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Properties

Label 30T49
Degree $30$
Order $180$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^2:C_{20}$

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Show commands: Magma

magma: G := TransitiveGroup(30, 49);
 

Group action invariants

Degree $n$:  $30$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $49$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3^2:C_{20}$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $5$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (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), (1,30,13,27,10,24,7,21,4,18)(2,29,15,25,11,23,9,19,5,17,3,28,14,26,12,22,8,20,6,16)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$
$5$:  $C_5$
$10$:  $C_{10}$
$20$:  20T1
$36$:  $C_3^2:C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 5: $C_5$

Degree 6: $C_3^2:C_4$

Degree 10: $C_{10}$

Degree 15: None

Low degree siblings

30T49, 45T25

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

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:  180.23
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A 3B 4A1 4A-1 5A1 5A-1 5A2 5A-2 10A1 10A-1 10A3 10A-3 15A1 15A-1 15A2 15A-2 15B1 15B-1 15B2 15B-2 20A1 20A-1 20A3 20A-3 20A7 20A-7 20A9 20A-9
Size 1 9 4 4 9 9 1 1 1 1 9 9 9 9 4 4 4 4 4 4 4 4 9 9 9 9 9 9 9 9
2 P 1A 1A 3A 3B 2A 2A 5A2 5A-2 5A-1 5A1 5A-2 5A2 5A-1 5A1 15A2 15A-1 15A1 15A-2 15B-2 15B-1 15B2 15B1 10A3 10A-3 10A3 10A-1 10A-3 10A1 10A-1 10A1
3 P 1A 2A 1A 1A 4A-1 4A1 5A-2 5A2 5A1 5A-1 10A3 10A-3 10A-1 10A1 5A1 5A2 5A-2 5A-1 5A-2 5A-1 5A2 5A1 20A-1 20A1 20A9 20A-3 20A-9 20A-7 20A7 20A3
5 P 1A 2A 3A 3B 4A1 4A-1 1A 1A 1A 1A 2A 2A 2A 2A 3A 3A 3A 3A 3B 3B 3B 3B 4A1 4A-1 4A-1 4A-1 4A1 4A-1 4A1 4A1
Type
180.23.1a 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
180.23.1b 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
180.23.1c1 C 1 −1 1 1 −i i 1 1 1 1 −1 −1 −1 −1 1 1 1 1 1 1 1 1 −i i i −i i −i −i i
180.23.1c2 C 1 −1 1 1 i −i 1 1 1 1 −1 −1 −1 −1 1 1 1 1 1 1 1 1 i −i −i i −i i i −i
180.23.1d1 C 1 1 1 1 1 1 ζ5−2 ζ52 ζ5 ζ5−1 ζ52 ζ5−2 ζ5 ζ5−1 ζ5 ζ5−1 ζ52 ζ5−2 ζ52 ζ5−2 ζ5−1 ζ5 ζ5 ζ5−1 ζ5−2 ζ52 ζ52 ζ5−2 ζ5−1 ζ5
180.23.1d2 C 1 1 1 1 1 1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−1 ζ5 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−1 ζ5 ζ52 ζ5−2 ζ5−2 ζ52 ζ5 ζ5−1
180.23.1d3 C 1 1 1 1 1 1 ζ5−1 ζ5 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−2 ζ52 ζ5−2 ζ52 ζ5 ζ5−1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−2 ζ52 ζ5−1 ζ5 ζ5 ζ5−1 ζ52 ζ5−2
180.23.1d4 C 1 1 1 1 1 1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−1 ζ5 ζ52 ζ5−2 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−1 ζ5 ζ5−2 ζ52 ζ52 ζ5−2 ζ5 ζ5−1 ζ5−1 ζ5 ζ5−2 ζ52
180.23.1e1 C 1 1 1 1 −1 −1 ζ5−2 ζ52 ζ5 ζ5−1 ζ52 ζ5−2 ζ5 ζ5−1 ζ5 ζ5−1 ζ52 ζ5−2 ζ52 ζ5−2 ζ5−1 ζ5 −ζ5 −ζ5−1 −ζ5−2 −ζ52 −ζ52 −ζ5−2 −ζ5−1 −ζ5
180.23.1e2 C 1 1 1 1 −1 −1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−1 ζ5 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−2 ζ52 ζ5 ζ5−1 −ζ5−1 −ζ5 −ζ52 −ζ5−2 −ζ5−2 −ζ52 −ζ5 −ζ5−1
180.23.1e3 C 1 1 1 1 −1 −1 ζ5−1 ζ5 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−2 ζ52 ζ5−2 ζ52 ζ5 ζ5−1 ζ5 ζ5−1 ζ52 ζ5−2 −ζ5−2 −ζ52 −ζ5−1 −ζ5 −ζ5 −ζ5−1 −ζ52 −ζ5−2
180.23.1e4 C 1 1 1 1 −1 −1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−1 ζ5 ζ52 ζ5−2 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−1 ζ5 ζ5−2 ζ52 −ζ52 −ζ5−2 −ζ5 −ζ5−1 −ζ5−1 −ζ5 −ζ5−2 −ζ52
180.23.1f1 C 1 −1 1 1 −ζ205 ζ205 −ζ202 ζ208 ζ204 −ζ206 −ζ208 ζ202 −ζ204 ζ206 ζ204 −ζ206 ζ208 −ζ202 ζ208 −ζ202 −ζ206 ζ204 −ζ209 ζ20 −ζ207 ζ203 −ζ203 ζ207 −ζ20 ζ209
180.23.1f2 C 1 −1 1 1 ζ205 −ζ205 ζ208 −ζ202 −ζ206 ζ204 ζ202 −ζ208 ζ206 −ζ204 −ζ206 ζ204 −ζ202 ζ208 −ζ202 ζ208 ζ204 −ζ206 ζ20 −ζ209 ζ203 −ζ207 ζ207 −ζ203 ζ209 −ζ20
180.23.1f3 C 1 −1 1 1 −ζ205 ζ205 ζ208 −ζ202 −ζ206 ζ204 ζ202 −ζ208 ζ206 −ζ204 −ζ206 ζ204 −ζ202 ζ208 −ζ202 ζ208 ζ204 −ζ206 −ζ20 ζ209 −ζ203 ζ207 −ζ207 ζ203 −ζ209 ζ20
180.23.1f4 C 1 −1 1 1 ζ205 −ζ205 −ζ202 ζ208 ζ204 −ζ206 −ζ208 ζ202 −ζ204 ζ206 ζ204 −ζ206 ζ208 −ζ202 ζ208 −ζ202 −ζ206 ζ204 ζ209 −ζ20 ζ207 −ζ203 ζ203 −ζ207 ζ20 −ζ209
180.23.1f5 C 1 −1 1 1 −ζ205 ζ205 −ζ206 ζ204 −ζ202 ζ208 −ζ204 ζ206 ζ202 −ζ208 −ζ202 ζ208 ζ204 −ζ206 ζ204 −ζ206 ζ208 −ζ202 ζ207 −ζ203 ζ20 −ζ209 ζ209 −ζ20 ζ203 −ζ207
180.23.1f6 C 1 −1 1 1 ζ205 −ζ205 ζ204 −ζ206 ζ208 −ζ202 ζ206 −ζ204 −ζ208 ζ202 ζ208 −ζ202 −ζ206 ζ204 −ζ206 ζ204 −ζ202 ζ208 −ζ203 ζ207 −ζ209 ζ20 −ζ20 ζ209 −ζ207 ζ203
180.23.1f7 C 1 −1 1 1 −ζ205 ζ205 ζ204 −ζ206 ζ208 −ζ202 ζ206 −ζ204 −ζ208 ζ202 ζ208 −ζ202 −ζ206 ζ204 −ζ206 ζ204 −ζ202 ζ208 ζ203 −ζ207 ζ209 −ζ20 ζ20 −ζ209 ζ207 −ζ203
180.23.1f8 C 1 −1 1 1 ζ205 −ζ205 −ζ206 ζ204 −ζ202 ζ208 −ζ204 ζ206 ζ202 −ζ208 −ζ202 ζ208 ζ204 −ζ206 ζ204 −ζ206 ζ208 −ζ202 −ζ207 ζ203 −ζ20 ζ209 −ζ209 ζ20 −ζ203 ζ207
180.23.4a R 4 0 −2 1 0 0 4 4 4 4 0 0 0 0 −2 −2 −2 −2 1 1 1 1 0 0 0 0 0 0 0 0
180.23.4b R 4 0 1 −2 0 0 4 4 4 4 0 0 0 0 1 1 1 1 −2 −2 −2 −2 0 0 0 0 0 0 0 0
180.23.4c1 C 4 0 −2 1 0 0 4ζ5−2 4ζ52 4ζ5 4ζ5−1 0 0 0 0 −2ζ5 −2ζ5−1 −2ζ52 −2ζ5−2 ζ52 ζ5−2 ζ5−1 ζ5 0 0 0 0 0 0 0 0
180.23.4c2 C 4 0 −2 1 0 0 4ζ52 4ζ5−2 4ζ5−1 4ζ5 0 0 0 0 −2ζ5−1 −2ζ5 −2ζ5−2 −2ζ52 ζ5−2 ζ52 ζ5 ζ5−1 0 0 0 0 0 0 0 0
180.23.4c3 C 4 0 −2 1 0 0 4ζ5−1 4ζ5 4ζ5−2 4ζ52 0 0 0 0 −2ζ5−2 −2ζ52 −2ζ5 −2ζ5−1 ζ5 ζ5−1 ζ52 ζ5−2 0 0 0 0 0 0 0 0
180.23.4c4 C 4 0 −2 1 0 0 4ζ5 4ζ5−1 4ζ52 4ζ5−2 0 0 0 0 −2ζ52 −2ζ5−2 −2ζ5−1 −2ζ5 ζ5−1 ζ5 ζ5−2 ζ52 0 0 0 0 0 0 0 0
180.23.4d1 C 4 0 1 −2 0 0 4ζ5−2 4ζ52 4ζ5 4ζ5−1 0 0 0 0 ζ5 ζ5−1 ζ52 ζ5−2 −2ζ52 −2ζ5−2 −2ζ5−1 −2ζ5 0 0 0 0 0 0 0 0
180.23.4d2 C 4 0 1 −2 0 0 4ζ52 4ζ5−2 4ζ5−1 4ζ5 0 0 0 0 ζ5−1 ζ5 ζ5−2 ζ52 −2ζ5−2 −2ζ52 −2ζ5 −2ζ5−1 0 0 0 0 0 0 0 0
180.23.4d3 C 4 0 1 −2 0 0 4ζ5−1 4ζ5 4ζ5−2 4ζ52 0 0 0 0 ζ5−2 ζ52 ζ5 ζ5−1 −2ζ5 −2ζ5−1 −2ζ52 −2ζ5−2 0 0 0 0 0 0 0 0
180.23.4d4 C 4 0 1 −2 0 0 4ζ5 4ζ5−1 4ζ52 4ζ5−2 0 0 0 0 ζ52 ζ5−2 ζ5−1 ζ5 −2ζ5−1 −2ζ5 −2ζ5−2 −2ζ52 0 0 0 0 0 0 0 0

magma: CharacterTable(G);