↑

Properties

Label 27T195
27T195 1 2 1->2 17 1->17 3 2->3 2->3 18 2->18 3->1 16 3->16 4 5 4->5 21 4->21 6 5->6 5->6 19 5->19 6->4 20 6->20 7 7->3 8 7->8 22 7->22 8->1 9 8->9 24 8->24 9->2 9->7 23 9->23 10 10->4 11 10->11 10->17 11->5 12 11->12 11->16 12->6 12->10 12->18 13 14 13->14 13->21 26 13->26 15 14->15 14->20 27 14->27 15->13 15->19 25 15->25 16->7 16->14 16->17 17->8 17->13 17->18 18->9 18->15 18->16 19->7 19->11 19->20 20->9 20->12 20->21 21->8 21->10 21->19 22->12 22->14 22->23 23->11 23->15 23->24 24->10 24->13 24->22 25->23 25->26 25->27 26->24 26->27 27->22 27->25
Degree $27$
Order $486$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^4:S_3$

Related objects

Downloads

Learn more

Show commands: Magma

Copy content magma:G := TransitiveGroup(27, 195);
 

Group invariants

Abstract group:  $C_3^4:S_3$
Copy content magma:IdentifyGroup(G);
 
Order:  $486=2 \cdot 3^{5}$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  yes
Copy content magma:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $27$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $195$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $3$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27)$, $(2,3)(5,6)(7,22,12,18,15,19)(8,24,10,17,13,21)(9,23,11,16,14,20)(25,27)$, $(1,17,8)(2,18,9)(3,16,7)(4,21,10)(5,19,11)(6,20,12)(13,26,24)(14,27,22)(15,25,23)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $S_3$ x 4, $C_6$
$18$:  $S_3\times C_3$ x 4, $C_3^2:C_2$
$54$:  $C_3^2 : C_6$ x 3, 18T23
$162$:  $C_3 \wr S_3 $, 27T61

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$ x 4

Degree 9: $C_3^2:C_2$, $C_3 \wr S_3 $

Low degree siblings

18T167 x 3, 27T194 x 6, 27T195 x 2

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{27}$ $1$ $1$ $0$ $()$
2A $2^{12},1^{3}$ $27$ $2$ $12$ $( 1, 8)( 2, 7)( 3, 9)( 4,10)( 5,12)( 6,11)(13,26)(14,25)(15,27)(16,18)(19,20)(22,23)$
3A1 $3^{9}$ $1$ $3$ $18$ $( 1, 4,26)( 2, 5,27)( 3, 6,25)( 7,12,15)( 8,10,13)( 9,11,14)(16,20,23)(17,21,24)(18,19,22)$
3A-1 $3^{9}$ $1$ $3$ $18$ $( 1,26, 4)( 2,27, 5)( 3,25, 6)( 7,15,12)( 8,13,10)( 9,14,11)(16,23,20)(17,24,21)(18,22,19)$
3B $3^{9}$ $2$ $3$ $18$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27)$
3C1 $3^{9}$ $2$ $3$ $18$ $( 1,27, 6)( 2,25, 4)( 3,26, 5)( 7,13,11)( 8,14,12)( 9,15,10)(16,24,19)(17,22,20)(18,23,21)$
3C-1 $3^{9}$ $2$ $3$ $18$ $( 1, 6,27)( 2, 4,25)( 3, 5,26)( 7,11,13)( 8,12,14)( 9,10,15)(16,19,24)(17,20,22)(18,21,23)$
3D1 $3^{3},1^{18}$ $3$ $3$ $6$ $( 1, 4,26)( 2, 5,27)( 3, 6,25)$
3D-1 $3^{3},1^{18}$ $3$ $3$ $6$ $( 1,26, 4)( 2,27, 5)( 3,25, 6)$
3E1 $3^{9}$ $3$ $3$ $18$ $( 1,26, 4)( 2,27, 5)( 3,25, 6)( 7,12,15)( 8,10,13)( 9,11,14)(16,23,20)(17,24,21)(18,22,19)$
3E-1 $3^{9}$ $3$ $3$ $18$ $( 1, 4,26)( 2, 5,27)( 3, 6,25)( 7,15,12)( 8,13,10)( 9,14,11)(16,20,23)(17,21,24)(18,19,22)$
3F1 $3^{6},1^{9}$ $3$ $3$ $12$ $( 1, 4,26)( 2, 5,27)( 3, 6,25)( 7,12,15)( 8,10,13)( 9,11,14)$
3F-1 $3^{6},1^{9}$ $3$ $3$ $12$ $( 1,26, 4)( 2,27, 5)( 3,25, 6)( 7,15,12)( 8,13,10)( 9,14,11)$
3G $3^{9}$ $6$ $3$ $18$ $( 1,25, 5)( 2,26, 6)( 3,27, 4)( 7,11,13)( 8,12,14)( 9,10,15)(16,18,17)(19,21,20)(22,24,23)$
3H $3^{9}$ $6$ $3$ $18$ $( 1, 3, 2)( 4, 6, 5)( 7,11,13)( 8,12,14)( 9,10,15)(16,22,21)(17,23,19)(18,24,20)(25,27,26)$
3I $3^{6},1^{9}$ $6$ $3$ $12$ $( 1,26, 4)( 2,27, 5)( 3,25, 6)(16,20,23)(17,21,24)(18,19,22)$
3J1 $3^{9}$ $6$ $3$ $18$ $( 1,27, 6)( 2,25, 4)( 3,26, 5)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)$
3J-1 $3^{9}$ $6$ $3$ $18$ $( 1, 6,27)( 2, 4,25)( 3, 5,26)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)(22,24,23)$
3K1 $3^{9}$ $6$ $3$ $18$ $( 1,25, 5)( 2,26, 6)( 3,27, 4)( 7,14,10)( 8,15,11)( 9,13,12)(16,19,24)(17,20,22)(18,21,23)$
3K-1 $3^{9}$ $6$ $3$ $18$ $( 1, 5,25)( 2, 6,26)( 3, 4,27)( 7,10,14)( 8,11,15)( 9,12,13)(16,24,19)(17,22,20)(18,23,21)$
3L1 $3^{9}$ $6$ $3$ $18$ $( 1,25, 5)( 2,26, 6)( 3,27, 4)( 7, 9, 8)(10,12,11)(13,15,14)(16,22,21)(17,23,19)(18,24,20)$
3L-1 $3^{9}$ $6$ $3$ $18$ $( 1, 5,25)( 2, 6,26)( 3, 4,27)( 7, 8, 9)(10,11,12)(13,14,15)(16,21,22)(17,19,23)(18,20,24)$
3M $3^{9}$ $18$ $3$ $18$ $( 1,16,14)( 2,17,15)( 3,18,13)( 4,20, 9)( 5,21, 7)( 6,19, 8)(10,25,22)(11,26,23)(12,27,24)$
3N $3^{9}$ $18$ $3$ $18$ $( 1,18, 7)( 2,16, 8)( 3,17, 9)( 4,19,12)( 5,20,10)( 6,21,11)(13,27,23)(14,25,24)(15,26,22)$
3O $3^{9}$ $18$ $3$ $18$ $( 1,13,21)( 2,14,19)( 3,15,20)( 4, 8,24)( 5, 9,22)( 6, 7,23)(10,17,26)(11,18,27)(12,16,25)$
6A1 $6^{4},3$ $27$ $6$ $22$ $( 1,10,26, 8, 4,13)( 2,12,27, 7, 5,15)( 3,11,25, 9, 6,14)(16,19,23,18,20,22)(17,21,24)$
6A-1 $6^{4},3$ $27$ $6$ $22$ $( 1,13, 4, 8,26,10)( 2,15, 5, 7,27,12)( 3,14, 6, 9,25,11)(16,22,20,18,23,19)(17,24,21)$
6B1 $6,3,2^{9}$ $27$ $6$ $16$ $( 1,26, 4)( 2,25, 5, 3,27, 6)( 7,19)( 8,21)( 9,20)(10,24)(11,23)(12,22)(13,17)(14,16)(15,18)$
6B-1 $6,3,2^{9}$ $27$ $6$ $16$ $( 1, 4,26)( 2, 6,27, 3, 5,25)( 7,19)( 8,21)( 9,20)(10,24)(11,23)(12,22)(13,17)(14,16)(15,18)$
6C1 $6^{4},3$ $27$ $6$ $22$ $( 1,20,26,16, 4,23)( 2,19,27,18, 5,22)( 3,21,25,17, 6,24)( 7,13,12, 8,15,10)( 9,14,11)$
6C-1 $6^{4},3$ $27$ $6$ $22$ $( 1,23, 4,16,26,20)( 2,22, 5,18,27,19)( 3,24, 6,17,25,21)( 7,10,15, 8,12,13)( 9,11,14)$
6D1 $6^{3},2^{3},1^{3}$ $27$ $6$ $18$ $( 1,15, 4, 7,26,12)( 2,14, 5, 9,27,11)( 3,13, 6, 8,25,10)(16,17)(20,21)(23,24)$
6D-1 $6^{3},2^{3},1^{3}$ $27$ $6$ $18$ $( 1,12,26, 7, 4,15)( 2,11,27, 9, 5,14)( 3,10,25, 8, 6,13)(16,17)(20,21)(23,24)$
9A1 $9^{3}$ $18$ $9$ $24$ $( 1,18,15, 4,19, 7,26,22,12)( 2,16,13, 5,20, 8,27,23,10)( 3,17,14, 6,21, 9,25,24,11)$
9A-1 $9^{3}$ $18$ $9$ $24$ $( 1,18,12,26,22, 7, 4,19,15)( 2,16,10,27,23, 8, 5,20,13)( 3,17,11,25,24, 9, 6,21,14)$
9B1 $9^{3}$ $18$ $9$ $24$ $( 1,16, 9,26,23,14, 4,20,11)( 2,17, 7,27,24,15, 5,21,12)( 3,18, 8,25,22,13, 6,19,10)$
9B-1 $9^{3}$ $18$ $9$ $24$ $( 1, 9,23, 4,11,16,26,14,20)( 2, 7,24, 5,12,17,27,15,21)( 3, 8,22, 6,10,18,25,13,19)$
9C1 $9^{3}$ $18$ $9$ $24$ $( 1,13,17, 4, 8,21,26,10,24)( 2,14,18, 5, 9,19,27,11,22)( 3,15,16, 6, 7,20,25,12,23)$
9C-1 $9^{3}$ $18$ $9$ $24$ $( 1,10,21,26, 8,17, 4,13,24)( 2,11,19,27, 9,18, 5,14,22)( 3,12,20,25, 7,16, 6,15,23)$

Malle's constant $a(G)$:     $1/6$

Copy content magma:ConjugacyClasses(G);
 

Character table

39 x 39 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed