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Properties

Label 30T12
Degree $30$
Order $60$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3\times C_{10}$

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Show commands: Magma

magma: G := TransitiveGroup(30, 12);
 

Group action invariants

Degree $n$:  $30$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $12$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_3\times C_{10}$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $10$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,24,25,17,19,11,14,5,7,29)(2,23,26,18,20,12,13,6,8,30)(3,16,28,9,22)(4,15,27,10,21), (1,2)(3,24)(4,23)(5,16)(6,15)(7,8)(9,29)(10,30)(11,22)(12,21)(13,14)(17,28)(18,27)(19,20)(25,26)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$5$:  $C_5$
$6$:  $S_3$
$10$:  $C_{10}$ x 3
$12$:  $D_{6}$
$20$:  20T3
$30$:  $S_3 \times C_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 5: $C_5$

Degree 6: $D_{6}$

Degree 10: $C_{10}$

Degree 15: $S_3 \times C_5$

Low degree siblings

30T12

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $60=2^{2} \cdot 3 \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  60.11
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 5A1 5A-1 5A2 5A-2 6A 10A1 10A-1 10A3 10A-3 10B1 10B-1 10B3 10B-3 10C1 10C-1 10C3 10C-3 15A1 15A-1 15A2 15A-2 30A1 30A-1 30A7 30A-7
Size 1 1 3 3 2 1 1 1 1 2 1 1 1 1 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2
2 P 1A 1A 1A 1A 3A 5A2 5A-2 5A-1 5A1 3A 5A-2 5A2 5A-1 5A1 5A-2 5A-1 5A-2 5A2 5A1 5A2 5A-1 5A1 15A2 15A-2 15A-1 15A1 15A1 15A-2 15A-1 15A2
3 P 1A 2A 2B 2C 1A 5A-2 5A2 5A1 5A-1 2A 10A3 10A-3 10A-1 10A1 10C-3 10C1 10B-1 10B1 10B3 10C3 10B-3 10C-1 5A-2 5A2 5A1 5A-1 10A1 10A3 10A-1 10A-3
5 P 1A 2A 2B 2C 3A 1A 1A 1A 1A 6A 2A 2A 2A 2A 2C 2C 2B 2B 2B 2C 2B 2C 3A 3A 3A 3A 6A 6A 6A 6A
Type
60.11.1a 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 1 1 1 1 1 1
60.11.1b 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 1 1 −1 −1 −1 −1
60.11.1c 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 1 1 −1 −1 −1 −1
60.11.1d 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 1 1 1 1 1 1
60.11.1e1 C 1 1 1 1 1 ζ5−2 ζ52 ζ5 ζ5−1 1 ζ52 ζ5−2 ζ5 ζ5−1 ζ5−1 ζ5 ζ52 ζ5−2 ζ5−2 ζ52 ζ5−1 ζ5 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−1 ζ5 ζ5−2 ζ52
60.11.1e2 C 1 1 1 1 1 ζ52 ζ5−2 ζ5−1 ζ5 1 ζ5−2 ζ52 ζ5−1 ζ5 ζ5 ζ5−1 ζ5−2 ζ52 ζ52 ζ5−2 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5 ζ5−1 ζ52 ζ5−2
60.11.1e3 C 1 1 1 1 1 ζ5−1 ζ5 ζ5−2 ζ52 1 ζ5 ζ5−1 ζ5−2 ζ52 ζ52 ζ5−2 ζ5 ζ5−1 ζ5−1 ζ5 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−2 ζ52 ζ52 ζ5−2 ζ5−1 ζ5
60.11.1e4 C 1 1 1 1 1 ζ5 ζ5−1 ζ52 ζ5−2 1 ζ5−1 ζ5 ζ52 ζ5−2 ζ5−2 ζ52 ζ5−1 ζ5 ζ5 ζ5−1 ζ5−2 ζ52 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−2 ζ52 ζ5 ζ5−1
60.11.1f1 C 1 −1 −1 1 1 ζ5−2 ζ52 ζ5 ζ5−1 −1 −ζ52 −ζ5−2 −ζ5 −ζ5−1 −ζ5−1 −ζ5 −ζ52 −ζ5−2 ζ5−2 ζ52 ζ5−1 ζ5 ζ5−2 ζ52 ζ5 ζ5−1 −ζ5−1 −ζ5 −ζ5−2 −ζ52
60.11.1f2 C 1 −1 −1 1 1 ζ52 ζ5−2 ζ5−1 ζ5 −1 −ζ5−2 −ζ52 −ζ5−1 −ζ5 −ζ5 −ζ5−1 −ζ5−2 −ζ52 ζ52 ζ5−2 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−1 ζ5 −ζ5 −ζ5−1 −ζ52 −ζ5−2
60.11.1f3 C 1 −1 −1 1 1 ζ5−1 ζ5 ζ5−2 ζ52 −1 −ζ5 −ζ5−1 −ζ5−2 −ζ52 −ζ52 −ζ5−2 −ζ5 −ζ5−1 ζ5−1 ζ5 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−2 ζ52 −ζ52 −ζ5−2 −ζ5−1 −ζ5
60.11.1f4 C 1 −1 −1 1 1 ζ5 ζ5−1 ζ52 ζ5−2 −1 −ζ5−1 −ζ5 −ζ52 −ζ5−2 −ζ5−2 −ζ52 −ζ5−1 −ζ5 ζ5 ζ5−1 ζ5−2 ζ52 ζ5 ζ5−1 ζ52 ζ5−2 −ζ5−2 −ζ52 −ζ5 −ζ5−1
60.11.1g1 C 1 −1 1 −1 1 ζ5−2 ζ52 ζ5 ζ5−1 −1 −ζ52 −ζ5−2 −ζ5 −ζ5−1 ζ5−1 ζ5 ζ52 ζ5−2 −ζ5−2 −ζ52 −ζ5−1 −ζ5 ζ5−2 ζ52 ζ5 ζ5−1 −ζ5−1 −ζ5 −ζ5−2 −ζ52
60.11.1g2 C 1 −1 1 −1 1 ζ52 ζ5−2 ζ5−1 ζ5 −1 −ζ5−2 −ζ52 −ζ5−1 −ζ5 ζ5 ζ5−1 ζ5−2 ζ52 −ζ52 −ζ5−2 −ζ5 −ζ5−1 ζ52 ζ5−2 ζ5−1 ζ5 −ζ5 −ζ5−1 −ζ52 −ζ5−2
60.11.1g3 C 1 −1 1 −1 1 ζ5−1 ζ5 ζ5−2 ζ52 −1 −ζ5 −ζ5−1 −ζ5−2 −ζ52 ζ52 ζ5−2 ζ5 ζ5−1 −ζ5−1 −ζ5 −ζ52 −ζ5−2 ζ5−1 ζ5 ζ5−2 ζ52 −ζ52 −ζ5−2 −ζ5−1 −ζ5
60.11.1g4 C 1 −1 1 −1 1 ζ5 ζ5−1 ζ52 ζ5−2 −1 −ζ5−1 −ζ5 −ζ52 −ζ5−2 ζ5−2 ζ52 ζ5−1 ζ5 −ζ5 −ζ5−1 −ζ5−2 −ζ52 ζ5 ζ5−1 ζ52 ζ5−2 −ζ5−2 −ζ52 −ζ5 −ζ5−1
60.11.1h1 C 1 1 −1 −1 1 ζ5−2 ζ52 ζ5 ζ5−1 1 ζ52 ζ5−2 ζ5 ζ5−1 −ζ5−1 −ζ5 −ζ52 −ζ5−2 −ζ5−2 −ζ52 −ζ5−1 −ζ5 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−1 ζ5 ζ5−2 ζ52
60.11.1h2 C 1 1 −1 −1 1 ζ52 ζ5−2 ζ5−1 ζ5 1 ζ5−2 ζ52 ζ5−1 ζ5 −ζ5 −ζ5−1 −ζ5−2 −ζ52 −ζ52 −ζ5−2 −ζ5 −ζ5−1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5 ζ5−1 ζ52 ζ5−2
60.11.1h3 C 1 1 −1 −1 1 ζ5−1 ζ5 ζ5−2 ζ52 1 ζ5 ζ5−1 ζ5−2 ζ52 −ζ52 −ζ5−2 −ζ5 −ζ5−1 −ζ5−1 −ζ5 −ζ52 −ζ5−2 ζ5−1 ζ5 ζ5−2 ζ52 ζ52 ζ5−2 ζ5−1 ζ5
60.11.1h4 C 1 1 −1 −1 1 ζ5 ζ5−1 ζ52 ζ5−2 1 ζ5−1 ζ5 ζ52 ζ5−2 −ζ5−2 −ζ52 −ζ5−1 −ζ5 −ζ5 −ζ5−1 −ζ5−2 −ζ52 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−2 ζ52 ζ5 ζ5−1
60.11.2a R 2 2 0 0 −1 2 2 2 2 −1 2 2 2 2 0 0 0 0 0 0 0 0 −1 −1 −1 −1 −1 −1 −1 −1
60.11.2b R 2 −2 0 0 −1 2 2 2 2 1 −2 −2 −2 −2 0 0 0 0 0 0 0 0 −1 −1 −1 −1 1 1 1 1
60.11.2c1 C 2 2 0 0 −1 2ζ5−2 2ζ52 2ζ5 2ζ5−1 −1 2ζ52 2ζ5−2 2ζ5 2ζ5−1 0 0 0 0 0 0 0 0 −ζ5−2 −ζ52 −ζ5 −ζ5−1 −ζ5−1 −ζ5 −ζ5−2 −ζ52
60.11.2c2 C 2 2 0 0 −1 2ζ52 2ζ5−2 2ζ5−1 2ζ5 −1 2ζ5−2 2ζ52 2ζ5−1 2ζ5 0 0 0 0 0 0 0 0 −ζ52 −ζ5−2 −ζ5−1 −ζ5 −ζ5 −ζ5−1 −ζ52 −ζ5−2
60.11.2c3 C 2 2 0 0 −1 2ζ5−1 2ζ5 2ζ5−2 2ζ52 −1 2ζ5 2ζ5−1 2ζ5−2 2ζ52 0 0 0 0 0 0 0 0 −ζ5−1 −ζ5 −ζ5−2 −ζ52 −ζ52 −ζ5−2 −ζ5−1 −ζ5
60.11.2c4 C 2 2 0 0 −1 2ζ5 2ζ5−1 2ζ52 2ζ5−2 −1 2ζ5−1 2ζ5 2ζ52 2ζ5−2 0 0 0 0 0 0 0 0 −ζ5 −ζ5−1 −ζ52 −ζ5−2 −ζ5−2 −ζ52 −ζ5 −ζ5−1
60.11.2d1 C 2 −2 0 0 −1 2ζ5−2 2ζ52 2ζ5 2ζ5−1 1 −2ζ52 −2ζ5−2 −2ζ5 −2ζ5−1 0 0 0 0 0 0 0 0 −ζ5−2 −ζ52 −ζ5 −ζ5−1 ζ5−1 ζ5 ζ5−2 ζ52
60.11.2d2 C 2 −2 0 0 −1 2ζ52 2ζ5−2 2ζ5−1 2ζ5 1 −2ζ5−2 −2ζ52 −2ζ5−1 −2ζ5 0 0 0 0 0 0 0 0 −ζ52 −ζ5−2 −ζ5−1 −ζ5 ζ5 ζ5−1 ζ52 ζ5−2
60.11.2d3 C 2 −2 0 0 −1 2ζ5−1 2ζ5 2ζ5−2 2ζ52 1 −2ζ5 −2ζ5−1 −2ζ5−2 −2ζ52 0 0 0 0 0 0 0 0 −ζ5−1 −ζ5 −ζ5−2 −ζ52 ζ52 ζ5−2 ζ5−1 ζ5
60.11.2d4 C 2 −2 0 0 −1 2ζ5 2ζ5−1 2ζ52 2ζ5−2 1 −2ζ5−1 −2ζ5 −2ζ52 −2ζ5−2 0 0 0 0 0 0 0 0 −ζ5 −ζ5−1 −ζ52 −ζ5−2 ζ5−2 ζ52 ζ5 ζ5−1

magma: CharacterTable(G);