# Properties

 Label 30T23 Degree $30$ Order $120$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $S_3\times F_5$

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magma: G := TransitiveGroup(30, 23);

## Group action invariants

 Degree $n$: $30$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $23$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $S_3\times F_5$ Parity: $-1$ magma: IsEven(G); Primitive: no magma: IsPrimitive(G); Nilpotency class: $-1$ (not nilpotent) magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $2$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,7,20,13)(2,8,19,14)(3,11,28,29)(4,12,27,30)(5,15)(6,16)(9,24,22,18)(10,23,21,17), (1,21,11,2,22,12)(3,6,20,27,24,25,9,17,13,16,29,8)(4,5,19,28,23,26,10,18,14,15,30,7) magma: Generators(G);

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$6$:  $S_3$
$8$:  $C_4\times C_2$
$12$:  $D_{6}$
$20$:  $F_5$
$24$:  $S_3 \times C_4$
$40$:  $F_{5}\times C_2$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 5: $F_5$

Degree 6: $D_{6}$

Degree 10: $F_{5}\times C_2$

Degree 15: $F_5 \times S_3$

## Low degree siblings

15T11, 30T24, 30T32

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $4, 4, 4, 4, 4, 4, 2, 2, 1, 1$ $15$ $4$ $( 3, 5, 9,18)( 4, 6,10,17)( 7,13,26,20)( 8,14,25,19)(11,22)(12,21) (15,29,28,24)(16,30,27,23)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1$ $5$ $2$ $( 3, 9)( 4,10)( 5,18)( 6,17)( 7,26)( 8,25)(13,20)(14,19)(15,28)(16,27)(23,30) (24,29)$ $4, 4, 4, 4, 4, 4, 2, 2, 1, 1$ $15$ $4$ $( 3,18, 9, 5)( 4,17,10, 6)( 7,20,26,13)( 8,19,25,14)(11,22)(12,21) (15,24,28,29)(16,23,27,30)$ $4, 4, 4, 4, 4, 4, 2, 2, 2$ $5$ $4$ $( 1, 2)( 3,16, 9,27)( 4,15,10,28)( 5,30,18,23)( 6,29,17,24)( 7,14,26,19) ( 8,13,25,20)(11,12)(21,22)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $3$ $2$ $( 1, 2)( 3,23)( 4,24)( 5,16)( 6,15)( 7, 8)( 9,30)(10,29)(11,21)(12,22)(13,14) (17,28)(18,27)(19,20)(25,26)$ $4, 4, 4, 4, 4, 4, 2, 2, 2$ $5$ $4$ $( 1, 2)( 3,27, 9,16)( 4,28,10,15)( 5,23,18,30)( 6,24,17,29)( 7,19,26,14) ( 8,20,25,13)(11,12)(21,22)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $15$ $2$ $( 1, 2)( 3,30)( 4,29)( 5,27)( 6,28)( 7,25)( 8,26)( 9,23)(10,24)(11,21)(12,22) (13,19)(14,20)(15,17)(16,18)$ $15, 15$ $8$ $15$ $( 1, 3, 5, 7, 9,11,13,15,18,20,22,24,26,28,29)( 2, 4, 6, 8,10,12,14,16,17,19, 21,23,25,27,30)$ $6, 6, 6, 6, 3, 3$ $10$ $6$ $( 1, 3,11,13,22,24)( 2, 4,12,14,21,23)( 5,20,15,29,26, 9)( 6,19,16,30,25,10) ( 7,28,18)( 8,27,17)$ $12, 12, 6$ $10$ $12$ $( 1, 4,18,25,22,23, 7,16,11,14,28, 6)( 2, 3,17,26,21,24, 8,15,12,13,27, 5) ( 9,30,20,10,29,19)$ $10, 10, 10$ $12$ $10$ $( 1, 4,26,27,20,21,13,16, 7,10)( 2, 3,25,28,19,22,14,15, 8, 9)( 5,17,29,12,24, 6,18,30,11,23)$ $12, 12, 6$ $10$ $12$ $( 1, 4,29, 8,22,23,20,27,11,14, 9,17)( 2, 3,30, 7,21,24,19,28,12,13,10,18) ( 5,25,15, 6,26,16)$ $5, 5, 5, 5, 5, 5$ $4$ $5$ $( 1, 7,13,20,26)( 2, 8,14,19,25)( 3, 9,15,22,28)( 4,10,16,21,27) ( 5,11,18,24,29)( 6,12,17,23,30)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3$ $2$ $3$ $( 1,11,22)( 2,12,21)( 3,13,24)( 4,14,23)( 5,15,26)( 6,16,25)( 7,18,28) ( 8,17,27)( 9,20,29)(10,19,30)$

magma: ConjugacyClasses(G);

## Group invariants

 Order: $120=2^{3} \cdot 3 \cdot 5$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Label: 120.36 magma: IdentifyGroup(G);
 Character table:  2 3 3 3 3 3 3 3 3 . 2 2 1 2 1 2 3 1 . 1 . 1 . 1 . 1 1 1 . 1 1 1 5 1 . . . . 1 . . 1 . . 1 . 1 1 1a 4a 2a 4b 4c 2b 4d 2c 15a 6a 12a 10a 12b 5a 3a 2P 1a 2a 1a 2a 2a 1a 2a 1a 15a 3a 6a 5a 6a 5a 3a 3P 1a 4b 2a 4a 4d 2b 4c 2c 5a 2a 4d 10a 4c 5a 1a 5P 1a 4a 2a 4b 4c 2b 4d 2c 3a 6a 12a 2b 12b 1a 3a 7P 1a 4b 2a 4a 4d 2b 4c 2c 15a 6a 12b 10a 12a 5a 3a 11P 1a 4b 2a 4a 4d 2b 4c 2c 15a 6a 12b 10a 12a 5a 3a 13P 1a 4a 2a 4b 4c 2b 4d 2c 15a 6a 12a 10a 12b 5a 3a X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 1 -1 -1 1 -1 1 1 1 -1 1 -1 1 1 X.3 1 -1 1 -1 1 -1 1 -1 1 1 1 -1 1 1 1 X.4 1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 -1 1 1 X.5 1 A -1 -A A 1 -A -1 1 -1 A 1 -A 1 1 X.6 1 -A -1 A -A 1 A -1 1 -1 -A 1 A 1 1 X.7 1 A -1 -A -A -1 A 1 1 -1 -A -1 A 1 1 X.8 1 -A -1 A A -1 -A 1 1 -1 A -1 -A 1 1 X.9 2 . 2 . -2 . -2 . -1 -1 1 . 1 2 -1 X.10 2 . 2 . 2 . 2 . -1 -1 -1 . -1 2 -1 X.11 2 . -2 . B . -B . -1 1 -A . A 2 -1 X.12 2 . -2 . -B . B . -1 1 A . -A 2 -1 X.13 4 . . . . -4 . . -1 . . 1 . -1 4 X.14 4 . . . . 4 . . -1 . . -1 . -1 4 X.15 8 . . . . . . . 1 . . . . -2 -4 A = -E(4) = -Sqrt(-1) = -i B = -2*E(4) = -2*Sqrt(-1) = -2i 

magma: CharacterTable(G);