↑

Properties

Label 30T188
30T188 1 17 1->17 25 1->25 2 18 2->18 26 2->26 3 3->26 4 4->25 5 5->4 29 5->29 6 6->3 30 6->30 7 7->2 12 7->12 8 8->1 11 8->11 9 20 9->20 10 19 10->19 11->6 27 11->27 12->5 28 12->28 13 13->5 13->7 14 14->6 14->8 15 15->14 16 16->13 17->11 22 17->22 18->12 21 18->21 19->13 19->29 20->14 20->30 21->8 22->7 23 23->16 23->17 24 24->15 24->18 25->20 25->23 26->19 26->24 27->2 28->1 29->9 29->24 30->10 30->23
Degree $30$
Order $750$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_5^3:C_6$

Related objects

Downloads

Learn more

Show commands: Magma

Copy content magma:G := TransitiveGroup(30, 188);
 

Group invariants

Abstract group:  $C_5^3:C_6$
Copy content magma:IdentifyGroup(G);
 
Order:  $750=2 \cdot 3 \cdot 5^{3}$
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 $t$:  $188$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $10$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,17,22,7,12,28)(2,18,21,8,11,27)(3,26,24,15,14,6)(4,25,23,16,13,5)(9,20,30,10,19,29)$, $(1,25,20,14,8)(2,26,19,13,7)(5,29,24,18,12)(6,30,23,17,11)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$10$:  $D_{5}$
$30$:  $D_5\times C_3$
$150$:  $(C_5^2 : C_3):C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 5: None

Degree 6: $C_6$

Degree 10: None

Degree 15: 15T30

Low degree siblings

15T30 x 8, 30T188 x 7

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 $2^{15}$ $125$ $2$ $15$ $( 1,13)( 2,14)( 3,21)( 4,22)( 5,11)( 6,12)( 7, 8)( 9,16)(10,15)(17,29)(18,30)(19,25)(20,26)(23,24)(27,28)$
3A1 $3^{10}$ $25$ $3$ $20$ $( 1,12,22)( 2,11,21)( 3,14,24)( 4,13,23)( 5,16,25)( 6,15,26)( 7,17,28)( 8,18,27)( 9,19,30)(10,20,29)$
3A-1 $3^{10}$ $25$ $3$ $20$ $( 1,22,12)( 2,21,11)( 3,24,14)( 4,23,13)( 5,25,16)( 6,26,15)( 7,28,17)( 8,27,18)( 9,30,19)(10,29,20)$
5A1 $5^{6}$ $2$ $5$ $24$ $( 1,20, 8,25,14)( 2,19, 7,26,13)( 3,22,10,27,16)( 4,21, 9,28,15)( 5,24,12,29,18)( 6,23,11,30,17)$
5A2 $5^{6}$ $2$ $5$ $24$ $( 1, 8,14,20,25)( 2, 7,13,19,26)( 3,10,16,22,27)( 4, 9,15,21,28)( 5,12,18,24,29)( 6,11,17,23,30)$
5B1 $5^{4},1^{10}$ $6$ $5$ $16$ $( 1,20, 8,25,14)( 2,19, 7,26,13)( 3,16,27,10,22)( 4,15,28, 9,21)$
5B2 $5^{4},1^{10}$ $6$ $5$ $16$ $( 3,10,16,22,27)( 4, 9,15,21,28)( 5,29,24,18,12)( 6,30,23,17,11)$
5C1 $5^{6}$ $6$ $5$ $24$ $( 1,20, 8,25,14)( 2,19, 7,26,13)( 3,22,10,27,16)( 4,21, 9,28,15)( 5,29,24,18,12)( 6,30,23,17,11)$
5C2 $5^{6}$ $6$ $5$ $24$ $( 1,25,20,14, 8)( 2,26,19,13, 7)( 3,16,27,10,22)( 4,15,28, 9,21)( 5,29,24,18,12)( 6,30,23,17,11)$
5D1 $5^{6}$ $6$ $5$ $24$ $( 1,25,20,14, 8)( 2,26,19,13, 7)( 3,22,10,27,16)( 4,21, 9,28,15)( 5,12,18,24,29)( 6,11,17,23,30)$
5D2 $5^{6}$ $6$ $5$ $24$ $( 1,14,25, 8,20)( 2,13,26, 7,19)( 3,22,10,27,16)( 4,21, 9,28,15)( 5,12,18,24,29)( 6,11,17,23,30)$
5E1 $5^{4},1^{10}$ $6$ $5$ $16$ $( 1, 8,14,20,25)( 2, 7,13,19,26)( 3,16,27,10,22)( 4,15,28, 9,21)$
5E2 $5^{4},1^{10}$ $6$ $5$ $16$ $( 1,14,25, 8,20)( 2,13,26, 7,19)( 3,27,22,16,10)( 4,28,21,15, 9)$
5F1 $5^{6}$ $6$ $5$ $24$ $( 1, 8,14,20,25)( 2, 7,13,19,26)( 3,22,10,27,16)( 4,21, 9,28,15)( 5,29,24,18,12)( 6,30,23,17,11)$
5F2 $5^{6}$ $6$ $5$ $24$ $( 1,20, 8,25,14)( 2,19, 7,26,13)( 3,27,22,16,10)( 4,28,21,15, 9)( 5,18,29,12,24)( 6,17,30,11,23)$
5G1 $5^{4},1^{10}$ $6$ $5$ $16$ $( 3,27,22,16,10)( 4,28,21,15, 9)( 5,18,29,12,24)( 6,17,30,11,23)$
5G2 $5^{4},1^{10}$ $6$ $5$ $16$ $( 3,16,27,10,22)( 4,15,28, 9,21)( 5,12,18,24,29)( 6,11,17,23,30)$
5H1 $5^{4},1^{10}$ $6$ $5$ $16$ $( 1,25,20,14, 8)( 2,26,19,13, 7)( 3,27,22,16,10)( 4,28,21,15, 9)$
5H2 $5^{4},1^{10}$ $6$ $5$ $16$ $( 1,20, 8,25,14)( 2,19, 7,26,13)( 3,22,10,27,16)( 4,21, 9,28,15)$
5I1 $5^{2},1^{20}$ $6$ $5$ $8$ $( 5,12,18,24,29)( 6,11,17,23,30)$
5I2 $5^{2},1^{20}$ $6$ $5$ $8$ $( 3,22,10,27,16)( 4,21, 9,28,15)$
5J1 $5^{6}$ $6$ $5$ $24$ $( 1,14,25, 8,20)( 2,13,26, 7,19)( 3,16,27,10,22)( 4,15,28, 9,21)( 5,29,24,18,12)( 6,30,23,17,11)$
5J2 $5^{6}$ $6$ $5$ $24$ $( 1, 8,14,20,25)( 2, 7,13,19,26)( 3,10,16,22,27)( 4, 9,15,21,28)( 5,18,29,12,24)( 6,17,30,11,23)$
5K1 $5^{6}$ $6$ $5$ $24$ $( 1,14,25, 8,20)( 2,13,26, 7,19)( 3,22,10,27,16)( 4,21, 9,28,15)( 5,24,12,29,18)( 6,23,11,30,17)$
5K2 $5^{6}$ $6$ $5$ $24$ $( 1, 8,14,20,25)( 2, 7,13,19,26)( 3,27,22,16,10)( 4,28,21,15, 9)( 5,12,18,24,29)( 6,11,17,23,30)$
6A1 $6^{5}$ $125$ $6$ $25$ $( 1,30,22,13,18, 4)( 2,29,21,14,17, 3)( 5,15,20,11,10,26)( 6,16,19,12, 9,25)( 7,24,28, 8,23,27)$
6A-1 $6^{5}$ $125$ $6$ $25$ $( 1, 4,18,13,22,30)( 2, 3,17,14,21,29)( 5,26,10,11,20,15)( 6,25, 9,12,19,16)( 7,27,23, 8,28,24)$
15A1 $15^{2}$ $50$ $15$ $28$ $( 1,27,24,20,16,12, 8, 3,29,25,22,18,14,10, 5)( 2,28,23,19,15,11, 7, 4,30,26,21,17,13, 9, 6)$
15A-1 $15^{2}$ $50$ $15$ $28$ $( 1, 5,10,14,18,22,25,29, 3, 8,12,16,20,24,27)( 2, 6, 9,13,17,21,26,30, 4, 7,11,15,19,23,28)$
15A2 $15^{2}$ $50$ $15$ $28$ $( 1,24,16, 8,29,22,14, 5,27,20,12, 3,25,18,10)( 2,23,15, 7,30,21,13, 6,28,19,11, 4,26,17, 9)$
15A-2 $15^{2}$ $50$ $15$ $28$ $( 1,10,18,25, 3,12,20,27, 5,14,22,29, 8,16,24)( 2, 9,17,26, 4,11,19,28, 6,13,21,30, 7,15,23)$

Malle's constant $a(G)$:     $1/8$

Copy content magma:ConjugacyClasses(G);
 

Character table

32 x 32 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed