LMFDB - Galois group: 15T49

Show commands: Magma

magma:G := TransitiveGroup(15, 49);

Group invariants

Abstract group:	$C_5^3:(C_4\times S_3)$	magma:IdentifyGroup(G);
Order:	$3000=2^{3} \cdot 3 \cdot 5^{3}$	magma:Order(G);
Cyclic:	no	magma:IsCyclic(G);
Abelian:	no	magma:IsAbelian(G);
Solvable:	yes	magma:IsSolvable(G);
Nilpotency class:	not nilpotent	magma:NilpotencyClass(G);

Group action invariants

Degree $n$:	$15$	magma:t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$49$	magma:t, n := TransitiveGroupIdentification(G); t;
CHM label:	$[5^{3}:4]S(3)$
Parity:	$-1$	magma:IsEven(G);
Primitive:	no	magma:IsPrimitive(G);
$\card{\Aut(F/K)}$:	$1$	magma:Order(Centralizer(SymmetricGroup(n), G));
Generators:	$(1,11)(2,7)(4,14)(5,10)(8,13)$, $(1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10,15)$, $(1,7,4,13)(2,14,8,11)(3,6,12,9)$, $(3,6,9,12,15)$	magma:Generators(G);

Low degree resolvents

$\card{(G/N)}$ Galois groups for stem field(s)
$2$: $C_2$ x 3
$4$: $C_4$ x 2, $C_2^2$
$6$: $S_3$
$8$: $C_4\times C_2$
$12$: $D_{6}$
$20$: $F_5$
$24$: $S_3 \times C_4$
$40$: $F_{5}\times C_2$
$120$: $F_5 \times S_3$
$600$: 15T27

Resolvents shown for degrees $\leq 47$

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_4$ x 2, $C_2^2$
$6$:	$S_3$
$8$:	$C_4\times C_2$
$12$:	$D_{6}$
$20$:	$F_5$
$24$:	$S_3 \times C_4$
$40$:	$F_{5}\times C_2$
$120$:	$F_5 \times S_3$
$600$:	15T27

Subfields

Degree 3: $S_3$
Degree 5: None

Low degree siblings

15T49 x 3, 30T433 x 4, 30T435 x 2, 30T438 x 4, 30T442 x 4
Siblings are shown with degree $\leq 47$

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

Conjugacy classes

Label	Cycle Type	Size	Order	Index	Representative
1A	$1^{15}$	$1$	$1$	$0$	$()$
2A	$2^{5},1^{5}$	$15$	$2$	$5$	$( 1,15)( 3, 4)( 6, 7)( 9,10)(12,13)$
2B	$2^{7},1$	$75$	$2$	$7$	$( 1, 3)( 2, 5)( 4,15)( 6,13)( 7,12)( 8,14)( 9,10)$
2C	$2^{6},1^{3}$	$125$	$2$	$6$	$( 1,13)( 2, 5)( 3, 9)( 4,10)( 8,14)(12,15)$
3A	$3^{5}$	$50$	$3$	$10$	$( 1, 3,14)( 2, 4, 6)( 5, 7, 9)( 8,10,12)(11,13,15)$
4A1	$4^{3},1^{3}$	$125$	$4$	$9$	$( 1,10,13, 4)( 2,14, 8,11)( 3,12,15, 6)$
4A-1	$4^{3},1^{3}$	$125$	$4$	$9$	$( 1, 4,13,10)( 2,11, 8,14)( 3, 6,15,12)$
4B1	$4^{3},2,1$	$375$	$4$	$10$	$( 1, 4,13,10)( 2, 9, 5, 3)( 6,11)( 8,12,14,15)$
4B-1	$4^{3},2,1$	$375$	$4$	$10$	$( 1,10,13, 4)( 2, 3, 5, 9)( 6,11)( 8,15,14,12)$
5A	$5^{3}$	$4$	$5$	$12$	$( 1, 7,13, 4,10)( 2, 8,14, 5,11)( 3, 9,15, 6,12)$
5B	$5^{2},1^{5}$	$12$	$5$	$8$	$( 1, 7,13, 4,10)( 3, 9,15, 6,12)$
5C	$5^{2},1^{5}$	$12$	$5$	$8$	$( 1,13,10, 7, 4)( 3, 6, 9,12,15)$
5D	$5^{3}$	$12$	$5$	$12$	$( 1, 4, 7,10,13)( 2,14,11, 8, 5)( 3, 6, 9,12,15)$
5E	$5^{3}$	$12$	$5$	$12$	$( 1,13,10, 7, 4)( 2,11, 5,14, 8)( 3,12, 6,15, 9)$
5F	$5^{3}$	$12$	$5$	$12$	$( 1,13,10, 7, 4)( 2, 8,14, 5,11)( 3, 9,15, 6,12)$
5G	$5,1^{10}$	$12$	$5$	$4$	$( 3,12, 6,15, 9)$
5H	$5^{2},1^{5}$	$24$	$5$	$8$	$( 2,14,11, 8, 5)( 3, 9,15, 6,12)$
5I	$5^{3}$	$24$	$5$	$12$	$( 1,13,10, 7, 4)( 2,11, 5,14, 8)( 3, 9,15, 6,12)$
6A	$6^{2},3$	$250$	$6$	$12$	$( 1,11, 3,13,14,15)( 2,12, 4, 8, 6,10)( 5, 9, 7)$
10A	$10,1^{5}$	$60$	$10$	$9$	$( 1, 3, 7, 9,13,15, 4, 6,10,12)$
10B	$10,5$	$60$	$10$	$13$	$( 1, 5, 7,11,13, 2, 4, 8,10,14)( 3, 6, 9,12,15)$
10C	$10,5$	$60$	$10$	$13$	$( 1, 6, 4, 9, 7,12,10,15,13, 3)( 2, 8,14, 5,11)$
10D	$10,5$	$60$	$10$	$13$	$( 1, 7,13, 4,10)( 2,12,11, 6, 5,15,14, 9, 8, 3)$
10E	$10,5$	$60$	$10$	$13$	$( 1, 7,13, 4,10)( 2,15, 8, 6,14,12, 5, 3,11, 9)$
10F	$5,2^{5}$	$60$	$10$	$9$	$( 1,14)( 2, 4)( 3,15,12, 9, 6)( 5, 7)( 8,10)(11,13)$
10G	$10,2^{2},1$	$300$	$10$	$11$	$( 1,12,13,15,10, 3, 7, 6, 4, 9)( 2, 5)( 8,14)$
12A1	$12,3$	$250$	$12$	$13$	$( 1, 6,11,10, 3, 2,13,12,14, 4,15, 8)( 5, 7, 9)$
12A-1	$12,3$	$250$	$12$	$13$	$( 1, 8,15, 4,14,12,13, 2, 3,10,11, 6)( 5, 9, 7)$
15A	$15$	$200$	$15$	$14$	$( 1,11,15, 4,14, 3, 7, 2, 6,10, 5, 9,13, 8,12)$

Malle's constant $a(G)$: $1/4$

magma:ConjugacyClasses(G);

Character table

		1A	2A	2B	2C	3A	4A1	4A-1	4B1	4B-1	5A	5B	5C	5D	5E	5F	5G	5H	5I	6A	10A	10B	10C	10D	10E	10F	10G	12A1	12A-1	15A
	Size	1	15	75	125	50	125	125	375	375	4	12	12	12	12	12	12	24	24	250	60	60	60	60	60	60	300	250	250	200
	2 P	1A	1A	1A	1A	3A	2C	2C	2C	2C	5A	5B	5C	5D	5E	5F	5G	5H	5I	3A	5B	5A	5D	5E	5F	5G	5C	6A	6A	15A
	3 P	1A	2A	2B	2C	1A	4A-1	4A1	4B-1	4B1	5A	5B	5C	5D	5E	5F	5G	5H	5I	2C	10A	10B	10C	10D	10E	10F	10G	4A1	4A-1	5A
	5 P	1A	2A	2B	2C	3A	4A1	4A-1	4B1	4B-1	1A	1A	1A	1A	1A	1A	1A	1A	1A	6A	2A	2A	2A	2A	2A	2A	2B	12A1	12A-1	3A
	Type
3000.bf.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
3000.bf.1b	R	$1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$
3000.bf.1c	R	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$1$
3000.bf.1d	R	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$
3000.bf.1e1	C	$1$	$- 1$	$1$	$- 1$	$1$	$- i$	$i$	$- i$	$i$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$i$	$- i$	$1$
3000.bf.1e2	C	$1$	$- 1$	$1$	$- 1$	$1$	$i$	$- i$	$i$	$- i$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$- i$	$i$	$1$
3000.bf.1f1	C	$1$	$1$	$- 1$	$- 1$	$1$	$- i$	$i$	$i$	$- i$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$i$	$- i$	$1$
3000.bf.1f2	C	$1$	$1$	$- 1$	$- 1$	$1$	$i$	$- i$	$- i$	$i$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- i$	$i$	$1$
3000.bf.2a	R	$2$	$0$	$0$	$2$	$- 1$	$2$	$2$	$0$	$0$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$- 1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 1$	$- 1$	$- 1$
3000.bf.2b	R	$2$	$0$	$0$	$2$	$- 1$	$- 2$	$- 2$	$0$	$0$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$- 1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$1$	$1$	$- 1$
3000.bf.2c1	C	$2$	$0$	$0$	$- 2$	$- 1$	$- 2 i$	$2 i$	$0$	$0$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- i$	$i$	$- 1$
3000.bf.2c2	C	$2$	$0$	$0$	$- 2$	$- 1$	$2 i$	$- 2 i$	$0$	$0$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$i$	$- i$	$- 1$
3000.bf.4a	R	$4$	$4$	$0$	$0$	$4$	$0$	$0$	$0$	$0$	$- 1$	$4$	$4$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$0$	$- 1$	$- 1$	$4$	$- 1$	$- 1$	$- 1$	$0$	$0$	$0$	$- 1$
3000.bf.4b	R	$4$	$- 4$	$0$	$0$	$4$	$0$	$0$	$0$	$0$	$- 1$	$4$	$4$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$0$	$1$	$1$	$- 4$	$1$	$1$	$1$	$0$	$0$	$0$	$- 1$
3000.bf.8a	R	$8$	$0$	$0$	$0$	$- 4$	$0$	$0$	$0$	$0$	$- 2$	$8$	$8$	$- 2$	$- 2$	$- 2$	$- 2$	$- 2$	$- 2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$1$
3000.bf.12a	R	$12$	$0$	$4$	$0$	$0$	$0$	$0$	$0$	$0$	$12$	$2$	$- 3$	$2$	$2$	$2$	$2$	$- 3$	$- 3$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 1$	$0$	$0$	$0$
3000.bf.12b	R	$12$	$4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$12$	$- 3$	$2$	$- 3$	$- 3$	$- 3$	$- 3$	$2$	$2$	$0$	$- 1$	$- 1$	$- 1$	$4$	$- 1$	$- 1$	$0$	$0$	$0$	$0$
3000.bf.12c	R	$12$	$4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 3$	$- 3$	$2$	$- 3$	$- 3$	$7$	$2$	$2$	$- 3$	$0$	$- 1$	$4$	$- 1$	$- 1$	$- 1$	$- 1$	$0$	$0$	$0$	$0$
3000.bf.12d	R	$12$	$4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 3$	$- 3$	$2$	$- 3$	$7$	$2$	$- 3$	$- 3$	$2$	$0$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$4$	$0$	$0$	$0$	$0$
3000.bf.12e	R	$12$	$4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 3$	$- 3$	$2$	$2$	$- 3$	$- 3$	$7$	$- 3$	$2$	$0$	$4$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$0$	$0$	$0$	$0$
3000.bf.12f	R	$12$	$4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 3$	$- 3$	$2$	$7$	$2$	$- 3$	$- 3$	$2$	$- 3$	$0$	$- 1$	$- 1$	$- 1$	$- 1$	$4$	$- 1$	$0$	$0$	$0$	$0$
3000.bf.12g	R	$12$	$- 4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$12$	$- 3$	$2$	$- 3$	$- 3$	$- 3$	$- 3$	$2$	$2$	$0$	$1$	$1$	$1$	$- 4$	$1$	$1$	$0$	$0$	$0$	$0$
3000.bf.12h	R	$12$	$0$	$- 4$	$0$	$0$	$0$	$0$	$0$	$0$	$12$	$2$	$- 3$	$2$	$2$	$2$	$2$	$- 3$	$- 3$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$1$	$0$	$0$	$0$
3000.bf.12i	R	$12$	$- 4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 3$	$- 3$	$2$	$- 3$	$- 3$	$7$	$2$	$2$	$- 3$	$0$	$1$	$- 4$	$1$	$1$	$1$	$1$	$0$	$0$	$0$	$0$
3000.bf.12j	R	$12$	$- 4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 3$	$- 3$	$2$	$- 3$	$7$	$2$	$- 3$	$- 3$	$2$	$0$	$1$	$1$	$1$	$1$	$1$	$- 4$	$0$	$0$	$0$	$0$
3000.bf.12k	R	$12$	$- 4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 3$	$- 3$	$2$	$2$	$- 3$	$- 3$	$7$	$- 3$	$2$	$0$	$- 4$	$1$	$1$	$1$	$1$	$1$	$0$	$0$	$0$	$0$
3000.bf.12l	R	$12$	$- 4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 3$	$- 3$	$2$	$7$	$2$	$- 3$	$- 3$	$2$	$- 3$	$0$	$1$	$1$	$1$	$1$	$- 4$	$1$	$0$	$0$	$0$	$0$
3000.bf.24a	R	$24$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 6$	$4$	$- 6$	$- 6$	$4$	$- 6$	$4$	$4$	$- 1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
3000.bf.24b	R	$24$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 6$	$4$	$- 6$	$4$	$- 6$	$4$	$- 6$	$- 1$	$4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$

magma:CharacterTable(G);

Regular extensions

Data not computed

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This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	15T49

Degree	$15$
Order	$3000$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	no
Group:	$C_5^3:(C_4\times S_3)$