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Properties

Label 30T42
Degree $30$
Order $180$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3\times D_{15}$

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Show commands: Magma

magma: G := TransitiveGroup(30, 42);
 

Group action invariants

Degree $n$:  $30$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $42$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_3\times D_{15}$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (16,18,17)(19,21,20)(22,24,23)(25,27,26)(28,30,29), (2,3)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)(16,28)(17,30)(18,29)(19,27)(20,26)(21,25)(22,23), (1,17,6,19,7,23,12,25,13,28)(2,18,4,20,8,24,10,26,14,29)(3,16,5,21,9,22,11,27,15,30)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$ x 2
$10$:  $D_{5}$
$12$:  $D_{6}$ x 2
$20$:  $D_{10}$
$30$:  $D_{15}$
$36$:  $S_3^2$
$60$:  $D_5\times S_3$, $D_{30}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 5: $D_{5}$

Degree 6: $S_3^2$

Degree 10: $D_{10}$

Degree 15: None

Low degree siblings

45T13

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 $ $1$ $1$ $()$
$ 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $3$ $(16,17,18)(19,20,21)(22,23,24)(25,26,27)(28,29,30)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $45$ $2$ $( 2, 3)( 4,15)( 5,14)( 6,13)( 7,12)( 8,11)( 9,10)(16,28)(17,30)(18,29)(19,27) (20,26)(21,25)(22,23)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)(28,29,30)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,18,17)(19,21,20) (22,24,23)(25,27,26)(28,30,29)$
$ 15, 15 $ $2$ $15$ $( 1, 4, 9,12,14, 3, 6, 8,11,13, 2, 5, 7,10,15)(16,19,24,27,28,18,21,23,26,30, 17,20,22,25,29)$
$ 15, 15 $ $4$ $15$ $( 1, 4, 9,12,14, 3, 6, 8,11,13, 2, 5, 7,10,15)(16,20,23,27,29,17,21,24,25,30, 18,19,22,26,28)$
$ 15, 5, 5, 5 $ $4$ $15$ $( 1, 4, 9,12,14, 3, 6, 8,11,13, 2, 5, 7,10,15)(16,21,22,27,30)(17,19,23,25,28) (18,20,24,26,29)$
$ 15, 15 $ $2$ $15$ $( 1, 5, 8,12,15, 2, 6, 9,10,13, 3, 4, 7,11,14)(16,20,23,27,29,17,21,24,25,30, 18,19,22,26,28)$
$ 15, 5, 5, 5 $ $4$ $15$ $( 1, 5, 8,12,15, 2, 6, 9,10,13, 3, 4, 7,11,14)(16,21,22,27,30)(17,19,23,25,28) (18,20,24,26,29)$
$ 5, 5, 5, 5, 5, 5 $ $2$ $5$ $( 1, 6, 7,12,13)( 2, 4, 8,10,14)( 3, 5, 9,11,15)(16,21,22,27,30) (17,19,23,25,28)(18,20,24,26,29)$
$ 5, 5, 5, 5, 5, 5 $ $2$ $5$ $( 1, 7,13, 6,12)( 2, 8,14, 4,10)( 3, 9,15, 5,11)(16,22,30,21,27) (17,23,28,19,25)(18,24,29,20,26)$
$ 15, 5, 5, 5 $ $4$ $15$ $( 1, 7,13, 6,12)( 2, 8,14, 4,10)( 3, 9,15, 5,11)(16,23,29,21,25,18,22,28,20, 27,17,24,30,19,26)$
$ 15, 5, 5, 5 $ $4$ $15$ $( 1, 7,13, 6,12)( 2, 8,14, 4,10)( 3, 9,15, 5,11)(16,24,28,21,26,17,22,29,19, 27,18,23,30,20,25)$
$ 15, 15 $ $2$ $15$ $( 1, 8,15, 6,10, 3, 7,14, 5,12, 2, 9,13, 4,11)(16,23,29,21,25,18,22,28,20,27, 17,24,30,19,26)$
$ 15, 15 $ $4$ $15$ $( 1, 8,15, 6,10, 3, 7,14, 5,12, 2, 9,13, 4,11)(16,24,28,21,26,17,22,29,19,27, 18,23,30,20,25)$
$ 15, 15 $ $2$ $15$ $( 1, 9,14, 6,11, 2, 7,15, 4,12, 3, 8,13, 5,10)(16,24,28,21,26,17,22,29,19,27, 18,23,30,20,25)$
$ 30 $ $6$ $30$ $( 1,16, 4,19, 9,24,12,27,14,28, 3,18, 6,21, 8,23,11,26,13,30, 2,17, 5,20, 7, 22,10,25,15,29)$
$ 30 $ $6$ $30$ $( 1,16, 5,20, 8,23,12,27,15,29, 2,17, 6,21, 9,24,10,25,13,30, 3,18, 4,19, 7, 22,11,26,14,28)$
$ 10, 10, 10 $ $6$ $10$ $( 1,16, 6,21, 7,22,12,27,13,30)( 2,17, 4,19, 8,23,10,25,14,28)( 3,18, 5,20, 9, 24,11,26,15,29)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $15$ $2$ $( 1,16)( 2,18)( 3,17)( 4,29)( 5,28)( 6,30)( 7,27)( 8,26)( 9,25)(10,24)(11,23) (12,22)(13,21)(14,20)(15,19)$
$ 6, 6, 6, 6, 6 $ $30$ $6$ $( 1,16, 2,18, 3,17)( 4,29, 5,28, 6,30)( 7,27, 8,26, 9,25)(10,24,11,23,12,22) (13,21,14,20,15,19)$
$ 30 $ $6$ $30$ $( 1,19,10,29, 5,22,13,17, 8,26, 3,21,12,28, 4,24,15,16, 7,25, 2,20,11,30, 6, 23,14,18, 9,27)$
$ 30 $ $6$ $30$ $( 1,19,11,30, 4,24,13,17, 9,27, 2,20,12,28, 5,22,14,18, 7,25, 3,21,10,29, 6, 23,15,16, 8,26)$
$ 10, 10, 10 $ $6$ $10$ $( 1,19,12,28, 6,23,13,17, 7,25)( 2,20,10,29, 4,24,14,18, 8,26)( 3,21,11,30, 5, 22,15,16, 9,27)$
$ 6, 6, 6, 6, 6 $ $6$ $6$ $( 1,22, 2,23, 3,24)( 4,25, 5,26, 6,27)( 7,30, 8,28, 9,29)(10,17,11,18,12,16) (13,21,14,19,15,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1,22)( 2,23)( 3,24)( 4,25)( 5,26)( 6,27)( 7,30)( 8,28)( 9,29)(10,17)(11,18) (12,16)(13,21)(14,19)(15,20)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $180=2^{2} \cdot 3^{2} \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  180.29
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 3B 3C 5A1 5A2 6A 6B 10A1 10A3 15A1 15A2 15A4 15A7 15B1 15B2 15C1 15C2 15C4 15C7 30A1 30A7 30A11 30A13
Size 1 3 15 45 2 2 4 2 2 6 30 6 6 2 2 2 2 4 4 4 4 4 4 6 6 6 6
2 P 1A 1A 1A 1A 3A 3B 3C 5A2 5A1 3B 3A 5A1 5A2 15A7 15A2 15A4 15A1 15B2 15B1 15C2 15C4 15C7 15C1 15A1 15A7 15A4 15A2
3 P 1A 2A 2B 2C 1A 1A 1A 5A2 5A1 2A 2B 10A3 10A1 5A1 5A1 5A2 5A2 5A1 5A2 5A1 5A2 5A1 5A2 10A1 10A3 10A1 10A3
5 P 1A 2A 2B 2C 3A 3B 3C 1A 1A 6A 6B 2A 2A 3B 3B 3B 3B 3A 3A 3C 3C 3C 3C 6A 6A 6A 6A
Type
180.29.1a 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
180.29.1b 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
180.29.1c 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
180.29.1d 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
180.29.2a R 2 0 2 0 −1 2 −1 2 2 0 −1 0 0 2 2 2 2 −1 −1 −1 −1 −1 −1 0 0 0 0
180.29.2b R 2 2 0 0 2 −1 −1 2 2 −1 0 2 2 −1 −1 −1 −1 2 2 −1 −1 −1 −1 −1 −1 −1 −1
180.29.2c R 2 −2 0 0 2 −1 −1 2 2 1 0 −2 −2 −1 −1 −1 −1 2 2 −1 −1 −1 −1 1 1 1 1
180.29.2d R 2 0 −2 0 −1 2 −1 2 2 0 1 0 0 2 2 2 2 −1 −1 −1 −1 −1 −1 0 0 0 0
180.29.2e1 R 2 2 0 0 2 2 2 ζ5−2+ζ52 ζ5−1+ζ5 2 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5
180.29.2e2 R 2 2 0 0 2 2 2 ζ5−1+ζ5 ζ5−2+ζ52 2 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52
180.29.2f1 R 2 −2 0 0 2 2 2 ζ5−2+ζ52 ζ5−1+ζ5 −2 0 −ζ5−1−ζ5 −ζ5−2−ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5
180.29.2f2 R 2 −2 0 0 2 2 2 ζ5−1+ζ5 ζ5−2+ζ52 −2 0 −ζ5−2−ζ52 −ζ5−1−ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52
180.29.2g1 R 2 2 0 0 2 −1 −1 ζ15−6+ζ156 ζ15−3+ζ153 −1 0 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−4+ζ154 ζ15−2+ζ152 ζ15−1+ζ15 ζ15−7+ζ157
180.29.2g2 R 2 2 0 0 2 −1 −1 ζ15−6+ζ156 ζ15−3+ζ153 −1 0 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−1+ζ15 ζ15−7+ζ157 ζ15−4+ζ154 ζ15−2+ζ152
180.29.2g3 R 2 2 0 0 2 −1 −1 ζ15−3+ζ153 ζ15−6+ζ156 −1 0 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−2+ζ152 ζ15−1+ζ15 ζ15−7+ζ157 ζ15−4+ζ154
180.29.2g4 R 2 2 0 0 2 −1 −1 ζ15−3+ζ153 ζ15−6+ζ156 −1 0 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−7+ζ157 ζ15−4+ζ154 ζ15−2+ζ152 ζ15−1+ζ15
180.29.2h1 R 2 −2 0 0 2 −1 −1 ζ15−6+ζ156 ζ15−3+ζ153 1 0 −ζ15−3−ζ153 −ζ15−6−ζ156 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−1+ζ15 −ζ15−4−ζ154 −ζ15−2−ζ152 −ζ15−1−ζ15 −ζ15−7−ζ157
180.29.2h2 R 2 −2 0 0 2 −1 −1 ζ15−6+ζ156 ζ15−3+ζ153 1 0 −ζ15−3−ζ153 −ζ15−6−ζ156 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−3+ζ153 ζ15−6+ζ156 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−4+ζ154 −ζ15−1−ζ15 −ζ15−7−ζ157 −ζ15−4−ζ154 −ζ15−2−ζ152
180.29.2h3 R 2 −2 0 0 2 −1 −1 ζ15−3+ζ153 ζ15−6+ζ156 1 0 −ζ15−6−ζ156 −ζ15−3−ζ153 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−7+ζ157 −ζ15−2−ζ152 −ζ15−1−ζ15 −ζ15−7−ζ157 −ζ15−4−ζ154
180.29.2h4 R 2 −2 0 0 2 −1 −1 ζ15−3+ζ153 ζ15−6+ζ156 1 0 −ζ15−6−ζ156 −ζ15−3−ζ153 ζ15−1+ζ15 ζ15−2+ζ152 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−6+ζ156 ζ15−3+ζ153 ζ15−4+ζ154 ζ15−7+ζ157 ζ15−1+ζ15 ζ15−2+ζ152 −ζ15−7−ζ157 −ζ15−4−ζ154 −ζ15−2−ζ152 −ζ15−1−ζ15
180.29.4a R 4 0 0 0 −2 −2 1 4 4 0 0 0 0 −2 −2 −2 −2 −2 −2 1 1 1 1 0 0 0 0
180.29.4b1 R 4 0 0 0 −2 4 −2 2ζ5−2+2ζ52 2ζ5−1+2ζ5 0 0 0 0 2ζ5−1+2ζ5 2ζ5−2+2ζ52 2ζ5−1+2ζ5 2ζ5−2+2ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 0 0 0 0
180.29.4b2 R 4 0 0 0 −2 4 −2 2ζ5−1+2ζ5 2ζ5−2+2ζ52 0 0 0 0 2ζ5−2+2ζ52 2ζ5−1+2ζ5 2ζ5−2+2ζ52 2ζ5−1+2ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 0 0 0 0
180.29.4c1 R 4 0 0 0 −2 −2 1 2ζ15−6+2ζ156 2ζ15−3+2ζ153 0 0 0 0 2ζ15−7+2ζ157 2ζ15−1+2ζ15 2ζ15−2+2ζ152 2ζ15−4+2ζ154 −ζ15−3−ζ153 −ζ15−6−ζ156 −ζ15−2−ζ152 −ζ15−4−ζ154 −ζ15−7−ζ157 −ζ15−1−ζ15 0 0 0 0
180.29.4c2 R 4 0 0 0 −2 −2 1 2ζ15−6+2ζ156 2ζ15−3+2ζ153 0 0 0 0 2ζ15−2+2ζ152 2ζ15−4+2ζ154 2ζ15−7+2ζ157 2ζ15−1+2ζ15 −ζ15−3−ζ153 −ζ15−6−ζ156 −ζ15−7−ζ157 −ζ15−1−ζ15 −ζ15−2−ζ152 −ζ15−4−ζ154 0 0 0 0
180.29.4c3 R 4 0 0 0 −2 −2 1 2ζ15−3+2ζ153 2ζ15−6+2ζ156 0 0 0 0 2ζ15−4+2ζ154 2ζ15−7+2ζ157 2ζ15−1+2ζ15 2ζ15−2+2ζ152 −ζ15−6−ζ156 −ζ15−3−ζ153 −ζ15−1−ζ15 −ζ15−2−ζ152 −ζ15−4−ζ154 −ζ15−7−ζ157 0 0 0 0
180.29.4c4 R 4 0 0 0 −2 −2 1 2ζ15−3+2ζ153 2ζ15−6+2ζ156 0 0 0 0 2ζ15−1+2ζ15 2ζ15−2+2ζ152 2ζ15−4+2ζ154 2ζ15−7+2ζ157 −ζ15−6−ζ156 −ζ15−3−ζ153 −ζ15−4−ζ154 −ζ15−7−ζ157 −ζ15−1−ζ15 −ζ15−2−ζ152 0 0 0 0

magma: CharacterTable(G);