↑

Properties

Label 27T1161
27T1161 1 10 1->10 18 1->18 2 4 2->4 24 2->24 3 17 3->17 20 3->20 21 4->21 23 4->23 5 14 5->14 26 5->26 6 6->2 25 6->25 7 7->3 7->18 8 8->4 12 8->12 9 15 9->15 9->20 11 10->11 13 10->13 11->1 11->3 12->7 12->17 13->1 22 13->22 14->7 27 14->27 15->2 15->21 16 16->9 19 16->19 17->11 17->24 18->12 18->13 19->8 19->14 20->16 20->26 21->22 21->27 22->10 22->15 23->6 23->8 24->6 24->16 25->5 25->23 26->5 26->25 27->9 27->19
Degree $27$
Order $51840$
Cyclic no
Abelian no
Solvable no
Primitive yes
$p$-group no
Group: $\SO(5,3)$

Related objects

Downloads

Learn more

Show commands: Magma

Copy content magma:G := TransitiveGroup(27, 1161);
 

Group invariants

Abstract group:  $\SO(5,3)$
Copy content magma:IdentifyGroup(G);
 
Order:  $51840=2^{7} \cdot 3^{4} \cdot 5$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  no
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 $t$:  $1161$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  yes
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,10,13)(2,24,6)(3,17,11)(4,23,8)(5,26,25)(7,18,12)(9,20,16)(14,27,19)(15,21,22)$, $(1,18,13,22,10,11)(2,4,21,27,9,15)(3,20,26,5,14,7)(6,25,23)(8,12,17,24,16,19)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 9: None

Low degree siblings

36T16210, 40T18793, 40T18794, 45T838

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
$1^{27}$ $1$ $1$ $0$ $()$
$2^{6},1^{15}$ $36$ $2$ $6$ $( 1,15)( 2,12)( 4,11)( 5,13)(16,26)(20,22)$
$3^{9}$ $480$ $3$ $18$ $( 1,26, 5)( 2, 4,22)( 3,24,27)( 6,25,23)( 7, 9, 8)(10,19,21)(11,20,12)(13,15,16)(14,18,17)$
$6^{2},3^{5}$ $1440$ $6$ $20$ $( 1,13,26,15, 5,16)( 2,20, 4,12,22,11)( 3,27,24)( 6,23,25)( 7, 8, 9)(10,21,19)(14,17,18)$
$2^{10},1^{7}$ $270$ $2$ $10$ $( 2, 7)( 3, 4)( 5, 9)(10,22)(11,23)(12,24)(13,25)(14,27)(15,26)(20,21)$
$3^{6},1^{9}$ $240$ $3$ $12$ $( 1, 8,17)( 3, 5,20)( 4, 9,21)( 6,16,18)(10,25,12)(13,24,22)$
$6^{2},3^{2},2^{4},1$ $2160$ $6$ $18$ $( 1,17, 8)( 2, 7)( 3,21, 5, 4,20, 9)( 6,18,16)(10,24,25,22,12,13)(11,23)(14,27)(15,26)$
$2^{12},1^{3}$ $540$ $2$ $12$ $( 1,22)( 2,26)( 3, 9)( 4, 5)( 7,27)( 8,24)(10,12)(11,19)(13,17)(14,15)(16,18)(20,21)$
$5^{5},1^{2}$ $5184$ $5$ $20$ $( 1,11,20,12,26)( 2,16,15, 4,22)( 3,24,25,14, 9)( 6,27, 8,21,10)( 7,18,17,23,19)$
$10,5^{3},2$ $5184$ $10$ $22$ $( 1, 2,11,16,20,15,12, 4,26,22)( 3,14,24, 9,25)( 5,13)( 6,21,27,10, 8)( 7,23,18,19,17)$
$4^{5},2^{3},1$ $540$ $4$ $18$ $( 1,10,15, 7)( 2,14,12, 8)( 3, 9)( 4,13, 5,11)(16,24,26,21)(17,19)(18,20,27,22)(23,25)$
$12,6,4^{2},1$ $4320$ $12$ $22$ $( 1,14,22,10,12,18,15, 8,20, 7, 2,27)( 3,19,25, 9,17,23)( 4,11, 5,13)(16,21,26,24)$
$2^{12},1^{3}$ $45$ $2$ $12$ $( 1, 4)( 2,26)( 3,24)( 5,22)( 6,25)( 8, 9)(10,18)(11,15)(12,16)(13,20)(14,19)(17,21)$
$3^{9}$ $80$ $3$ $18$ $( 1,13,10)( 2,11,14)( 3, 6, 9)( 4,20,18)( 5,16,21)( 7,27,23)( 8,24,25)(12,17,22)(15,19,26)$
$6^{4},3$ $720$ $6$ $22$ $( 1,18,13, 4,10,20)( 2,19,11,26,14,15)( 3, 8, 6,24, 9,25)( 5,17,16,22,21,12)( 7,23,27)$
$4^{6},1^{3}$ $540$ $4$ $18$ $( 1, 9, 4, 8)( 2,16,26,12)( 3,20,24,13)( 5,19,22,14)( 6,18,25,10)(11,21,15,17)$
$12^{2},3$ $4320$ $12$ $24$ $( 1,19, 6, 9,22,18, 4,14,25, 8, 5,10)( 2,15,13,16,17, 3,26,11,20,12,21,24)( 7,27,23)$
$6^{4},3$ $4320$ $6$ $22$ $( 1, 2,20,13,26, 4)( 3,17,27,21,24, 7)( 5,25,11, 9,12,19)( 6,22,14,16, 8,15)(10,23,18)$
$4^{5},2,1^{5}$ $1620$ $4$ $16$ $( 1,27, 8,26)( 2,22, 7,24)( 3, 5, 9, 4)(10,20,12,21)(14,18,15,16)(23,25)$
$6^{4},3$ $1440$ $6$ $22$ $( 1, 2,26, 4, 5,22)( 3,23,24, 6,27,25)( 7, 8, 9)(10,17,19,14,21,18)(11,13,20,15,12,16)$
$6^{3},2^{3},1^{3}$ $1440$ $6$ $18$ $( 1,17,15, 4,21,11)( 2,19,12, 5,18,13)( 3, 6)(10,20,26,14,16,22)(23,27)(24,25)$
$4^{5},2^{3},1$ $3240$ $4$ $18$ $( 1,24)( 2,27)( 4,10,16,25)( 5,23,20,14)( 6,18, 9,21)( 7,17,19, 8)(11,22,26,13)(12,15)$
$8^{3},2,1$ $6480$ $8$ $22$ $( 1,18,17,25, 4,10,21, 6)( 2, 8,12,15,26, 9,16,11)( 3,19,22,20,24,14, 5,13)( 7,23)$
$9^{3}$ $5760$ $9$ $24$ $( 1, 6, 2,19, 5,26,16, 4,23)( 3,17, 8,15,21,24,12, 7,25)( 9,20,18,10,13,11,14,27,22)$
$6,3^{4},2^{3},1^{3}$ $1440$ $6$ $16$ $( 1, 5,26)( 3, 8,23)( 4,22)( 6, 9, 7,27,24,25)(10,21,19)(11,20)(13,16,15)(14,17)$

Malle's constant $a(G)$:     $1/6$

Copy content magma:ConjugacyClasses(G);
 

Character table

Character table not computed

Copy content magma:CharacterTable(G);
 

Regular extensions

$f_{ 1 } =$ $\left(x^{3}+6 x^{2}-8\right)^{9}-2^{4} 3^{12} x^{6} \left(x^{2}+5 x+4\right)^{4} \left(x-2\right) t$ Copy content Toggle raw display