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Properties

Label 30T624
30T624 1 11 1->11 28 1->28 2 12 2->12 27 2->27 3 19 3->19 25 3->25 4 20 4->20 26 4->26 5 15 5->15 6 16 6->16 7 7->5 7->16 8 8->6 8->15 9 9->1 9->1 10 10->2 10->2 11->9 17 11->17 12->10 18 12->18 13 13->4 29 13->29 14 14->3 30 14->30 15->7 15->13 16->8 16->14 17->3 17->29 18->4 18->30 22 19->22 23 19->23 21 20->21 24 20->24 21->13 21->25 22->14 22->26 23->11 23->27 24->12 24->28 25->10 25->17 26->9 26->18 27->7 27->19 28->8 28->20 29->21 29->23 30->22 30->24
Degree $30$
Order $6000$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_5^3.S_3$

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Copy content magma:G := TransitiveGroup(30, 624);
 

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 $t$:  $624$
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:  $(1,28,8,15,13,4,20,21,25,10,2,27,7,16,14,3,19,22,26,9)(11,17,29,23)(12,18,30,24)$, $(1,11,9)(2,12,10)(3,25,17)(4,26,18)(5,15,7)(6,16,8)(13,29,21)(14,30,22)(19,23,27)(20,24,28)$
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: None

Degree 3: $S_3$

Degree 5: None

Degree 6: $S_4$

Degree 10: None

Degree 15: 15T58

Low degree siblings

15T58, 20T360, 30T591, 30T604, 30T607, 30T616, 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 $2^{14},1^{2}$ $15$ $2$ $14$ $( 1, 2)( 3, 4)( 5,29)( 6,30)( 7, 8)( 9,10)(11,23)(12,24)(13,14)(15,16)(19,20)(21,22)(25,26)(27,28)$
2B $2^{10},1^{10}$ $75$ $2$ $10$ $( 1, 2)( 3,28)( 4,27)( 7,26)( 8,25)( 9,22)(10,21)(13,20)(14,19)(15,16)$
2C $2^{12},1^{6}$ $125$ $2$ $12$ $( 1, 7)( 2, 8)( 5,23)( 6,24)( 9,27)(10,28)(11,17)(12,18)(13,25)(14,26)(15,21)(16,22)$
3A $3^{10}$ $200$ $3$ $20$ $( 1,22,12)( 2,21,11)( 3,29,20)( 4,30,19)( 5,26,27)( 6,25,28)( 7,16,18)( 8,15,17)( 9,23,14)(10,24,13)$
4A1 $4^{7},1^{2}$ $150$ $4$ $21$ $( 1,19,25, 7)( 2,20,26, 8)( 3,24, 4,23)( 5, 9, 6,10)(11,27,12,28)(15,18,16,17)(21,30,22,29)$
4A-1 $4^{7},1^{2}$ $150$ $4$ $21$ $( 1, 7,25,19)( 2, 8,26,20)( 3,23, 4,24)( 5,10, 6, 9)(11,28,12,27)(15,17,16,18)(21,29,22,30)$
4B1 $4^{6},2^{3}$ $750$ $4$ $21$ $( 1, 3,19,27)( 2, 4,20,28)( 5,12,23,18)( 6,11,24,17)( 7,21,13, 9)( 8,22,14,10)(15,25)(16,26)(29,30)$
4B-1 $4^{6},2^{3}$ $750$ $4$ $21$ $( 1,27,19, 3)( 2,28,20, 4)( 5,18,23,12)( 6,17,24,11)( 7, 9,13,21)( 8,10,14,22)(15,25)(16,26)(29,30)$
5A $5^{2},1^{20}$ $12$ $5$ $8$ $( 1,13,25, 7,19)( 2,14,26, 8,20)$
5B1 $5^{4},1^{10}$ $12$ $5$ $16$ $( 3,15,27, 9,21)( 4,16,28,10,22)( 5,29,23,17,11)( 6,30,24,18,12)$
5B2 $5^{4},1^{10}$ $12$ $5$ $16$ $( 3,27,21,15, 9)( 4,28,22,16,10)( 5,23,11,29,17)( 6,24,12,30,18)$
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 $5^{4},1^{10}$ $24$ $5$ $16$ $( 1, 7,13,19,25)( 2, 8,14,20,26)( 5,29,23,17,11)( 6,30,24,18,12)$
5E $5^{6}$ $48$ $5$ $24$ $( 1,13,25, 7,19)( 2,14,26, 8,20)( 3, 9,15,21,27)( 4,10,16,22,28)( 5,29,23,17,11)( 6,30,24,18,12)$
6A $6^{4},3^{2}$ $1000$ $6$ $24$ $( 1,18,22, 7,12,16)( 2,17,21, 8,11,15)( 3,20,29)( 4,19,30)( 5, 9,26,23,27,14)( 6,10,25,24,28,13)$
10A1 $10,2^{9},1^{2}$ $60$ $10$ $18$ $( 1, 8,13,20,25, 2, 7,14,19,26)( 3, 4)( 5,29)( 6,30)( 9,10)(11,23)(12,24)(15,16)(21,22)(27,28)$
10A3 $10,2^{9},1^{2}$ $60$ $10$ $18$ $( 1, 8,13,20,25, 2, 7,14,19,26)( 3,21)( 4,22)( 5, 6)( 9,15)(10,16)(11,12)(17,18)(23,24)(29,30)$
10B1 $10^{2},2^{4},1^{2}$ $60$ $10$ $22$ $( 1,25)( 2,26)( 3,10,15,22,27, 4, 9,16,21,28)( 5,18,29,12,23, 6,17,30,11,24)( 7,19)( 8,20)$
10B3 $10^{2},2^{4},1^{2}$ $60$ $10$ $22$ $( 1,25)( 2,26)( 3,22, 9,28,15, 4,21,10,27,16)( 5,12,17,24,29, 6,11,18,23,30)( 7,19)( 8,20)$
10C $10^{2},2^{4},1^{2}$ $120$ $10$ $22$ $( 1,20, 7,26,13, 2,19, 8,25,14)( 3,27)( 4,28)( 5,18,29,12,23, 6,17,30,11,24)( 9,21)(10,22)$
10D $5^{2},2^{10}$ $300$ $10$ $18$ $( 1, 2)( 3,28)( 4,27)( 5,17,29,11,23)( 6,18,30,12,24)( 7,26)( 8,25)( 9,22)(10,21)(13,20)(14,19)(15,16)$
15A1 $15^{2}$ $400$ $15$ $28$ $( 1,30,28,13,12,16,25,24, 4, 7, 6,22,19,18,10)( 2,29,27,14,11,15,26,23, 3, 8, 5,21,20,17, 9)$
15A-1 $15^{2}$ $400$ $15$ $28$ $( 1,10,18,19,22, 6, 7, 4,24,25,16,12,13,28,30)( 2, 9,17,20,21, 5, 8, 3,23,26,15,11,14,27,29)$
20A1 $20,4^{2},1^{2}$ $300$ $20$ $25$ $( 1,19,25, 7)( 2,20,26, 8)( 3,30,10,11,15,24,22, 5,27,18, 4,29, 9,12,16,23,21, 6,28,17)$
20A-1 $20,4^{2},1^{2}$ $300$ $20$ $25$ $( 1, 7,25,19)( 2, 8,26,20)( 3,17,28, 6,21,23,16,12, 9,29, 4,18,27, 5,22,24,15,11,10,30)$
20A3 $20,4^{2},1^{2}$ $300$ $20$ $25$ $( 1, 7,25,19)( 2, 8,26,20)( 3,11,22,18, 9,23,28,30,15, 5, 4,12,21,17,10,24,27,29,16, 6)$
20A-3 $20,4^{2},1^{2}$ $300$ $20$ $25$ $( 1,19,25, 7)( 2,20,26, 8)( 3, 6,16,29,27,24,10,17,21,12, 4, 5,15,30,28,23, 9,18,22,11)$

Malle's constant $a(G)$:     $1/8$

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