LMFDB - Galois group: 27T494

Show commands: Magma

magma:G := TransitiveGroup(27, 494);

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 $t$:	$494$	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:	$(1,6,18,13,23,7)(2,5,16,15,24,9)(3,4,17,14,22,8)(10,19,27)(11,21,25,12,20,26)$, $(1,8,10,18,14,27)(2,9,11,16,15,25)(3,7,12,17,13,26)(4,21,24)(5,19,22)(6,20,23)$, $(1,19,9,3,21,8,2,20,7)(4,17,10,6,16,12,5,18,11)(13,23,27,15,22,26,14,24,25)$	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: $S_3$ x 2
Degree 9: $S_3^2$

Low degree siblings

27T466, 27T471
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	$2^{12},1^{3}$	$27$	$2$	$12$	$( 1,15)( 2,14)( 3,13)( 4,22)( 5,24)( 6,23)( 7,16)( 8,18)( 9,17)(10,12)(20,21)(25,27)$
2B	$2^{9},1^{9}$	$27$	$2$	$9$	$( 4, 8)( 5, 9)( 6, 7)(16,24)(17,22)(18,23)(19,27)(20,25)(21,26)$
2C	$2^{13},1$	$81$	$2$	$13$	$( 1, 3)( 4,25)( 5,27)( 6,26)( 7,19)( 8,21)( 9,20)(10,15)(11,14)(12,13)(16,23)(17,22)(18,24)$
3A	$3^{9}$	$2$	$3$	$18$	$( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27)$
3B1	$3^{6},1^{9}$	$3$	$3$	$12$	$( 1, 2, 3)( 4, 6, 5)( 7, 9, 8)(13,15,14)(16,17,18)(22,23,24)$
3B-1	$3^{6},1^{9}$	$3$	$3$	$12$	$( 1, 3, 2)( 4, 5, 6)( 7, 8, 9)(13,14,15)(16,18,17)(22,24,23)$
3C	$3^{6},1^{9}$	$6$	$3$	$12$	$( 4, 6, 5)( 7, 8, 9)(16,17,18)(19,21,20)(22,24,23)(25,26,27)$
3D1	$3^{6},1^{9}$	$6$	$3$	$12$	$( 4, 6, 5)(10,12,11)(13,14,15)(16,18,17)(22,23,24)(25,26,27)$
3D-1	$3^{6},1^{9}$	$6$	$3$	$12$	$( 4, 5, 6)(10,11,12)(13,15,14)(16,17,18)(22,24,23)(25,27,26)$
3E	$3^{4},1^{15}$	$18$	$3$	$8$	$( 4, 5, 6)(10,11,12)(13,15,14)(19,21,20)$
3F	$3^{9}$	$18$	$3$	$18$	$( 1,15,11)( 2,13,12)( 3,14,10)( 4,21,23)( 5,19,24)( 6,20,22)( 7,26,16)( 8,27,17)( 9,25,18)$
3G	$3^{9}$	$18$	$3$	$18$	$( 1,17,23)( 2,18,24)( 3,16,22)( 4,15, 8)( 5,13, 9)( 6,14, 7)(10,26,20)(11,27,21)(12,25,19)$
3H1	$3^{9}$	$18$	$3$	$18$	$( 1,12,15)( 2,10,13)( 3,11,14)( 4,22,20)( 5,23,21)( 6,24,19)( 7,17,26)( 8,18,27)( 9,16,25)$
3H-1	$3^{9}$	$18$	$3$	$18$	$( 1,15,12)( 2,13,10)( 3,14,11)( 4,20,22)( 5,21,23)( 6,19,24)( 7,26,17)( 8,27,18)( 9,25,16)$
3I	$3^{9}$	$36$	$3$	$18$	$( 1, 9,21)( 2, 7,19)( 3, 8,20)( 4,12,17)( 5,10,18)( 6,11,16)(13,25,22)(14,26,23)(15,27,24)$
3J	$3^{7},1^{6}$	$36$	$3$	$14$	$( 1, 2, 3)( 7, 9, 8)(10,11,12)(13,14,15)(16,17,18)(19,21,20)(22,23,24)$
3K	$3^{9}$	$36$	$3$	$18$	$( 1,18,24)( 2,16,22)( 3,17,23)( 4,14, 8)( 5,15, 9)( 6,13, 7)(10,27,21)(11,25,19)(12,26,20)$
3L1	$3^{9}$	$36$	$3$	$18$	$( 1, 7,21)( 2, 8,19)( 3, 9,20)( 4,12,18)( 5,10,16)( 6,11,17)(13,26,22)(14,27,23)(15,25,24)$
3L-1	$3^{9}$	$36$	$3$	$18$	$( 1, 8,21)( 2, 9,19)( 3, 7,20)( 4,12,16)( 5,10,17)( 6,11,18)(13,27,22)(14,25,23)(15,26,24)$
3M	$3^{9}$	$54$	$3$	$18$	$( 1,17,23)( 2,18,24)( 3,16,22)( 4,13, 8)( 5,14, 9)( 6,15, 7)(10,27,21)(11,25,19)(12,26,20)$
3N	$3^{9}$	$54$	$3$	$18$	$( 1,16,22)( 2,17,23)( 3,18,24)( 4,13, 7)( 5,14, 8)( 6,15, 9)(10,26,19)(11,27,20)(12,25,21)$
3O	$3^{9}$	$108$	$3$	$18$	$( 1,11,15)( 2,12,13)( 3,10,14)( 4,22,20)( 5,23,21)( 6,24,19)( 7,17,27)( 8,18,25)( 9,16,26)$
6A1	$6^{3},2^{3},1^{3}$	$27$	$6$	$18$	$( 1,13, 2,15, 3,14)( 4,24, 6,22, 5,23)( 7,18, 9,16, 8,17)(10,12)(20,21)(25,27)$
6A-1	$6^{3},2^{3},1^{3}$	$27$	$6$	$18$	$( 1, 3)( 4,21, 6,19, 5,20)( 7,27, 9,25, 8,26)(10,13,11,15,12,14)(16,18)(23,24)$
6B	$6^{3},3^{3}$	$54$	$6$	$21$	$( 1, 3, 2)( 4, 7, 5, 8, 6, 9)(10,12,11)(13,15,14)(16,23,17,24,18,22)(19,26,20,27,21,25)$
6C1	$6^{3},2^{4},1$	$81$	$6$	$19$	$( 1, 3)( 4,27, 6,25, 5,26)( 7,21, 9,19, 8,20)(10,13,11,15,12,14)(16,23)(17,22)(18,24)$
6C-1	$6^{3},2^{4},1$	$81$	$6$	$19$	$( 1, 3)( 4,26, 5,25, 6,27)( 7,20, 8,19, 9,21)(10,14,12,15,11,13)(16,23)(17,22)(18,24)$
6D1	$6^{2},3^{2},2^{3},1^{3}$	$81$	$6$	$17$	$( 1,18)( 2,16)( 3,17)( 4, 6, 5)( 7,14, 8,15, 9,13)(10,27,12,26,11,25)(19,20,21)$
6D-1	$6^{2},3^{2},2^{3},1^{3}$	$81$	$6$	$17$	$( 1,18)( 2,16)( 3,17)( 4, 5, 6)( 7,13, 9,15, 8,14)(10,25,11,26,12,27)(19,21,20)$
6E	$6^{2},2^{6},1^{3}$	$162$	$6$	$16$	$( 2, 3)( 4,19, 5,21, 6,20)( 7,25)( 8,27)( 9,26)(10,14,11,13,12,15)(16,18)(22,24)$
6F	$6^{3},3^{3}$	$162$	$6$	$21$	$( 1,26,15,16,11, 7)( 2,27,13,17,12, 8)( 3,25,14,18,10, 9)( 4,23,21)( 5,24,19)( 6,22,20)$
6G	$6^{4},3$	$162$	$6$	$22$	$( 1,19,17,12,23,25)( 2,21,18,11,24,27)( 3,20,16,10,22,26)( 4, 9,15, 5, 8,13)( 6, 7,14)$
6H	$6^{4},3$	$162$	$6$	$22$	$( 1,22,17, 3,23,16)( 2,24,18)( 4,26,13,20, 8,12)( 5,25,14,19, 9,11)( 6,27,15,21, 7,10)$
6I	$6^{4},3$	$162$	$6$	$22$	$( 1,21,16,12,22,25)( 2,20,17,11,23,27)( 3,19,18,10,24,26)( 4, 9,13, 6, 7,15)( 5, 8,14)$
6J	$6^{3},2^{4},1$	$162$	$6$	$19$	$( 1,14)( 2,13)( 3,15)( 4,17, 6,18, 5,16)( 7,22, 8,24, 9,23)(10,11)(19,27,21,25,20,26)$
6K1	$6^{3},3^{3}$	$162$	$6$	$21$	$( 1, 9,12,16,15,25)( 2, 7,10,17,13,26)( 3, 8,11,18,14,27)( 4,20,22)( 5,21,23)( 6,19,24)$
6K-1	$6^{3},3^{3}$	$162$	$6$	$21$	$( 1,25,15,16,12, 9)( 2,26,13,17,10, 7)( 3,27,14,18,11, 8)( 4,22,20)( 5,23,21)( 6,24,19)$
6L1	$6^{3},2^{4},1$	$162$	$6$	$19$	$( 1, 7)( 2, 9)( 3, 8)( 4,24, 6,22, 5,23)(10,26,12,27,11,25)(13,16,14,18,15,17)(20,21)$
6L-1	$6^{3},2^{4},1$	$162$	$6$	$19$	$( 1, 7)( 2, 9)( 3, 8)( 4,23, 5,22, 6,24)(10,25,11,27,12,26)(13,17,15,18,14,16)(20,21)$
9A	$9^{3}$	$108$	$9$	$24$	$( 1,19, 8, 2,20, 9, 3,21, 7)( 4,17,12, 5,18,10, 6,16,11)(13,23,25,14,24,26,15,22,27)$
9B	$9^{3}$	$108$	$9$	$24$	$( 1, 9,21, 2, 7,19, 3, 8,20)( 4,11,18, 5,12,16, 6,10,17)(13,26,23,14,27,24,15,25,22)$

Malle's constant $a(G)$: $1/8$

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	27T494

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