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Properties

Label 45T297
45T297 1 18 1->18 30 1->30 2 17 2->17 28 2->28 3 16 3->16 29 3->29 4 4->17 22 4->22 5 5->18 24 5->24 6 6->16 23 6->23 7 21 7->21 7->23 8 19 8->19 8->22 9 20 9->20 9->24 10 10->19 26 10->26 11 11->20 27 11->27 12 12->21 25 12->25 13 14 13->14 13->26 14->27 15 15->25 16->14 17->15 18->13 19->1 20->3 21->2 22->7 23->9 24->8 25->6 26->4 27->5 28->12 45 28->45 29->11 43 29->43 30->10 44 30->44 31 31->45 32 33 32->33 32->43 33->44 34 34->33 38 34->38 35 35->32 37 35->37 36 36->31 39 36->39 38->39 40 40->34 41 41->35 42 41->42 42->36 43->42 44->41 45->40
Degree $45$
Order $3240$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $\He_3:S_5$

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Copy content magma:G := TransitiveGroup(45, 297);
 

Group invariants

Abstract group:  $\He_3:S_5$
Copy content magma:IdentifyGroup(G);
 
Order:  $3240=2^{3} \cdot 3^{4} \cdot 5$
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 $n$:  $45$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $297$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,18)(2,17)(3,16)(4,22)(5,24)(6,23)(7,21)(8,19)(9,20)(10,26)(11,27)(12,25)(13,14)(28,45)(29,43)(30,44)(32,33)(34,38)(35,37)(36,39)(41,42)$, $(1,30,10,19)(2,28,12,21)(3,29,11,20)(4,17,15,25,6,16,14,27,5,18,13,26)(7,23,9,24,8,22)(31,45,40,34,33,44,41,35,32,43,42,36)(38,39)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $S_3$, $C_6$
$18$:  $S_3\times C_3$
$54$:  $C_3^2 : C_6$
$120$:  $S_5$
$360$:  $\GL(2,4):C_2$, $S_5 \times C_3$
$1080$:  30T221

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 5: $S_5$

Degree 9: None

Degree 15: $\GL(2,4):C_2$

Low degree siblings

45T296, 45T297, 45T307, 45T308 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{45}$ $1$ $1$ $0$ $()$
2A $2^{18},1^{9}$ $15$ $2$ $18$ $( 1,19)( 2,20)( 3,21)( 4,31)( 5,32)( 6,33)( 7,43)( 8,45)( 9,44)(13,23)(14,24)(15,22)(16,36)(17,35)(18,34)(28,38)(29,39)(30,37)$
2B $2^{21},1^{3}$ $90$ $2$ $21$ $( 1,18)( 2,17)( 3,16)( 4,15)( 5,14)( 6,13)( 7,39)( 8,38)( 9,37)(10,25)(11,26)(12,27)(19,45)(20,44)(21,43)(23,24)(28,34)(29,36)(30,35)(32,33)(40,42)$
3A $3^{15}$ $2$ $3$ $30$ $( 1, 3, 2)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,18,17)(19,21,20)(22,23,24)(25,27,26)(28,29,30)(31,32,33)(34,35,36)(37,38,39)(40,42,41)(43,45,44)$
3B1 $3^{10},1^{15}$ $3$ $3$ $20$ $( 4, 5, 6)( 7, 9, 8)(13,14,15)(16,17,18)(22,23,24)(25,26,27)(31,32,33)(34,36,35)(40,42,41)(43,44,45)$
3B-1 $3^{10},1^{15}$ $3$ $3$ $20$ $( 4, 6, 5)( 7, 8, 9)(13,15,14)(16,18,17)(22,24,23)(25,27,26)(31,33,32)(34,35,36)(40,41,42)(43,45,44)$
3C $3^{15}$ $6$ $3$ $30$ $( 1,31,18)( 2,33,16)( 3,32,17)( 4,34,19)( 5,35,21)( 6,36,20)( 7,37,24)( 8,38,22)( 9,39,23)(10,41,25)(11,40,27)(12,42,26)(13,44,29)(14,43,30)(15,45,28)$
3D1 $3^{15}$ $6$ $3$ $30$ $( 1,17,31)( 2,18,33)( 3,16,32)( 4,19,35)( 5,21,36)( 6,20,34)( 7,23,39)( 8,24,37)( 9,22,38)(10,27,41)(11,26,40)(12,25,42)(13,29,43)(14,30,45)(15,28,44)$
3D-1 $3^{15}$ $6$ $3$ $30$ $( 1,31,17)( 2,33,18)( 3,32,16)( 4,35,19)( 5,36,21)( 6,34,20)( 7,39,23)( 8,37,24)( 9,38,22)(10,41,27)(11,40,26)(12,42,25)(13,43,29)(14,45,30)(15,44,28)$
3E $3^{15}$ $20$ $3$ $30$ $( 1,10,38)( 2,12,37)( 3,11,39)( 4, 6, 5)( 7,16,26)( 8,18,25)( 9,17,27)(13,14,15)(19,20,21)(22,31,41)(23,32,40)(24,33,42)(28,29,30)(34,36,35)(43,45,44)$
3F $3^{12},1^{9}$ $40$ $3$ $24$ $( 1,21,37)( 2,19,39)( 3,20,38)( 4,23,33)( 5,24,31)( 6,22,32)( 7,18,35)( 8,17,36)( 9,16,34)(13,14,15)(28,29,30)(43,45,44)$
3G1 $3^{13},1^{6}$ $60$ $3$ $26$ $( 1,37,29)( 2,39,28)( 3,38,30)( 7,45,17)( 8,44,16)( 9,43,18)(10,11,12)(13,32,23)(14,33,24)(15,31,22)(19,20,21)(34,35,36)(40,41,42)$
3G-1 $3^{13},1^{6}$ $60$ $3$ $26$ $( 1,29,37)( 2,28,39)( 3,30,38)( 7,17,45)( 8,16,44)( 9,18,43)(10,12,11)(13,23,32)(14,24,33)(15,22,31)(19,21,20)(34,36,35)(40,42,41)$
3H $3^{15}$ $120$ $3$ $30$ $( 1,25, 6)( 2,26, 5)( 3,27, 4)( 7,23,38)( 8,24,39)( 9,22,37)(10,36,31)(11,34,32)(12,35,33)(13,29,44)(14,30,43)(15,28,45)(16,42,21)(17,40,19)(18,41,20)$
3I1 $3^{15}$ $120$ $3$ $30$ $( 1, 6,45)( 2, 5,43)( 3, 4,44)( 7,37,23)( 8,38,24)( 9,39,22)(10,41,26)(11,40,25)(12,42,27)(13,16,20)(14,18,19)(15,17,21)(28,33,34)(29,31,35)(30,32,36)$
3I-1 $3^{15}$ $120$ $3$ $30$ $( 1,45, 6)( 2,43, 5)( 3,44, 4)( 7,23,37)( 8,24,38)( 9,22,39)(10,26,41)(11,25,40)(12,27,42)(13,20,16)(14,19,18)(15,21,17)(28,34,33)(29,35,31)(30,36,32)$
4A $4^{9},2^{4},1$ $270$ $4$ $31$ $( 2, 3)( 4, 7,42,44)( 5, 9,41,45)( 6, 8,40,43)(10,28,21,39)(11,30,20,38)(12,29,19,37)(13,34,24,26)(14,36,22,27)(15,35,23,25)(16,32)(17,33)(18,31)$
5A $5^{9}$ $24$ $5$ $36$ $( 1,29,21,38,12)( 2,28,19,37,11)( 3,30,20,39,10)( 4,24,40,33,15)( 5,22,42,31,13)( 6,23,41,32,14)( 7,27,16,45,34)( 8,26,18,44,35)( 9,25,17,43,36)$
6A $6^{6},3^{3}$ $30$ $6$ $36$ $( 1,20, 3,19, 2,21)( 4,33, 5,31, 6,32)( 7,44, 8,43, 9,45)(10,12,11)(13,22,14,23,15,24)(16,35,18,36,17,34)(25,26,27)(28,37,29,38,30,39)(40,41,42)$
6B1 $6^{4},3^{2},2^{6},1^{3}$ $45$ $6$ $30$ $( 4,41, 6,42, 5,40)( 7,45, 8,44, 9,43)(10,21)(11,20)(12,19)(13,22,15,24,14,23)(16,17,18)(25,34,27,35,26,36)(28,39)(29,37)(30,38)(31,32,33)$
6B-1 $6^{4},3^{2},2^{6},1^{3}$ $45$ $6$ $30$ $( 4,40, 5,42, 6,41)( 7,43, 9,44, 8,45)(10,21)(11,20)(12,19)(13,23,14,24,15,22)(16,18,17)(25,36,26,35,27,34)(28,39)(29,37)(30,38)(31,33,32)$
6C $6^{6},3^{3}$ $90$ $6$ $36$ $( 1,36,31,20,18, 6)( 2,35,33,21,16, 5)( 3,34,32,19,17, 4)( 7,24,37)( 8,22,38)( 9,23,39)(10,44,41,29,25,13)(11,43,40,30,27,14)(12,45,42,28,26,15)$
6D1 $6^{6},3^{3}$ $90$ $6$ $36$ $( 1,33,16)( 2,32,17)( 3,31,18)( 4, 7,21,24,34,38)( 5, 8,20,22,35,39)( 6, 9,19,23,36,37)(10,13,26,30,42,44)(11,14,25,28,41,43)(12,15,27,29,40,45)$
6D-1 $6^{6},3^{3}$ $90$ $6$ $36$ $( 1,16,33)( 2,17,32)( 3,18,31)( 4,38,34,24,21, 7)( 5,39,35,22,20, 8)( 6,37,36,23,19, 9)(10,44,42,30,26,13)(11,43,41,28,25,14)(12,45,40,29,27,15)$
6E1 $6^{5},2^{6},1^{3}$ $90$ $6$ $31$ $( 2, 3)( 4,36, 5,35, 6,34)( 7,24, 9,22, 8,23)(10,29)(11,28)(12,30)(13,27,14,25,15,26)(16,31,17,32,18,33)(19,21)(37,38)(40,44,42,45,41,43)$
6E-1 $6^{5},2^{6},1^{3}$ $90$ $6$ $31$ $( 2, 3)( 4,34, 6,35, 5,36)( 7,23, 8,22, 9,24)(10,29)(11,28)(12,30)(13,26,15,25,14,27)(16,33,18,32,17,31)(19,21)(37,38)(40,43,41,45,42,44)$
6F $6^{7},3$ $180$ $6$ $37$ $( 1, 8,10,18,38,25)( 2, 9,12,17,37,27)( 3, 7,11,16,39,26)( 4,14, 6,15, 5,13)(19,43,20,45,21,44)(22,41,31)(23,42,32,24,40,33)(28,35,29,34,30,36)$
6G1 $6^{6},3,2^{3}$ $180$ $6$ $35$ $( 1,29,37)( 2,30,39, 3,28,38)( 4,26)( 5,27)( 6,25)( 7,33,45,24,17,14)( 8,32,44,23,16,13)( 9,31,43,22,18,15)(10,19,11,20,12,21)(34,41,35,42,36,40)$
6G-1 $6^{6},3,2^{3}$ $180$ $6$ $35$ $( 1,37,29)( 2,38,28, 3,39,30)( 4,26)( 5,27)( 6,25)( 7,14,17,24,45,33)( 8,13,16,23,44,32)( 9,15,18,22,43,31)(10,21,12,20,11,19)(34,40,36,42,35,41)$
12A1 $12^{2},6,4^{3},2,1$ $270$ $12$ $37$ $( 2, 3)( 4,43,41, 7, 6,45,42, 8, 5,44,40, 9)(10,39,21,28)(11,38,20,30)(12,37,19,29)(13,25,22,34,15,27,24,35,14,26,23,36)(16,31,17,32,18,33)$
12A-1 $12^{2},6,4^{3},2,1$ $270$ $12$ $37$ $( 2, 3)( 4, 9,40,44, 5, 8,42,45, 6, 7,41,43)(10,28,21,39)(11,30,20,38)(12,29,19,37)(13,36,23,26,14,35,24,27,15,34,22,25)(16,33,18,32,17,31)$
15A1 $15^{3}$ $24$ $15$ $42$ $( 1,19,10,29,37, 3,21,11,30,38, 2,20,12,28,39)( 4,41,13,24,32, 5,40,14,22,33, 6,42,15,23,31)( 7,17,35,27,43, 8,16,36,26,45, 9,18,34,25,44)$
15A-1 $15^{3}$ $24$ $15$ $42$ $( 1,20,11,29,39, 2,21,10,28,38, 3,19,12,30,37)( 4,42,14,24,31, 6,40,13,23,33, 5,41,15,22,32)( 7,18,36,27,44, 9,16,35,25,45, 8,17,34,26,43)$
15B1 $15^{2},5^{3}$ $72$ $15$ $40$ $( 1,37,20,28,12)( 2,39,21,30,11)( 3,38,19,29,10)( 4,15,40,32,24, 5,13,42,33,22, 6,14,41,31,23)( 7,34,43,26,17, 9,36,44,27,18, 8,35,45,25,16)$
15B-1 $15^{2},5^{3}$ $72$ $15$ $40$ $( 1,12,28,20,37)( 2,11,30,21,39)( 3,10,29,19,38)( 4,23,31,41,14, 6,22,33,42,13, 5,24,32,40,15)( 7,16,25,45,35, 8,18,27,44,36, 9,17,26,43,34)$
15C1 $15^{3}$ $72$ $15$ $42$ $( 1,14,36,10,22,17,28, 4,27,39,33,44,21,42, 7)( 2,13,35,12,24,18,30, 6,25,38,32,45,19,40, 9)( 3,15,34,11,23,16,29, 5,26,37,31,43,20,41, 8)$
15C-1 $15^{3}$ $72$ $15$ $42$ $( 1, 7,42,21,44,33,39,27, 4,28,17,22,10,36,14)( 2, 9,40,19,45,32,38,25, 6,30,18,24,12,35,13)( 3, 8,41,20,43,31,37,26, 5,29,16,23,11,34,15)$
15D1 $15^{3}$ $72$ $15$ $42$ $( 1,14, 8,19,41,17,30,24,35,10,31,45,37, 4,27)( 2,13, 7,20,42,18,29,23,34,12,33,43,39, 6,25)( 3,15, 9,21,40,16,28,22,36,11,32,44,38, 5,26)$
15D-1 $15^{3}$ $72$ $15$ $42$ $( 1,27, 4,37,45,31,10,35,24,30,17,41,19, 8,14)( 2,25, 6,39,43,33,12,34,23,29,18,42,20, 7,13)( 3,26, 5,38,44,32,11,36,22,28,16,40,21, 9,15)$
15D2 $15^{3}$ $72$ $15$ $42$ $( 1, 8,41,30,35,31,37,27,14,19,17,24,10,45, 4)( 2, 7,42,29,34,33,39,25,13,20,18,23,12,43, 6)( 3, 9,40,28,36,32,38,26,15,21,16,22,11,44, 5)$
15D-2 $15^{3}$ $72$ $15$ $42$ $( 1, 4,45,10,24,17,19,14,27,37,31,35,30,41, 8)( 2, 6,43,12,23,18,20,13,25,39,33,34,29,42, 7)( 3, 5,44,11,22,16,21,15,26,38,32,36,28,40, 9)$

Malle's constant $a(G)$:     $1/18$

Copy content magma:ConjugacyClasses(G);
 

Character table

41 x 41 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed