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Properties

Label 28T312
28T312 1 2 1->2 22 1->22 21 2->21 3 25 3->25 26 3->26 4 4->25 4->26 5 20 5->20 27 5->27 6 19 6->19 28 6->28 7 7->19 23 7->23 8 8->20 24 8->24 9 9->4 9->8 10 10->3 10->7 11 11->4 15 11->15 12 12->3 16 12->16 13 13->10 14 13->14 14->9 15->8 15->16 16->7 17 17->11 17->12 18 18->11 18->12 19->14 20->13 21->5 21->10 22->6 22->9 23->17 23->21 24->18 24->22 25->2 25->17 26->1 26->18 27->24 27->28 28->23
Degree $28$
Order $5376$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $C_2^5:\GL(3,2)$

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Copy content magma:G := TransitiveGroup(28, 312);
 

Group invariants

Abstract group:  $C_2^5:\GL(3,2)$
Copy content magma:IdentifyGroup(G);
 
Order:  $5376=2^{8} \cdot 3 \cdot 7$
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 $n$:  $28$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $312$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $4$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,22,6,28,23,17,11,15,8,20,13,10,3,26)(2,21,5,27,24,18,12,16,7,19,14,9,4,25)$, $(1,2)(3,25,17,12)(4,26,18,11)(5,20)(6,19)(7,23,21,10)(8,24,22,9)(13,14)(15,16)(27,28)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$168$:  $\GL(3,2)$
$336$:  14T17 x 3
$672$:  28T84
$1344$:  $C_2^3:\GL(3,2)$
$2688$:  14T43 x 3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: None

Degree 7: $\GL(3,2)$

Degree 14: $\GL(3,2) \times C_2$, 14T43 x 2

Low degree siblings

28T311 x 6, 28T312 x 5, 32T397089 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 $1^{28}$ $1$ $1$ $0$ $()$
2A $2^{14}$ $1$ $2$ $14$ $( 1,16)( 2,15)( 3,18)( 4,17)( 5,20)( 6,19)( 7,22)( 8,21)( 9,23)(10,24)(11,25)(12,26)(13,27)(14,28)$
2B $2^{14}$ $1$ $2$ $14$ $( 1,15)( 2,16)( 3,17)( 4,18)( 5,19)( 6,20)( 7,21)( 8,22)( 9,24)(10,23)(11,26)(12,25)(13,28)(14,27)$
2C $2^{14}$ $1$ $2$ $14$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)$
2D $2^{8},1^{12}$ $7$ $2$ $8$ $( 1,15)( 2,16)( 5,19)( 6,20)( 7,21)( 8,22)( 9,24)(10,23)$
2E $2^{6},1^{16}$ $7$ $2$ $6$ $( 1,15)( 2,16)( 9,24)(10,23)(11,26)(12,25)$
2F $2^{14}$ $7$ $2$ $14$ $( 1,16)( 2,15)( 3,18)( 4,17)( 5, 6)( 7, 8)( 9,23)(10,24)(11,12)(13,27)(14,28)(19,20)(21,22)(25,26)$
2G $2^{14}$ $7$ $2$ $14$ $( 1,16)( 2,15)( 3,18)( 4,17)( 5, 6)( 7,22)( 8,21)( 9,10)(11,12)(13,14)(19,20)(23,24)(25,26)(27,28)$
2H $2^{14}$ $42$ $2$ $14$ $( 1,17)( 2,18)( 3,15)( 4,16)( 5,25)( 6,26)( 7,21)( 8,22)( 9,24)(10,23)(11,20)(12,19)(13,28)(14,27)$
2I $2^{14}$ $42$ $2$ $14$ $( 1,18)( 2,17)( 3,16)( 4,15)( 5,26)( 6,25)( 7,22)( 8,21)( 9,23)(10,24)(11,19)(12,20)(13,27)(14,28)$
2J $2^{14}$ $42$ $2$ $14$ $( 1, 4)( 2, 3)( 5,11)( 6,12)( 7, 8)( 9,10)(13,14)(15,18)(16,17)(19,26)(20,25)(21,22)(23,24)(27,28)$
2K $2^{10},1^{8}$ $42$ $2$ $10$ $( 1,13)( 2,14)( 5,19)( 6,20)( 7,25)( 8,26)(11,22)(12,21)(15,28)(16,27)$
2L $2^{14}$ $42$ $2$ $14$ $( 1,14)( 2,13)( 3, 4)( 5,20)( 6,19)( 7,26)( 8,25)( 9,10)(11,21)(12,22)(15,27)(16,28)(17,18)(23,24)$
2M $2^{14}$ $42$ $2$ $14$ $( 1,27)( 2,28)( 3,18)( 4,17)( 5, 6)( 7,11)( 8,12)( 9,23)(10,24)(13,16)(14,15)(19,20)(21,26)(22,25)$
2N $2^{8},1^{12}$ $42$ $2$ $8$ $( 1,26)( 2,25)( 7,27)( 8,28)(11,15)(12,16)(13,22)(14,21)$
2O $2^{12},1^{4}$ $42$ $2$ $12$ $( 1,15)( 2,16)( 3,23)( 4,24)( 7,25)( 8,26)( 9,18)(10,17)(11,22)(12,21)(13,28)(14,27)$
3A $3^{8},1^{4}$ $224$ $3$ $16$ $( 1,23,17)( 2,24,18)( 3,15,10)( 4,16, 9)( 5, 7,25)( 6, 8,26)(11,20,22)(12,19,21)$
4A $4^{4},2^{6}$ $84$ $4$ $18$ $( 1,16)( 2,15)( 3, 5,17,19)( 4, 6,18,20)( 7,28,21,13)( 8,27,22,14)( 9,23)(10,24)(11,12)(25,26)$
4B $4^{4},2^{4},1^{4}$ $84$ $4$ $16$ $( 1,28,15,13)( 2,27,16,14)( 5,19)( 6,20)( 7,25,21,12)( 8,26,22,11)( 9,24)(10,23)$
4C $4^{4},2^{6}$ $84$ $4$ $18$ $( 1,19,15, 5)( 2,20,16, 6)( 3,12,17,25)( 4,11,18,26)( 7, 8)( 9,23)(10,24)(13,14)(21,22)(27,28)$
4D $4^{4},2^{2},1^{8}$ $84$ $4$ $14$ $( 1,15)( 2,16)( 3,28,17,13)( 4,27,18,14)( 5, 7,19,21)( 6, 8,20,22)$
4E $4^{4},2^{4},1^{4}$ $168$ $4$ $16$ $( 1,22,26,13)( 2,21,25,14)( 3,20)( 4,19)( 5,18)( 6,17)( 7,12,27,16)( 8,11,28,15)$
4F $4^{6},2^{2}$ $168$ $4$ $20$ $( 1,27,15,14)( 2,28,16,13)( 3,21,23,12)( 4,22,24,11)( 5, 6)( 7,10,25,17)( 8, 9,26,18)(19,20)$
4G $4^{4},2^{6}$ $168$ $4$ $18$ $( 1,22)( 2,21)( 3,17)( 4,18)( 5,25,24,27)( 6,26,23,28)( 7,16)( 8,15)( 9,14,19,12)(10,13,20,11)$
4H $4^{6},2^{2}$ $168$ $4$ $20$ $( 1,21, 6,24)( 2,22, 5,23)( 3,12,17,25)( 4,11,18,26)( 7,20, 9,15)( 8,19,10,16)(13,27)(14,28)$
4I $4^{4},2^{6}$ $168$ $4$ $18$ $( 1,24, 6, 7)( 2,23, 5, 8)( 3,12)( 4,11)( 9,20,21,15)(10,19,22,16)(13,14)(17,25)(18,26)(27,28)$
4J $4^{4},2^{6}$ $168$ $4$ $18$ $( 1, 7,20, 9)( 2, 8,19,10)( 3,12)( 4,11)( 5,23,16,22)( 6,24,15,21)(13,27)(14,28)(17,25)(18,26)$
4K $4^{6},2^{2}$ $168$ $4$ $20$ $( 1,15)( 2,16)( 3,26, 8,23)( 4,25, 7,24)( 5,27,19,14)( 6,28,20,13)( 9,18,12,21)(10,17,11,22)$
4L $4^{6},1^{4}$ $168$ $4$ $18$ $( 1,23, 8,20)( 2,24, 7,19)( 5,16, 9,21)( 6,15,10,22)(11,28,26,13)(12,27,25,14)$
6A $6^{4},2^{2}$ $224$ $6$ $22$ $( 1, 4,23,16,17, 9)( 2, 3,24,15,18,10)( 5,11, 7,20,25,22)( 6,12, 8,19,26,21)(13,27)(14,28)$
6B $6^{2},3^{4},2^{2}$ $224$ $6$ $20$ $( 1,23, 6,15,10,20)( 2,24, 5,16, 9,19)( 3,28,11)( 4,27,12)( 7,21)( 8,22)(13,26,17)(14,25,18)$
6C $6^{2},3^{4},1^{4}$ $224$ $6$ $18$ $( 1,11,23,15,26,10)( 2,12,24,16,25, 9)( 3,22,20)( 4,21,19)( 5,18, 7)( 6,17, 8)$
6D $6^{4},2^{2}$ $224$ $6$ $22$ $( 1, 3, 8,15,17,22)( 2, 4, 7,16,18,21)( 5,14,25,19,27,12)( 6,13,26,20,28,11)( 9,24)(10,23)$
6E $6^{4},2^{2}$ $224$ $6$ $22$ $( 1, 2)( 3, 5,10, 4, 6, 9)( 7,13,25, 8,14,26)(11,21,28,12,22,27)(15,16)(17,19,23,18,20,24)$
6F $6^{4},2^{2}$ $224$ $6$ $22$ $( 1, 4,28,16,17,14)( 2, 3,27,15,18,13)( 5, 8,25, 6, 7,26)( 9,23)(10,24)(11,19,22,12,20,21)$
6G $6^{4},2^{2}$ $224$ $6$ $22$ $( 1, 4, 8,16,17,21)( 2, 3, 7,15,18,22)( 5,23,27, 6,24,28)( 9,13,19,10,14,20)(11,12)(25,26)$
7A1 $7^{4}$ $192$ $7$ $24$ $( 1,13,10,26,17, 6,22)( 2,14, 9,25,18, 5,21)( 3,20, 8,15,28,23,11)( 4,19, 7,16,27,24,12)$
7A-1 $7^{4}$ $192$ $7$ $24$ $( 1,22, 6,17,26,10,13)( 2,21, 5,18,25, 9,14)( 3,11,23,28,15, 8,20)( 4,12,24,27,16, 7,19)$
14A1 $14^{2}$ $192$ $14$ $26$ $( 1, 4,13,19,10, 7,26,16,17,27, 6,24,22,12)( 2, 3,14,20, 9, 8,25,15,18,28, 5,23,21,11)$
14A-1 $14^{2}$ $192$ $14$ $26$ $( 1,12,22,24, 6,27,17,16,26, 7,10,19,13, 4)( 2,11,21,23, 5,28,18,15,25, 8, 9,20,14, 3)$
14B1 $14^{2}$ $192$ $14$ $26$ $( 1,13, 8, 6,10,11, 3,15,28,22,20,23,26,17)( 2,14, 7, 5, 9,12, 4,16,27,21,19,24,25,18)$
14B-1 $14^{2}$ $192$ $14$ $26$ $( 1,17,26,23,20,22,28,15, 3,11,10, 6, 8,13)( 2,18,25,24,19,21,27,16, 4,12, 9, 5, 7,14)$
14C1 $14^{2}$ $192$ $14$ $26$ $( 1, 9,22,25,13, 5,17, 2,10,21,26,14, 6,18)( 3,16,23, 7,11,27,20, 4,15,24, 8,12,28,19)$
14C-1 $14^{2}$ $192$ $14$ $26$ $( 1,18, 6,14,26,21,10, 2,17, 5,13,25,22, 9)( 3,19,28,12, 8,24,15, 4,20,27,11, 7,23,16)$

Malle's constant $a(G)$:     $1/6$

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Character table

44 x 44 character table

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Regular extensions

Data not computed