (20,21)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$6$	$2$	$( 1, 3)( 2, 4)( 5,23)( 6,24)( 7,21)( 8,22)( 9,19)(10,20)(11,17)(12,18)(13,15) (14,16)$
	$ 6, 6, 6, 6 $	$8$	$6$	$( 1, 3, 9,23, 5, 8)( 2, 4,10,24, 6, 7)(11,17,19,14,16,22)(12,18,20,13,15,21)$
	$ 6, 6, 6, 6 $	$8$	$6$	$( 1, 3, 9,11,17,19)( 2, 4,10,12,18,20)( 5, 8,14,16,22,23)( 6, 7,13,15,21,24)$
	$ 4, 4, 4, 4, 2, 2, 2, 2 $	$6$	$4$	$( 1, 3,14,16)( 2, 4,13,15)( 5,11,17,23)( 6,12,18,24)( 7,10)( 8, 9)(19,22) (20,21)$
	$ 4, 4, 4, 4, 2, 2, 2, 2 $	$6$	$4$	$( 1, 3,14,16)( 2, 4,13,15)( 5,11,17,23)( 6,12,18,24)( 7,21)( 8,22)( 9,19) (10,20)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$6$	$2$	$( 1, 4)( 2, 3)( 5,24)( 6,23)( 7, 9)( 8,10)(11,18)(12,17)(13,16)(14,15)(19,21) (20,22)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$6$	$2$	$( 1, 4)( 2, 3)( 5,24)( 6,23)( 7,22)( 8,21)( 9,20)(10,19)(11,18)(12,17)(13,16) (14,15)$
	$ 6, 6, 6, 6 $	$8$	$6$	$( 1, 4, 9,24, 5, 7)( 2, 3,10,23, 6, 8)(11,18,19,13,16,21)(12,17,20,14,15,22)$
	$ 6, 6, 6, 6 $	$8$	$6$	$( 1, 4, 9,12,17,20)( 2, 3,10,11,18,19)( 5, 7,14,15,22,24)( 6, 8,13,16,21,23)$
	$ 4, 4, 4, 4, 2, 2, 2, 2 $	$6$	$4$	$( 1, 4,14,15)( 2, 3,13,16)( 5,12,17,24)( 6,11,18,23)( 7, 9)( 8,10)(19,21) (20,22)$
	$ 4, 4, 4, 4, 2, 2, 2, 2 $	$6$	$4$	$( 1, 4,14,15)( 2, 3,13,16)( 5,12,17,24)( 6,11,18,23)( 7,22)( 8,21)( 9,20) (10,19)$
	$ 6, 6, 6, 6 $	$8$	$6$	$( 1, 5, 9,14,17,22)( 2, 6,10,13,18,21)( 3, 8,11,16,19,23)( 4, 7,12,15,20,24)$
	$ 3, 3, 3, 3, 3, 3, 3, 3 $	$8$	$3$	$( 1, 5, 9)( 2, 6,10)( 3, 8,23)( 4, 7,24)(11,16,19)(12,15,20)(13,18,21) (14,17,22)$
	$ 4, 4, 4, 4, 2, 2, 2, 2 $	$6$	$4$	$( 1, 5,14,17)( 2, 6,13,18)( 3,11,16,23)( 4,12,15,24)( 7,20)( 8,19)( 9,22) (10,21)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$6$	$2$	$( 1, 5)( 2, 6)( 3,23)( 4,24)( 7,20)( 8,19)( 9,22)(10,21)(11,16)(12,15)(13,18) (14,17)$
	$ 6, 6, 6, 6 $	$8$	$6$	$( 1, 6, 9,13,17,21)( 2, 5,10,14,18,22)( 3, 7,11,15,19,24)( 4, 8,12,16,20,23)$
	$ 6, 6, 6, 6 $	$8$	$6$	$( 1, 6, 9, 2, 5,10)( 3, 7,23, 4, 8,24)(11,15,19,12,16,20)(13,17,21,14,18,22)$
	$ 4, 4, 4, 4, 2, 2, 2, 2 $	$6$	$4$	$( 1, 6,14,18)( 2, 5,13,17)( 3,12,16,24)( 4,11,15,23)( 7,19)( 8,20)( 9,21) (10,22)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$6$	$2$	$( 1, 6)( 2, 5)( 3,24)( 4,23)( 7,19)( 8,20)( 9,21)(10,22)(11,15)(12,16)(13,17) (14,18)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$3$	$2$	$( 1,11)( 2,12)( 3, 5)( 4, 6)( 7,10)( 8, 9)(13,24)(14,23)(15,18)(16,17)(19,22) (20,21)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$3$	$2$	$( 1,11)( 2,12)( 3, 5)( 4, 6)( 7,21)( 8,22)( 9,19)(10,20)(13,24)(14,23)(15,18) (16,17)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1,11)( 2,12)( 3,17)( 4,18)( 5,16)( 6,15)( 7,21)( 8,22)( 9,19)(10,20)(13,24) (14,23)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$3$	$2$	$( 1,12)( 2,11)( 3, 6)( 4, 5)( 7, 9)( 8,10)(13,23)(14,24)(15,17)(16,18)(19,21) (20,22)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$3$	$2$	$( 1,12)( 2,11)( 3, 6)( 4, 5)( 7,22)( 8,21)( 9,20)(10,19)(13,23)(14,24)(15,17) (16,18)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1,12)( 2,11)( 3,18)( 4,17)( 5,15)( 6,16)( 7,22)( 8,21)( 9,20)(10,19)(13,23) (14,24)$
	$ 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 $	$1$	$2$	$( 1,14)( 2,13)( 3,16)( 4,15)( 5,17)( 6,18)( 7,20)( 8,19)( 9,22)(10,21)(11,23) (12,24)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1,23)( 2,24)( 3, 5)( 4, 6)( 7,10)( 8, 9)(11,14)(12,13)(15,18)(16,17)(19,22) (20,21)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1,24)( 2,23)( 3, 6)( 4, 5)( 7, 9)( 8,10)(11,13)(12,14)(15,17)(16,18)(19,21) (20,22)$

magma: ConjugacyClasses(G);

Group invariants

Order:	$192=2^{6} \cdot 3$	magma: Order(G);
Cyclic:	no	magma: IsCyclic(G);
Abelian:	no	magma: IsAbelian(G);
Solvable:	yes	magma: IsSolvable(G);
Nilpotency class:	not nilpotent
Label:	192.1537	magma: IdentifyGroup(G);
Character table:	40 x 40 character table

magma: CharacterTable(G);

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Group action invariants

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Group invariants

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	24T400
Degree	$24$
Order	$192$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	no
Group:	$C_2^3\times S_4$