# Properties

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

# Related objects

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); Nilpotency class: $-1$ (not nilpotent) 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 $\leq 47$

## 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

 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); Label: 192.1537 magma: IdentifyGroup(G);
 Character table: not available.

magma: CharacterTable(G);