↑

Properties

Label 30T604
30T604 1 11 1->11 16 1->16 2 12 2->12 15 2->15 3 3->2 13 3->13 4 4->1 14 4->14 5 5->15 30 5->30 6 6->16 29 6->29 7 7->4 17 7->17 8 8->3 18 8->18 9 19 9->19 20 9->20 10 10->19 10->20 11->12 21 11->21 22 12->22 13->22 23 13->23 14->21 24 14->24 15->8 25 15->25 16->7 26 16->26 17->24 27 17->27 18->23 28 18->28 19->29 20->30 21->1 21->26 22->2 22->25 23->3 23->6 24->4 24->5 25->5 25->28 26->6 26->27 27->7 27->14 28->8 28->13 29->9 29->18 30->10 30->17
Degree $30$
Order $6000$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_5^3.S_3$

Related objects

Downloads

Learn more

Show commands: Magma

Copy content magma:G := TransitiveGroup(30, 604);
 

Group invariants

Abstract group:  $D_5^3.S_3$
Copy content magma:IdentifyGroup(G);
 
Order:  $6000=2^{4} \cdot 3 \cdot 5^{3}$
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 $n$:  $30$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $604$
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:  $(1,11,21)(2,12,22)(3,13,23)(4,14,24)(5,15,25)(6,16,26)(7,17,27)(8,18,28)(9,19,29)(10,20,30)$, $(1,16,7,4)(2,15,8,3)(5,30,17,24)(6,29,18,23)(9,20)(10,19)(11,12)(13,22,25,28)(14,21,26,27)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$
$6$:  $S_3$
$12$:  $C_3 : C_4$
$24$:  $S_4$
$48$:  12T27

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 5: None

Degree 6: $S_3$

Degree 10: None

Degree 15: 15T58

Low degree siblings

15T58, 20T360, 30T591, 30T607, 30T616, 30T624, 40T5196

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{30}$ $1$ $1$ $0$ $()$
2A $2^{4},1^{22}$ $15$ $2$ $4$ $( 9,27)(10,28)(15,21)(16,22)$
2B $2^{8},1^{14}$ $75$ $2$ $8$ $( 1, 7)( 2, 8)( 5,17)( 6,18)(13,25)(14,26)(23,29)(24,30)$
2C $2^{12},1^{6}$ $125$ $2$ $12$ $( 3, 9)( 4,10)( 5,23)( 6,24)( 7,25)( 8,26)(11,17)(12,18)(13,19)(14,20)(15,27)(16,28)$
3A $3^{10}$ $200$ $3$ $20$ $( 1,21,29)( 2,22,30)( 3,17,19)( 4,18,20)( 5, 7,15)( 6, 8,16)( 9,11,13)(10,12,14)(23,25,27)(24,26,28)$
4A1 $4^{2},2^{11}$ $150$ $4$ $17$ $( 1,28)( 2,27)( 3,20)( 4,19)( 5,18,11,30)( 6,17,12,29)( 7,10)( 8, 9)(13,22)(14,21)(15,26)(16,25)(23,24)$
4A-1 $4^{2},2^{11}$ $150$ $4$ $17$ $( 1,28)( 2,27)( 3,20)( 4,19)( 5,30,11,18)( 6,29,12,17)( 7,10)( 8, 9)(13,22)(14,21)(15,26)(16,25)(23,24)$
4B1 $4^{6},2^{3}$ $750$ $4$ $21$ $( 1,30, 7,12)( 2,29, 8,11)( 3,16, 9,28)( 4,15,10,27)( 5,20)( 6,19)(13,24,25,18)(14,23,26,17)(21,22)$
4B-1 $4^{6},2^{3}$ $750$ $4$ $21$ $( 1,12, 7,30)( 2,11, 8,29)( 3,28, 9,16)( 4,27,10,15)( 5,20)( 6,19)(13,18,25,24)(14,17,26,23)(21,22)$
5A $5^{2},1^{20}$ $12$ $5$ $8$ $( 5,17,29,11,23)( 6,18,30,12,24)$
5B1 $5^{4},1^{10}$ $12$ $5$ $16$ $( 1,19, 7,25,13)( 2,20, 8,26,14)( 3, 9,15,21,27)( 4,10,16,22,28)$
5B2 $5^{4},1^{10}$ $12$ $5$ $16$ $( 1, 7,13,19,25)( 2, 8,14,20,26)( 3,15,27, 9,21)( 4,16,28,10,22)$
5C $5^{6}$ $16$ $5$ $24$ $( 1,13,25, 7,19)( 2,14,26, 8,20)( 3,21, 9,27,15)( 4,22,10,28,16)( 5,17,29,11,23)( 6,18,30,12,24)$
5D $5^{4},1^{10}$ $24$ $5$ $16$ $( 3, 9,15,21,27)( 4,10,16,22,28)( 5,11,17,23,29)( 6,12,18,24,30)$
5E $5^{6}$ $48$ $5$ $24$ $( 1,25,19,13, 7)( 2,26,20,14, 8)( 3, 9,15,21,27)( 4,10,16,22,28)( 5,17,29,11,23)( 6,18,30,12,24)$
6A $6^{4},3^{2}$ $1000$ $6$ $24$ $( 1,29,21)( 2,30,22)( 3,13,17, 9,19,11)( 4,14,18,10,20,12)( 5,27, 7,23,15,25)( 6,28, 8,24,16,26)$
10A1 $5^{2},2^{4},1^{12}$ $60$ $10$ $12$ $( 5,11,17,23,29)( 6,12,18,24,30)( 9,27)(10,28)(15,21)(16,22)$
10A3 $5^{2},2^{4},1^{12}$ $60$ $10$ $12$ $( 1,13)( 2,14)( 5,11,17,23,29)( 6,12,18,24,30)(19,25)(20,26)$
10B1 $5^{4},2^{4},1^{2}$ $60$ $10$ $20$ $( 1,25,19,13, 7)( 2,26,20,14, 8)( 3,21, 9,27,15)( 4,22,10,28,16)( 5,11)( 6,12)(17,29)(18,30)$
10B3 $5^{4},2^{4},1^{2}$ $60$ $10$ $20$ $( 1,13,25, 7,19)( 2,14,26, 8,20)( 3,27,21,15, 9)( 4,28,22,16,10)( 5,11)( 6,12)(17,29)(18,30)$
10C $5^{4},2^{4},1^{2}$ $120$ $10$ $20$ $( 1, 7)( 2, 8)( 3,21, 9,27,15)( 4,22,10,28,16)( 5,23,11,29,17)( 6,24,12,30,18)(13,25)(14,26)$
10D $5^{2},2^{8},1^{4}$ $300$ $10$ $16$ $( 1, 7)( 2, 8)( 3,21, 9,27,15)( 4,22,10,28,16)( 5,17)( 6,18)(13,25)(14,26)(23,29)(24,30)$
15A1 $15^{2}$ $400$ $15$ $28$ $( 1,17,15,13,29, 3,25,11,21, 7,23, 9,19, 5,27)( 2,18,16,14,30, 4,26,12,22, 8,24,10,20, 6,28)$
15A-1 $15^{2}$ $400$ $15$ $28$ $( 1,27, 5,19, 9,23, 7,21,11,25, 3,29,13,15,17)( 2,28, 6,20,10,24, 8,22,12,26, 4,30,14,16,18)$
20A1 $10^{2},4^{2},2$ $300$ $20$ $25$ $( 1,22,25,10,19,28,13,16, 7, 4)( 2,21,26, 9,20,27,14,15, 8, 3)( 5,18,11,30)( 6,17,12,29)(23,24)$
20A-1 $10^{2},4^{2},2$ $300$ $20$ $25$ $( 1, 4, 7,16,13,28,19,10,25,22)( 2, 3, 8,15,14,27,20, 9,26,21)( 5,30,11,18)( 6,29,12,17)(23,24)$
20A3 $10^{2},4^{2},2$ $300$ $20$ $25$ $( 1,10,13, 4,25,28, 7,22,19,16)( 2, 9,14, 3,26,27, 8,21,20,15)( 5,30,11,18)( 6,29,12,17)(23,24)$
20A-3 $10^{2},4^{2},2$ $300$ $20$ $25$ $( 1,16,19,22, 7,28,25, 4,13,10)( 2,15,20,21, 8,27,26, 3,14, 9)( 5,18,11,30)( 6,17,12,29)(23,24)$

Malle's constant $a(G)$:     $1/4$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 3A 4A1 4A-1 4B1 4B-1 5A 5B1 5B2 5C 5D 5E 6A 10A1 10A3 10B1 10B3 10C 10D 15A1 15A-1 20A1 20A-1 20A3 20A-3
Size 1 15 75 125 200 150 150 750 750 12 12 12 16 24 48 1000 60 60 60 60 120 300 400 400 300 300 300 300
2 P 1A 1A 1A 1A 3A 2A 2A 2C 2C 5A 5B2 5B1 5C 5D 5E 3A 5A 5A 5B1 5B2 5D 5A 15A1 15A-1 10B1 10B1 10B3 10B3
3 P 1A 2A 2B 2C 1A 4A-1 4A1 4B-1 4B1 5A 5B2 5B1 5C 5D 5E 2C 10A3 10A1 10B3 10B1 10C 10D 5C 5C 20A3 20A-3 20A1 20A-1
5 P 1A 2A 2B 2C 3A 4A1 4A-1 4B1 4B-1 1A 1A 1A 1A 1A 1A 6A 2A 2A 2A 2A 2A 2B 3A 3A 4A1 4A-1 4A-1 4A1
Type
6000.bm.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
6000.bm.1b R 1 1 1 1 1 −1 −1 −1 −1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1
6000.bm.1c1 C 1 −1 1 −1 1 −i i i −i 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 1 1 1 i −i −i i
6000.bm.1c2 C 1 −1 1 −1 1 i −i −i i 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 1 1 1 −i i i −i
6000.bm.2a R 2 2 2 2 −1 0 0 0 0 2 2 2 2 2 2 −1 2 2 2 2 2 2 −1 −1 0 0 0 0
6000.bm.2b S 2 −2 2 −2 −1 0 0 0 0 2 2 2 2 2 2 1 −2 −2 −2 −2 −2 2 −1 −1 0 0 0 0
6000.bm.3a R 3 −1 −1 3 0 −1 −1 1 1 3 3 3 3 3 3 0 −1 −1 −1 −1 −1 −1 0 0 −1 −1 −1 −1
6000.bm.3b R 3 −1 −1 3 0 1 1 −1 −1 3 3 3 3 3 3 0 −1 −1 −1 −1 −1 −1 0 0 1 1 1 1
6000.bm.3c1 C 3 1 −1 −3 0 −i i −i i 3 3 3 3 3 3 0 1 1 1 1 1 −1 0 0 i −i −i i
6000.bm.3c2 C 3 1 −1 −3 0 i −i i −i 3 3 3 3 3 3 0 1 1 1 1 1 −1 0 0 −i i i −i
6000.bm.12a R 12 8 4 0 0 0 0 0 0 7 2 2 −3 2 −3 0 3 3 −2 −2 −2 −1 0 0 0 0 0 0
6000.bm.12b R 12 −8 4 0 0 0 0 0 0 7 2 2 −3 2 −3 0 −3 −3 2 2 2 −1 0 0 0 0 0 0
6000.bm.12c1 R 12 4 0 0 0 2 2 0 0 2 5ζ5−2+2+5ζ52 −5ζ5−2−3−5ζ52 −3 −3 2 0 2ζ5−2+2ζ52 2ζ5−1+2ζ5 −ζ5−2+1−ζ52 ζ5−2+2+ζ52 −1 0 0 0 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52
6000.bm.12c2 R 12 4 0 0 0 2 2 0 0 2 −5ζ5−2−3−5ζ52 5ζ5−2+2+5ζ52 −3 −3 2 0 2ζ5−1+2ζ5 2ζ5−2+2ζ52 ζ5−2+2+ζ52 −ζ5−2+1−ζ52 −1 0 0 0 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5
6000.bm.12d1 R 12 0 −4 0 0 0 0 0 0 7 2 2 −3 2 −3 0 2ζ5−2+1+2ζ52 −2ζ5−2−1−2ζ52 4ζ5−2+2+4ζ52 −4ζ5−2−2−4ζ52 0 1 0 0 0 0 0 0
6000.bm.12d2 R 12 0 −4 0 0 0 0 0 0 7 2 2 −3 2 −3 0 −2ζ5−2−1−2ζ52 2ζ5−2+1+2ζ52 −4ζ5−2−2−4ζ52 4ζ5−2+2+4ζ52 0 1 0 0 0 0 0 0
6000.bm.12e1 R 12 4 0 0 0 −2 −2 0 0 2 5ζ5−2+2+5ζ52 −5ζ5−2−3−5ζ52 −3 −3 2 0 2ζ5−2+2ζ52 2ζ5−1+2ζ5 −ζ5−2+1−ζ52 ζ5−2+2+ζ52 −1 0 0 0 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52
6000.bm.12e2 R 12 4 0 0 0 −2 −2 0 0 2 −5ζ5−2−3−5ζ52 5ζ5−2+2+5ζ52 −3 −3 2 0 2ζ5−1+2ζ5 2ζ5−2+2ζ52 ζ5−2+2+ζ52 −ζ5−2+1−ζ52 −1 0 0 0 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5
6000.bm.12f1 C 12 −4 0 0 0 −2ζ205 2ζ205 0 0 2 −3−5ζ204+5ζ206 2+5ζ204−5ζ206 −3 −3 2 0 2ζ20−2+2ζ202 −2ζ20−4−2ζ204 −2−ζ204+ζ206 −1+ζ204−ζ206 1 0 0 0 ζ203−ζ205+ζ207 −ζ203+ζ205−ζ207 ζ203+ζ207 −ζ203−ζ207
6000.bm.12f2 C 12 −4 0 0 0 2ζ205 −2ζ205 0 0 2 −3−5ζ204+5ζ206 2+5ζ204−5ζ206 −3 −3 2 0 2ζ20−2+2ζ202 −2ζ20−4−2ζ204 −2−ζ204+ζ206 −1+ζ204−ζ206 1 0 0 0 −ζ203+ζ205−ζ207 ζ203−ζ205+ζ207 −ζ203−ζ207 ζ203+ζ207
6000.bm.12f3 C 12 −4 0 0 0 −2ζ205 2ζ205 0 0 2 2+5ζ204−5ζ206 −3−5ζ204+5ζ206 −3 −3 2 0 −2ζ20−4−2ζ204 2ζ20−2+2ζ202 −1+ζ204−ζ206 −2−ζ204+ζ206 1 0 0 0 −ζ203−ζ207 ζ203+ζ207 −ζ203+ζ205−ζ207 ζ203−ζ205+ζ207
6000.bm.12f4 C 12 −4 0 0 0 2ζ205 −2ζ205 0 0 2 2+5ζ204−5ζ206 −3−5ζ204+5ζ206 −3 −3 2 0 −2ζ20−4−2ζ204 2ζ20−2+2ζ202 −1+ζ204−ζ206 −2−ζ204+ζ206 1 0 0 0 ζ203+ζ207 −ζ203−ζ207 ζ203−ζ205+ζ207 −ζ203+ζ205−ζ207
6000.bm.16a R 16 0 0 0 4 0 0 0 0 −4 −4 −4 −4 6 1 0 0 0 0 0 0 0 −1 −1 0 0 0 0
6000.bm.16b1 C 16 0 0 0 −2 0 0 0 0 −4 −4 −4 −4 6 1 0 0 0 0 0 0 0 2−2ζ15−ζ152+ζ153−2ζ154+ζ155−ζ157 −1+2ζ15+ζ152−ζ153+2ζ154−ζ155+ζ157 0 0 0 0
6000.bm.16b2 C 16 0 0 0 −2 0 0 0 0 −4 −4 −4 −4 6 1 0 0 0 0 0 0 0 −1+2ζ15+ζ152−ζ153+2ζ154−ζ155+ζ157 2−2ζ15−ζ152+ζ153−2ζ154+ζ155−ζ157 0 0 0 0
6000.bm.24a R 24 8 0 0 0 0 0 0 0 4 −6 −6 9 −1 −1 0 −2 −2 −2 −2 3 0 0 0 0 0 0 0
6000.bm.24b R 24 −8 0 0 0 0 0 0 0 4 −6 −6 9 −1 −1 0 2 2 2 2 −3 0 0 0 0 0 0 0
6000.bm.48a R 48 0 0 0 0 0 0 0 0 −12 8 8 3 −2 −2 0 0 0 0 0 0 0 0 0 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed