LMFDB - Galois group: 24T1521

Show commands: Magma

magma:G := TransitiveGroup(24, 1521);

Group invariants

Abstract group:	$C_3^3:D_{12}$	magma:IdentifyGroup(G);
Order:	$648=2^{3} \cdot 3^{4}$	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 $n$:	$24$	magma:t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$1521$	magma:t, n := TransitiveGroupIdentification(G); t;
Parity:	$1$	magma:IsEven(G);
Primitive:	no	magma:IsPrimitive(G);
$\card{\Aut(F/K)}$:	$6$	magma:Order(Centralizer(SymmetricGroup(n), G));
Generators:	$(1,22)(2,21)(3,4)(5,18)(6,17)(7,8)(9,14)(10,13)(11,20)(12,19)(15,24)(16,23)$, $(1,3)(2,4)(5,24,21,8,13,16)(6,23,22,7,14,15)(9,19)(10,20)(11,17)(12,18)$	magma:Generators(G);

Low degree resolvents

$\card{(G/N)}$ Galois groups for stem field(s)
$2$: $C_2$ x 3
$4$: $C_2^2$
$6$: $S_3$ x 2
$8$: $D_{4}$
$12$: $D_{6}$ x 2
$24$: $D_{12}$, $(C_6\times C_2):C_2$
$36$: $S_3^2$
$72$: $C_3^2:D_4$, 12T38
$216$: 12T116, 12T118

Resolvents shown for degrees $\leq 47$

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_2^2$
$6$:	$S_3$ x 2
$8$:	$D_{4}$
$12$:	$D_{6}$ x 2
$24$:	$D_{12}$, $(C_6\times C_2):C_2$
$36$:	$S_3^2$
$72$:	$C_3^2:D_4$, 12T38
$216$:	12T116, 12T118

Subfields

Degree 2: $C_2$ x 3
Degree 3: None
Degree 4: $C_2^2$, $D_{4}$ x 2
Degree 6: None
Degree 8: $D_4$
Degree 12: 12T169

Low degree siblings

12T169 x 2, 18T210 x 2, 24T1521, 24T1533, 36T1105 x 2, 36T1106 x 2, 36T1107 x 2, 36T1153, 36T1154, 36T1160, 36T1167, 36T1173, 36T1190 x 2, 36T1195 x 2, 36T1225 x 2
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	$1^{24}$	$1$	$1$	$0$	$()$
2A	$2^{12}$	$9$	$2$	$12$	$( 1, 5)( 2, 6)( 3,24)( 4,23)( 7,12)( 8,11)( 9,13)(10,14)(15,20)(16,19)(17,21)(18,22)$
2B	$2^{12}$	$18$	$2$	$12$	$( 1,11)( 2,12)( 3, 9)( 4,10)( 5, 8)( 6, 7)(13,24)(14,23)(15,22)(16,21)(17,19)(18,20)$
2C	$2^{12}$	$54$	$2$	$12$	$( 1, 2)( 3, 7)( 4, 8)( 5,14)( 6,13)( 9,18)(10,17)(11,23)(12,24)(15,19)(16,20)(21,22)$
3A	$3^{8}$	$2$	$3$	$16$	$( 1,17, 9)( 2,18,10)( 3,11,19)( 4,12,20)( 5,21,13)( 6,22,14)( 7,15,23)( 8,16,24)$
3B	$3^{8}$	$2$	$3$	$16$	$( 1,17, 9)( 2,18,10)( 3,19,11)( 4,20,12)( 5,21,13)( 6,22,14)( 7,23,15)( 8,24,16)$
3C	$3^{4},1^{12}$	$4$	$3$	$8$	$( 3,19,11)( 4,20,12)( 7,15,23)( 8,16,24)$
3D	$3^{8}$	$4$	$3$	$16$	$( 1, 9,17)( 2,10,18)( 3,11,19)( 4,12,20)( 5,21,13)( 6,22,14)( 7,23,15)( 8,24,16)$
3E	$3^{4},1^{12}$	$4$	$3$	$8$	$( 3,11,19)( 4,12,20)( 7,15,23)( 8,16,24)$
3F1	$3^{4},1^{12}$	$4$	$3$	$8$	$( 3,11,19)( 4,12,20)( 5,21,13)( 6,22,14)$
3F-1	$3^{4},1^{12}$	$4$	$3$	$8$	$( 3,19,11)( 4,20,12)( 5,13,21)( 6,14,22)$
3G	$3^{8}$	$8$	$3$	$16$	$( 1,17, 9)( 2,18,10)( 3,11,19)( 4,12,20)( 5,13,21)( 6,14,22)( 7,15,23)( 8,16,24)$
3H	$3^{4},1^{12}$	$8$	$3$	$8$	$( 5,21,13)( 6,22,14)( 7,23,15)( 8,24,16)$
3I	$3^{6},1^{6}$	$8$	$3$	$12$	$( 1, 9,17)( 2,10,18)( 3,11,19)( 4,12,20)( 5,13,21)( 6,14,22)$
3J	$3^{6},1^{6}$	$8$	$3$	$12$	$( 1, 9,17)( 2,10,18)( 5,13,21)( 6,14,22)( 7,23,15)( 8,24,16)$
3K	$3^{2},1^{18}$	$8$	$3$	$4$	$( 1, 9,17)( 2,10,18)$
3L1	$3^{6},1^{6}$	$8$	$3$	$12$	$( 3,19,11)( 4,20,12)( 5,21,13)( 6,22,14)( 7,15,23)( 8,16,24)$
3L-1	$3^{6},1^{6}$	$8$	$3$	$12$	$( 1,17, 9)( 2,18,10)( 3,19,11)( 4,20,12)( 7,15,23)( 8,16,24)$
4A	$4^{6}$	$54$	$4$	$18$	$( 1,15,13,12)( 2,16,14,11)( 3,18, 8, 6)( 4,17, 7, 5)( 9,23,21,20)(10,24,22,19)$
6A	$6^{4}$	$18$	$6$	$20$	$( 1,13,17, 5, 9,21)( 2,14,18, 6,10,22)( 3,16,11,24,19, 8)( 4,15,12,23,20, 7)$
6B	$6^{4}$	$18$	$6$	$20$	$( 1,21,17,13, 9, 5)( 2,22,18,14,10, 6)( 3,16,19, 8,11,24)( 4,15,20, 7,12,23)$
6C1	$6^{4}$	$18$	$6$	$20$	$( 1, 3,17,11, 9,19)( 2, 4,18,12,10,20)( 5,24,21, 8,13,16)( 6,23,22, 7,14,15)$
6C-1	$6^{4}$	$18$	$6$	$20$	$( 1, 3, 9,19,17,11)( 2, 4,10,20,18,12)( 5,24,13,16,21, 8)( 6,23,14,15,22, 7)$
6D	$6^{4}$	$36$	$6$	$20$	$( 1, 8, 9,24,17,16)( 2, 7,10,23,18,15)( 3, 5,11,21,19,13)( 4, 6,12,22,20,14)$
6E	$6^{2},2^{6}$	$36$	$6$	$16$	$( 1,21)( 2,22)( 3,16,11,24,19, 8)( 4,15,12,23,20, 7)( 5, 9)( 6,10)(13,17)(14,18)$
6F1	$6^{2},2^{6}$	$36$	$6$	$16$	$( 1,16)( 2,15)( 3,13,11, 5,19,21)( 4,14,12, 6,20,22)( 7,10)( 8, 9)(17,24)(18,23)$
6F-1	$6^{2},2^{6}$	$36$	$6$	$16$	$( 1,16)( 2,15)( 3,21,19, 5,11,13)( 4,22,20, 6,12,14)( 7,10)( 8, 9)(17,24)(18,23)$
6G	$6^{2},2^{6}$	$108$	$6$	$16$	$( 1, 2)( 3,23,19, 7,11,15)( 4,24,20, 8,12,16)( 5,14)( 6,13)( 9,18)(10,17)(21,22)$
12A1	$12^{2}$	$54$	$12$	$22$	$( 1, 4,21,15,17,20,13, 7, 9,12, 5,23)( 2, 3,22,16,18,19,14, 8,10,11, 6,24)$
12A5	$12^{2}$	$54$	$12$	$22$	$( 1,20, 5,15, 9, 4,13,23,17,12,21, 7)( 2,19, 6,16,10, 3,14,24,18,11,22, 8)$

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

magma:ConjugacyClasses(G);

Character table

		1A	2A	2B	2C	3A	3B	3C	3D	3E	3F1	3F-1	3G	3H	3I	3J	3K	3L1	3L-1	4A	6A	6B	6C1	6C-1	6D	6E	6F1	6F-1	6G	12A1	12A5
	Size	1	9	18	54	2	2	4	4	4	4	4	8	8	8	8	8	8	8	54	18	18	18	18	36	36	36	36	108	54	54
	2 P	1A	1A	1A	1A	3A	3B	3C	3D	3E	3F-1	3F1	3G	3H	3I	3J	3K	3L-1	3L1	2A	3A	3B	3A	3A	3D	3E	3F1	3F-1	3C	6B	6B
	3 P	1A	2A	2B	2C	1A	1A	1A	1A	1A	1A	1A	1A	1A	1A	1A	1A	1A	1A	4A	2A	2A	2B	2B	2B	2A	2B	2B	2C	4A	4A
	Type
648.723.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
648.723.1b	R	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$1$	$- 1$	$1$	$- 1$	$1$	$- 1$	$- 1$	$- 1$	$1$	$1$
648.723.1c	R	$1$	$1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$1$	$- 1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$- 1$
648.723.1d	R	$1$	$1$	$1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$
648.723.2a	R	$2$	$2$	$0$	$0$	$2$	$- 1$	$2$	$2$	$- 1$	$2$	$2$	$- 1$	$2$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$2$	$0$	$2$	$0$	$- 1$	$0$	$- 1$	$0$	$0$	$0$	$- 1$	$- 1$
648.723.2b	R	$2$	$2$	$2$	$0$	$- 1$	$2$	$2$	$2$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$2$	$- 1$	$2$	$- 1$	$0$	$- 1$	$- 1$	$- 1$	$2$	$2$	$- 1$	$- 1$	$- 1$	$0$	$0$	$0$
648.723.2c	R	$2$	$- 2$	$0$	$0$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$0$	$0$	$- 2$	$0$	$- 2$	$0$	$- 2$	$0$	$0$	$0$	$0$	$0$
648.723.2d	R	$2$	$2$	$- 2$	$0$	$- 1$	$2$	$2$	$2$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$2$	$- 1$	$2$	$- 1$	$0$	$1$	$- 1$	$1$	$2$	$- 2$	$- 1$	$1$	$1$	$0$	$0$	$0$
648.723.2e	R	$2$	$2$	$0$	$0$	$2$	$- 1$	$2$	$2$	$- 1$	$2$	$2$	$- 1$	$2$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 2$	$0$	$2$	$0$	$- 1$	$0$	$- 1$	$0$	$0$	$0$	$1$	$1$
648.723.2f1	R	$2$	$- 2$	$0$	$0$	$2$	$- 1$	$2$	$2$	$- 1$	$2$	$2$	$- 1$	$2$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$0$	$0$	$- 2$	$0$	$1$	$0$	$1$	$0$	$0$	$0$	$- ζ_{12}^{- 1} - ζ_{12}$	$ζ_{12}^{- 1} + ζ_{12}$
648.723.2f2	R	$2$	$- 2$	$0$	$0$	$2$	$- 1$	$2$	$2$	$- 1$	$2$	$2$	$- 1$	$2$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$0$	$0$	$- 2$	$0$	$1$	$0$	$1$	$0$	$0$	$0$	$ζ_{12}^{- 1} + ζ_{12}$	$- ζ_{12}^{- 1} - ζ_{12}$
648.723.2g1	C	$2$	$- 2$	$0$	$0$	$- 1$	$2$	$2$	$2$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$2$	$- 1$	$2$	$- 1$	$0$	$- 1 - 2 ζ_{3}$	$1$	$1 + 2 ζ_{3}$	$- 2$	$0$	$1$	$1 + 2 ζ_{3}$	$- 1 - 2 ζ_{3}$	$0$	$0$	$0$
648.723.2g2	C	$2$	$- 2$	$0$	$0$	$- 1$	$2$	$2$	$2$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$2$	$- 1$	$2$	$- 1$	$0$	$1 + 2 ζ_{3}$	$1$	$- 1 - 2 ζ_{3}$	$- 2$	$0$	$1$	$- 1 - 2 ζ_{3}$	$1 + 2 ζ_{3}$	$0$	$0$	$0$
648.723.4a	R	$4$	$4$	$0$	$0$	$- 2$	$- 2$	$4$	$4$	$1$	$- 2$	$- 2$	$1$	$- 2$	$1$	$- 2$	$1$	$- 2$	$1$	$0$	$0$	$- 2$	$0$	$- 2$	$0$	$1$	$0$	$0$	$0$	$0$	$0$
648.723.4b	R	$4$	$0$	$0$	$2$	$4$	$4$	$1$	$- 2$	$4$	$- 2$	$- 2$	$1$	$1$	$- 2$	$1$	$- 2$	$- 2$	$1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 1$	$0$	$0$
648.723.4c	R	$4$	$0$	$2$	$0$	$4$	$4$	$- 2$	$1$	$4$	$1$	$1$	$- 2$	$- 2$	$1$	$- 2$	$1$	$1$	$- 2$	$0$	$2$	$0$	$2$	$0$	$- 1$	$0$	$- 1$	$- 1$	$0$	$0$	$0$
648.723.4d	R	$4$	$0$	$- 2$	$0$	$4$	$4$	$- 2$	$1$	$4$	$1$	$1$	$- 2$	$- 2$	$1$	$- 2$	$1$	$1$	$- 2$	$0$	$- 2$	$0$	$- 2$	$0$	$1$	$0$	$1$	$1$	$0$	$0$	$0$
648.723.4e	R	$4$	$0$	$0$	$- 2$	$4$	$4$	$1$	$- 2$	$4$	$- 2$	$- 2$	$1$	$1$	$- 2$	$1$	$- 2$	$- 2$	$1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$1$	$0$	$0$
648.723.4f	R	$4$	$- 4$	$0$	$0$	$- 2$	$- 2$	$4$	$4$	$1$	$- 2$	$- 2$	$1$	$- 2$	$1$	$- 2$	$1$	$- 2$	$1$	$0$	$0$	$2$	$0$	$2$	$0$	$- 1$	$0$	$0$	$0$	$0$	$0$
648.723.4g1	C	$4$	$0$	$2$	$0$	$- 2$	$4$	$- 2$	$1$	$- 2$	$- 2 - 3 ζ_{3}$	$1 + 3 ζ_{3}$	$1$	$1$	$1 + 3 ζ_{3}$	$- 2$	$- 2 - 3 ζ_{3}$	$1$	$1$	$0$	$2 ζ_{3}$	$0$	$2 ζ_{3}^{- 1}$	$0$	$- 1$	$0$	$- ζ_{3}^{- 1}$	$- ζ_{3}$	$0$	$0$	$0$
648.723.4g2	C	$4$	$0$	$2$	$0$	$- 2$	$4$	$- 2$	$1$	$- 2$	$1 + 3 ζ_{3}$	$- 2 - 3 ζ_{3}$	$1$	$1$	$- 2 - 3 ζ_{3}$	$- 2$	$1 + 3 ζ_{3}$	$1$	$1$	$0$	$2 ζ_{3}^{- 1}$	$0$	$2 ζ_{3}$	$0$	$- 1$	$0$	$- ζ_{3}$	$- ζ_{3}^{- 1}$	$0$	$0$	$0$
648.723.4h1	C	$4$	$0$	$- 2$	$0$	$- 2$	$4$	$- 2$	$1$	$- 2$	$- 2 - 3 ζ_{3}$	$1 + 3 ζ_{3}$	$1$	$1$	$1 + 3 ζ_{3}$	$- 2$	$- 2 - 3 ζ_{3}$	$1$	$1$	$0$	$- 2 ζ_{3}$	$0$	$- 2 ζ_{3}^{- 1}$	$0$	$1$	$0$	$ζ_{3}^{- 1}$	$ζ_{3}$	$0$	$0$	$0$
648.723.4h2	C	$4$	$0$	$- 2$	$0$	$- 2$	$4$	$- 2$	$1$	$- 2$	$1 + 3 ζ_{3}$	$- 2 - 3 ζ_{3}$	$1$	$1$	$- 2 - 3 ζ_{3}$	$- 2$	$1 + 3 ζ_{3}$	$1$	$1$	$0$	$- 2 ζ_{3}^{- 1}$	$0$	$- 2 ζ_{3}$	$0$	$1$	$0$	$ζ_{3}$	$ζ_{3}^{- 1}$	$0$	$0$	$0$
648.723.8a	R	$8$	$0$	$0$	$0$	$8$	$- 4$	$- 4$	$2$	$- 4$	$2$	$2$	$2$	$- 4$	$- 1$	$2$	$- 1$	$- 1$	$2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
648.723.8b	R	$8$	$0$	$0$	$0$	$8$	$- 4$	$2$	$- 4$	$- 4$	$- 4$	$- 4$	$- 1$	$2$	$2$	$- 1$	$2$	$2$	$- 1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
648.723.8c	R	$8$	$0$	$0$	$0$	$- 4$	$8$	$2$	$- 4$	$- 4$	$2$	$2$	$- 1$	$- 1$	$2$	$2$	$2$	$- 4$	$- 1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
648.723.8d	R	$8$	$0$	$0$	$0$	$- 4$	$- 4$	$2$	$- 4$	$2$	$2$	$2$	$- 4$	$- 1$	$- 1$	$- 1$	$- 1$	$2$	$5$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
648.723.8e	R	$8$	$0$	$0$	$0$	$- 4$	$- 4$	$2$	$- 4$	$2$	$2$	$2$	$5$	$- 1$	$- 1$	$- 1$	$- 1$	$2$	$- 4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
648.723.8f1	C	$8$	$0$	$0$	$0$	$- 4$	$- 4$	$- 4$	$2$	$2$	$- 4 - 6 ζ_{3}$	$2 + 6 ζ_{3}$	$- 1$	$2$	$- 1 - 3 ζ_{3}$	$2$	$2 + 3 ζ_{3}$	$- 1$	$- 1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
648.723.8f2	C	$8$	$0$	$0$	$0$	$- 4$	$- 4$	$- 4$	$2$	$2$	$2 + 6 ζ_{3}$	$- 4 - 6 ζ_{3}$	$- 1$	$2$	$2 + 3 ζ_{3}$	$2$	$- 1 - 3 ζ_{3}$	$- 1$	$- 1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$

magma:CharacterTable(G);

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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	24T1521

Degree	$24$
Order	$648$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	no
Group:	$C_3^3:D_{12}$