LMFDB - Galois group: 27T471

Show commands: Magma

magma:G := TransitiveGroup(27, 471);

Group invariants

Abstract group:	$C_3^4:S_3^2$	magma:IdentifyGroup(G);
Order:	$2916=2^{2} \cdot 3^{6}$	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 $n$:	$27$	magma:t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$471$	magma:t, n := TransitiveGroupIdentification(G); t;
Parity:	$-1$	magma:IsEven(G);
Primitive:	no	magma:IsPrimitive(G);
$\card{\Aut(F/K)}$:	$1$	magma:Order(Centralizer(SymmetricGroup(n), G));
Generators:	$(1,8,2,7,3,9)(4,5)(10,26)(11,25)(12,27)(13,23,15,24,14,22)(16,21,17,20,18,19)$, $(1,19,5,25,7,22)(2,21,6,27,8,24)(3,20,4,26,9,23)(10,16,11,18,12,17)(14,15)$, $(1,12,8,16,5,13)(2,11,9,18,6,15)(3,10,7,17,4,14)(19,24,26,20,23,27)(21,22,25)$	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 5
$12$: $D_{6}$ x 5
$18$: $C_3^2:C_2$
$36$: $S_3^2$ x 4, 18T12
$54$: $(C_3^2:C_3):C_2$
$108$: $C_3^2 : D_{6} $, 18T52, 18T58
$324$: 18T133, 18T135
$972$: 18T244

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 5
$12$:	$D_{6}$ x 5
$18$:	$C_3^2:C_2$
$36$:	$S_3^2$ x 4, 18T12
$54$:	$(C_3^2:C_3):C_2$
$108$:	$C_3^2 : D_{6} $, 18T52, 18T58
$324$:	18T133, 18T135
$972$:	18T244

Subfields

Degree 3: $S_3$
Degree 9: $C_3^2 : D_{6} $

Low degree siblings

27T466, 27T494
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	$1^{27}$	$1$	$1$	$0$	$()$
2A	$2^{13},1$	$27$	$2$	$13$	$( 1,27)( 2,26)( 3,25)( 4,22)( 5,24)( 6,23)( 7,21)( 8,20)( 9,19)(10,17)(11,16)(12,18)(13,15)$
2B	$2^{9},1^{9}$	$27$	$2$	$9$	$( 1, 9)( 2, 7)( 3, 8)(13,18)(14,16)(15,17)(19,22)(20,23)(21,24)$
2C	$2^{12},1^{3}$	$81$	$2$	$12$	$( 1, 2)( 4, 5)( 7, 9)(10,24)(11,23)(12,22)(13,26)(14,25)(15,27)(16,20)(17,19)(18,21)$
3A	$3^{9}$	$2$	$3$	$18$	$( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)(22,24,23)(25,27,26)$
3B1	$3^{6},1^{9}$	$3$	$3$	$12$	$( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(19,20,21)(22,23,24)(25,26,27)$
3B-1	$3^{6},1^{9}$	$3$	$3$	$12$	$( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(19,21,20)(22,24,23)(25,27,26)$
3C	$3^{9}$	$6$	$3$	$18$	$( 1, 4, 9)( 2, 5, 7)( 3, 6, 8)(10,13,18)(11,14,16)(12,15,17)(19,22,27)(20,23,25)(21,24,26)$
3D1	$3^{9}$	$6$	$3$	$18$	$( 1, 9, 4)( 2, 7, 5)( 3, 8, 6)(10,16,15)(11,17,13)(12,18,14)(19,26,23)(20,27,24)(21,25,22)$
3D-1	$3^{9}$	$6$	$3$	$18$	$( 1, 4, 9)( 2, 5, 7)( 3, 6, 8)(10,15,16)(11,13,17)(12,14,18)(19,23,26)(20,24,27)(21,22,25)$
3E	$3^{6},1^{9}$	$18$	$3$	$12$	$(10,17,14)(11,18,15)(12,16,13)(19,23,26)(20,24,27)(21,22,25)$
3F	$3^{9}$	$18$	$3$	$18$	$( 1,19,11)( 2,20,12)( 3,21,10)( 4,22,14)( 5,23,15)( 6,24,13)( 7,25,17)( 8,26,18)( 9,27,16)$
3G	$3^{6},1^{9}$	$18$	$3$	$12$	$( 1, 2, 3)( 4, 6, 5)(10,11,12)(13,15,14)(19,20,21)(22,24,23)$
3H1	$3^{9}$	$18$	$3$	$18$	$( 1,13,27)( 2,14,25)( 3,15,26)( 4,18,19)( 5,16,20)( 6,17,21)( 7,11,23)( 8,12,24)( 9,10,22)$
3H-1	$3^{9}$	$18$	$3$	$18$	$( 1,27,13)( 2,25,14)( 3,26,15)( 4,19,18)( 5,20,16)( 6,21,17)( 7,23,11)( 8,24,12)( 9,22,10)$
3I	$3^{9}$	$36$	$3$	$18$	$( 1,26,15)( 2,27,13)( 3,25,14)( 4,19,16)( 5,20,17)( 6,21,18)( 7,24,11)( 8,22,12)( 9,23,10)$
3J	$3^{9}$	$36$	$3$	$18$	$( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,17,14)(11,18,15)(12,16,13)(19,24,25)(20,22,26)(21,23,27)$
3K	$3^{9}$	$36$	$3$	$18$	$( 1, 8, 4)( 2, 9, 5)( 3, 7, 6)(10,17,13)(11,18,14)(12,16,15)(19,26,22)(20,27,23)(21,25,24)$
3L1	$3^{9}$	$36$	$3$	$18$	$( 1,21,10)( 2,19,11)( 3,20,12)( 4,22,15)( 5,23,13)( 6,24,14)( 7,26,17)( 8,27,18)( 9,25,16)$
3L-1	$3^{9}$	$36$	$3$	$18$	$( 1,24,16)( 2,22,17)( 3,23,18)( 4,27,10)( 5,25,11)( 6,26,12)( 7,21,13)( 8,19,14)( 9,20,15)$
3M	$3^{8},1^{3}$	$54$	$3$	$16$	$( 1, 2, 3)( 4, 6, 5)(10,17,13)(11,18,14)(12,16,15)(19,22,25)(20,23,26)(21,24,27)$
3N	$3^{8},1^{3}$	$54$	$3$	$16$	$( 1, 4, 7)( 2, 5, 8)( 3, 6, 9)(10,18,15)(11,16,13)(12,17,14)(22,23,24)(25,27,26)$
3O	$3^{9}$	$108$	$3$	$18$	$( 1,11,27)( 2,12,25)( 3,10,26)( 4,14,19)( 5,15,20)( 6,13,21)( 7,17,23)( 8,18,24)( 9,16,22)$
6A1	$6^{3},2^{4},1$	$27$	$6$	$19$	$( 1,26, 3,27, 2,25)( 4,24, 6,22, 5,23)( 7,20, 9,21, 8,19)(10,17)(11,16)(12,18)(13,15)$
6A-1	$6^{3},2^{4},1$	$27$	$6$	$19$	$( 1, 7)( 2, 9)( 3, 8)( 4, 5)(10,25,12,26,11,27)(13,23,15,24,14,22)(16,19,18,20,17,21)$
6B	$6^{3},3^{3}$	$54$	$6$	$21$	$( 1, 7, 3, 9, 2, 8)( 4, 5, 6)(10,11,12)(13,16,15,18,14,17)(19,23,21,22,20,24)(25,26,27)$
6C1	$6^{3},2^{3},1^{3}$	$81$	$6$	$18$	$( 1, 2)( 4, 5)( 7, 9)(10,23,12,24,11,22)(13,25,15,26,14,27)(16,19,18,20,17,21)$
6C-1	$6^{3},2^{3},1^{3}$	$81$	$6$	$18$	$( 1, 2)( 4, 5)( 7, 9)(10,22,11,24,12,23)(13,27,14,26,15,25)(16,21,17,20,18,19)$
6D1	$6^{2},3^{2},2^{3},1^{3}$	$81$	$6$	$17$	$( 4, 8)( 5, 9)( 6, 7)(10,14,11,15,12,13)(16,18,17)(19,26,21,25,20,27)(22,23,24)$
6D-1	$6^{2},3^{2},2^{3},1^{3}$	$81$	$6$	$17$	$( 4, 8)( 5, 9)( 6, 7)(10,13,12,15,11,14)(16,17,18)(19,27,20,25,21,26)(22,24,23)$
6E	$6^{3},2^{4},1$	$162$	$6$	$19$	$( 1, 4)( 2, 6)( 3, 5)( 7, 8)(10,19,17,23,14,26)(11,21,18,22,15,25)(12,20,16,24,13,27)$
6F	$6^{3},3^{3}$	$162$	$6$	$21$	$( 1,17,19, 7,11,25)( 2,18,20, 8,12,26)( 3,16,21, 9,10,27)( 4,14,22)( 5,15,23)( 6,13,24)$
6G	$6^{3},2^{4},1$	$162$	$6$	$19$	$( 1,14, 2,13, 3,15)( 4,12, 6,10, 5,11)( 7,17)( 8,16)( 9,18)(19,22,20,24,21,23)(26,27)$
6H	$6^{4},2,1$	$162$	$6$	$21$	$( 1, 6, 2, 5, 3, 4)( 7, 9)(10,21,17,24,13,27)(11,20,18,23,14,26)(12,19,16,22,15,25)$
6I	$6^{4},2,1$	$162$	$6$	$21$	$( 1,15, 4,10, 7,18)( 2,14, 5,12, 8,17)( 3,13, 6,11, 9,16)(19,21)(22,27,23,26,24,25)$
6J	$6^{4},3$	$162$	$6$	$22$	$( 1,20, 4,23, 9,25)( 2,19, 5,22, 7,27)( 3,21, 6,24, 8,26)(10,18,13)(11,17,14,12,16,15)$
6K1	$6^{3},3^{3}$	$162$	$6$	$21$	$( 1,23,13, 7,27,11)( 2,24,14, 8,25,12)( 3,22,15, 9,26,10)( 4,19,18)( 5,20,16)( 6,21,17)$
6K-1	$6^{3},3^{3}$	$162$	$6$	$21$	$( 1,11,27, 7,13,23)( 2,12,25, 8,14,24)( 3,10,26, 9,15,22)( 4,18,19)( 5,16,20)( 6,17,21)$
6L1	$6^{4},3$	$162$	$6$	$22$	$( 1,22, 9,21, 4,25)( 2,24, 7,20, 5,27)( 3,23, 8,19, 6,26)(10,13,16,11,15,17)(12,14,18)$
6L-1	$6^{4},3$	$162$	$6$	$22$	$( 1,25, 4,21, 9,22)( 2,27, 5,20, 7,24)( 3,26, 6,19, 8,23)(10,17,15,11,16,13)(12,18,14)$
9A	$9^{3}$	$108$	$9$	$24$	$( 1,12,24, 3,11,23, 2,10,22)( 4,14,27, 6,13,26, 5,15,25)( 7,16,21, 9,18,20, 8,17,19)$
9B	$9^{3}$	$108$	$9$	$24$	$( 1,26,10, 3,25,12, 2,27,11)( 4,19,15, 6,21,14, 5,20,13)( 7,24,17, 9,23,16, 8,22,18)$

Malle's constant $a(G)$: $1/9$

magma:ConjugacyClasses(G);

Character table

42 x 42 character table

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	27T471

Degree	$27$
Order	$2916$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	no
Group:	$C_3^4:S_3^2$