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Properties

Label 18T427
18T427 1 15 1->15 16 1->16 2 2->15 2->16 3 10 3->10 18 3->18 4 9 4->9 17 4->17 5 5->3 13 5->13 6 6->4 14 6->14 7 7->3 7->13 8 8->4 8->14 9->2 9->7 10->1 10->8 11 11->6 12 11->12 12->5 13->11 14->12 15->7 16->8 17->9 18->10
Degree $18$
Order $3024$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $\SL(2,8):C_6$

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Copy content magma:G := TransitiveGroup(18, 427);
 

Group invariants

Abstract group:  $\SL(2,8):C_6$
Copy content magma:IdentifyGroup(G);
 
Order:  $3024=2^{4} \cdot 3^{3} \cdot 7$
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 $n$:  $18$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $427$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $2$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,15,7,3,18,10)(2,16,8,4,17,9)(5,13)(6,14)(11,12)$, $(1,16)(2,15)(3,10,8,14,12,5)(4,9,7,13,11,6)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$1512$:  $\mathrm{P}\Gamma\mathrm{L}(2,8)$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 6: None

Degree 9: $\mathrm{P}\Gamma\mathrm{L}(2,8)$

Low degree siblings

There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{18}$ $1$ $1$ $0$ $()$
2A $2^{9}$ $1$ $2$ $9$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$
2B $2^{8},1^{2}$ $63$ $2$ $8$ $( 1, 9)( 2,10)( 5, 8)( 6, 7)(11,18)(12,17)(13,16)(14,15)$
2C $2^{9}$ $63$ $2$ $9$ $( 1, 3)( 2, 4)( 5,11)( 6,12)( 7, 8)( 9,15)(10,16)(13,17)(14,18)$
3A $3^{6}$ $56$ $3$ $12$ $( 1, 9,11)( 2,10,12)( 3,15,17)( 4,16,18)( 5, 8,14)( 6, 7,13)$
3B1 $3^{4},1^{6}$ $84$ $3$ $8$ $( 1,13, 6)( 2,14, 5)( 7, 9,16)( 8,10,15)$
3B-1 $3^{4},1^{6}$ $84$ $3$ $8$ $( 1, 6,13)( 2, 5,14)( 7,16, 9)( 8,15,10)$
6A $6^{3}$ $56$ $6$ $15$ $( 1,12, 9, 2,11,10)( 3,18,15, 4,17,16)( 5,13, 8, 6,14, 7)$
6B1 $6^{2},2^{3}$ $84$ $6$ $13$ $( 1,17,11, 2,18,12)( 3, 6,10, 4, 5, 9)( 7, 8)(13,14)(15,16)$
6B-1 $6^{2},2^{3}$ $84$ $6$ $13$ $( 1,12,18, 2,11,17)( 3, 9, 5, 4,10, 6)( 7, 8)(13,14)(15,16)$
6C1 $6^{2},2^{2},1^{2}$ $252$ $6$ $12$ $( 1, 7,13, 9, 6,16)( 2, 8,14,10, 5,15)(11,18)(12,17)$
6C-1 $6^{2},2^{2},1^{2}$ $252$ $6$ $12$ $( 1,16, 6, 9,13, 7)( 2,15, 5,10,14, 8)(11,18)(12,17)$
6D1 $6^{2},2^{3}$ $252$ $6$ $13$ $( 1,10, 6, 3,16,12)( 2, 9, 5, 4,15,11)( 7, 8)(13,17)(14,18)$
6D-1 $6^{2},2^{3}$ $252$ $6$ $13$ $( 1,12,16, 3, 6,10)( 2,11,15, 4, 5, 9)( 7, 8)(13,17)(14,18)$
7A $7^{2},1^{4}$ $216$ $7$ $12$ $( 1,13,11, 4, 6, 9,16)( 2,14,12, 3, 5,10,15)$
9A $9^{2}$ $168$ $9$ $16$ $( 1,16, 6, 9,18, 7,11, 4,13)( 2,15, 5,10,17, 8,12, 3,14)$
9B1 $9^{2}$ $168$ $9$ $16$ $( 1, 9, 4, 6,13,16,11,18, 7)( 2,10, 3, 5,14,15,12,17, 8)$
9B-1 $9^{2}$ $168$ $9$ $16$ $( 1, 7,18,11,16,13, 6, 4, 9)( 2, 8,17,12,15,14, 5, 3,10)$
14A $14,2^{2}$ $216$ $14$ $15$ $( 1, 5,13,10,11,15, 4, 2, 6,14, 9,12,16, 3)( 7, 8)(17,18)$
18A $18$ $168$ $18$ $17$ $( 1, 8,16,12, 6, 3, 9,14,18, 2, 7,15,11, 5, 4,10,13,17)$
18B1 $18$ $168$ $18$ $17$ $( 1,15, 9,12, 4,17, 6, 8,13, 2,16,10,11, 3,18, 5, 7,14)$
18B-1 $18$ $168$ $18$ $17$ $( 1,14, 7, 5,18, 3,11,10,16, 2,13, 8, 6,17, 4,12, 9,15)$

Malle's constant $a(G)$:     $1/8$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 3A 3B1 3B-1 6A 6B1 6B-1 6C1 6C-1 6D1 6D-1 7A 9A 9B1 9B-1 14A 18A 18B1 18B-1
Size 1 1 63 63 56 84 84 56 84 84 252 252 252 252 216 168 168 168 216 168 168 168
2 P 1A 1A 1A 1A 3A 3B-1 3B1 3A 3B-1 3B1 3B1 3B-1 3B-1 3B1 7A 9A 9B-1 9B1 7A 9A 9B1 9B-1
3 P 1A 2A 2B 2C 1A 1A 1A 2A 2A 2A 2B 2B 2C 2C 7A 3A 3A 3A 14A 6A 6A 6A
7 P 1A 2A 2B 2C 3A 3B1 3B-1 6A 6B1 6B-1 6C1 6C-1 6D1 6D-1 1A 9A 9B1 9B-1 2A 18A 18B1 18B-1
Type
3024.r.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
3024.r.1b R 1 −1 1 −1 1 1 1 −1 −1 −1 1 1 −1 −1 1 1 1 1 −1 −1 −1 −1
3024.r.1c1 C 1 1 1 1 1 ζ3−1 ζ3 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 1 1 ζ3−1 ζ3 1 1 ζ3 ζ3−1
3024.r.1c2 C 1 1 1 1 1 ζ3 ζ3−1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 ζ3 ζ3−1 1 1 ζ3−1 ζ3
3024.r.1d1 C 1 −1 1 −1 1 ζ3−1 ζ3 −1 −ζ3−1 −ζ3 ζ3 ζ3−1 −ζ3−1 −ζ3 1 1 ζ3−1 ζ3 −1 −1 −ζ3 −ζ3−1
3024.r.1d2 C 1 −1 1 −1 1 ζ3 ζ3−1 −1 −ζ3 −ζ3−1 ζ3−1 ζ3 −ζ3 −ζ3−1 1 1 ζ3 ζ3−1 −1 −1 −ζ3−1 −ζ3
3024.r.7a R 7 −7 −1 1 −2 1 1 2 −1 −1 −1 −1 1 1 0 1 1 1 0 −1 −1 −1
3024.r.7b R 7 7 −1 −1 −2 1 1 −2 1 1 −1 −1 −1 −1 0 1 1 1 0 1 1 1
3024.r.7c1 C 7 −7 −1 1 −2 ζ3−1 ζ3 2 −ζ3−1 −ζ3 −ζ3 −ζ3−1 ζ3−1 ζ3 0 1 ζ3−1 ζ3 0 −1 −ζ3 −ζ3−1
3024.r.7c2 C 7 −7 −1 1 −2 ζ3 ζ3−1 2 −ζ3 −ζ3−1 −ζ3−1 −ζ3 ζ3 ζ3−1 0 1 ζ3 ζ3−1 0 −1 −ζ3−1 −ζ3
3024.r.7d1 C 7 7 −1 −1 −2 ζ3−1 ζ3 −2 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3 0 1 ζ3−1 ζ3 0 1 ζ3 ζ3−1
3024.r.7d2 C 7 7 −1 −1 −2 ζ3 ζ3−1 −2 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 0 1 ζ3 ζ3−1 0 1 ζ3−1 ζ3
3024.r.8a R 8 8 0 0 −1 2 2 −1 2 2 0 0 0 0 1 −1 −1 −1 1 −1 −1 −1
3024.r.8b R 8 −8 0 0 −1 2 2 1 −2 −2 0 0 0 0 1 −1 −1 −1 −1 1 1 1
3024.r.8c1 C 8 8 0 0 −1 2ζ3−1 2ζ3 −1 2ζ3−1 2ζ3 0 0 0 0 1 −1 −ζ3−1 −ζ3 1 −1 −ζ3 −ζ3−1
3024.r.8c2 C 8 8 0 0 −1 2ζ3 2ζ3−1 −1 2ζ3 2ζ3−1 0 0 0 0 1 −1 −ζ3 −ζ3−1 1 −1 −ζ3−1 −ζ3
3024.r.8d1 C 8 −8 0 0 −1 2ζ3−1 2ζ3 1 −2ζ3−1 −2ζ3 0 0 0 0 1 −1 −ζ3−1 −ζ3 −1 1 ζ3 ζ3−1
3024.r.8d2 C 8 −8 0 0 −1 2ζ3 2ζ3−1 1 −2ζ3 −2ζ3−1 0 0 0 0 1 −1 −ζ3 −ζ3−1 −1 1 ζ3−1 ζ3
3024.r.21a R 21 −21 −3 3 3 0 0 −3 0 0 0 0 0 0 0 0 0 0 0 0 0 0
3024.r.21b R 21 21 −3 −3 3 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0
3024.r.27a R 27 27 3 3 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 −1 0 0 0
3024.r.27b R 27 −27 3 −3 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 1 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed