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Properties

Label 24T2199
24T2199 1 12 1->12 24 1->24 2 11 2->11 23 2->23 3 10 3->10 21 3->21 4 9 4->9 22 4->22 5 15 5->15 20 5->20 6 16 6->16 19 6->19 7 13 7->13 17 7->17 8 14 8->14 18 8->18 9->3 9->8 10->4 10->7 11->5 12->6 13->4 13->8 14->3 14->7 15->1 16->2 17->16 17->19 18->15 18->20 19->13 19->18 20->14 20->17 21->11 21->24 22->12 22->23 23->10 23->21 24->9 24->22
Degree $24$
Order $768$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^5:S_4$

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Copy content magma:G := TransitiveGroup(24, 2199);
 

Group invariants

Abstract group:  $C_2^5:S_4$
Copy content magma:IdentifyGroup(G);
 
Order:  $768=2^{8} \cdot 3$
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 $n$:  $24$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $2199$
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:  $(1,12)(2,11)(3,10,4,9)(5,15)(6,16)(7,13,8,14)(17,19,18,20)(21,24,22,23)$, $(1,24,9,8,18,15)(2,23,10,7,17,16)(3,21,11,5,20,14)(4,22,12,6,19,13)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$
$12$:  $D_{6}$
$24$:  $S_4$ x 3
$48$:  $S_4\times C_2$ x 3
$96$:  $V_4^2:S_3$
$192$:  $V_4^2:(S_3\times C_2)$ x 2, 12T100

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: $S_3$

Degree 4: $C_2^2$

Degree 6: $S_3$, $D_{6}$ x 2

Degree 8: None

Degree 12: $D_6$

Low degree siblings

16T1068 x 2, 24T1632 x 2, 24T1639 x 4, 24T2199, 24T2248 x 2, 24T2355 x 2, 24T2357 x 4, 24T2365 x 2, 24T2373 x 4, 24T2378 x 2, 24T2382 x 2, 24T2392 x 2, 24T2405 x 8, 24T2406 x 4, 24T2407 x 8, 24T2408 x 4, 24T2417 x 4, 24T2419 x 2, 24T2421 x 4, 24T2423 x 2, 24T2426 x 2, 24T2429 x 2, 32T34622 x 4, 32T34628

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{24}$ $1$ $1$ $0$ $()$
2A $2^{8},1^{8}$ $3$ $2$ $8$ $( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)$
2B $2^{8},1^{8}$ $3$ $2$ $8$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(15,16)(19,20)(21,22)$
2C $2^{8},1^{8}$ $3$ $2$ $8$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)(11,12)(13,14)(17,18)(23,24)$
2D $2^{12}$ $4$ $2$ $12$ $( 1, 7)( 2, 8)( 3, 6)( 4, 5)( 9,16)(10,15)(11,13)(12,14)(17,23)(18,24)(19,22)(20,21)$
2E $2^{4},1^{16}$ $6$ $2$ $4$ $( 1, 2)( 7, 8)( 9,10)(15,16)$
2F $2^{8},1^{8}$ $6$ $2$ $8$ $( 3, 4)( 7, 8)( 9,10)(13,14)(17,18)(19,20)(21,22)(23,24)$
2G $2^{4},1^{16}$ $6$ $2$ $4$ $( 1, 2)( 5, 6)(17,18)(21,22)$
2H $2^{4},1^{16}$ $6$ $2$ $4$ $( 5, 6)( 7, 8)(13,14)(15,16)$
2I $2^{8},1^{8}$ $6$ $2$ $8$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(21,22)(23,24)$
2J $2^{6},1^{12}$ $12$ $2$ $6$ $( 1, 2)( 7, 8)( 9,10)(13,14)(21,22)(23,24)$
2K $2^{12}$ $12$ $2$ $12$ $( 1, 8)( 2, 7)( 3, 5)( 4, 6)( 9,15)(10,16)(11,13)(12,14)(17,23)(18,24)(19,21)(20,22)$
2L $2^{6},1^{12}$ $12$ $2$ $6$ $( 5, 6)( 7, 8)(11,12)(13,14)(19,20)(23,24)$
2M $2^{12}$ $24$ $2$ $12$ $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,20)(10,19)(11,18)(12,17)(13,24)(14,23)(15,22)(16,21)$
2N $2^{12}$ $24$ $2$ $12$ $( 1, 5)( 2, 6)( 3, 8)( 4, 7)( 9,21)(10,22)(11,24)(12,23)(13,18)(14,17)(15,19)(16,20)$
3A $3^{8}$ $128$ $3$ $16$ $( 1,10,18)( 2, 9,17)( 3,11,20)( 4,12,19)( 5,14,22)( 6,13,21)( 7,15,24)( 8,16,23)$
4A $4^{4},2^{4}$ $12$ $4$ $16$ $( 1, 7)( 2, 8)( 3, 6)( 4, 5)( 9,15,10,16)(11,14,12,13)(17,24,18,23)(19,21,20,22)$
4B $4^{4},2^{4}$ $12$ $4$ $16$ $( 1, 8)( 2, 7)( 3, 5)( 4, 6)( 9,16,10,15)(11,14,12,13)(17,24,18,23)(19,22,20,21)$
4C $4^{4},2^{4}$ $24$ $4$ $16$ $( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,19,10,20)(11,17)(12,18)(13,23)(14,24)(15,21,16,22)$
4D $4^{4},2^{4}$ $24$ $4$ $16$ $( 1, 5, 2, 6)( 3, 7, 4, 8)( 9,22,10,21)(11,23)(12,24)(13,17)(14,18)(15,20,16,19)$
4E $4^{4},2^{4}$ $24$ $4$ $16$ $( 1, 8)( 2, 7)( 3, 6)( 4, 5)( 9,15,10,16)(11,13,12,14)(17,23,18,24)(19,22,20,21)$
4F $4^{4},2^{4}$ $24$ $4$ $16$ $( 1, 5)( 2, 6)( 3, 8)( 4, 7)( 9,22,10,21)(11,23,12,24)(13,18,14,17)(15,19,16,20)$
4G $4^{4},2^{4}$ $24$ $4$ $16$ $( 1, 5, 2, 6)( 3, 7, 4, 8)( 9,21)(10,22)(11,24,12,23)(13,17,14,18)(15,20)(16,19)$
4H $4^{4},2^{4}$ $24$ $4$ $16$ $( 1,11, 2,12)( 3,10, 4, 9)( 5,16, 6,15)( 7,13, 8,14)(17,20)(18,19)(21,23)(22,24)$
4I $4^{4},2^{4}$ $24$ $4$ $16$ $( 1,11)( 2,12)( 3, 9, 4,10)( 5,15, 6,16)( 7,13)( 8,14)(17,19,18,20)(21,23,22,24)$
4J $4^{4},2^{4}$ $48$ $4$ $16$ $( 1,11)( 2,12)( 3,10, 4, 9)( 5,15)( 6,16)( 7,14, 8,13)(17,20,18,19)(21,23,22,24)$
4K $4^{2},2^{8}$ $48$ $4$ $14$ $( 1,21, 2,22)( 3,23)( 4,24)( 5,18, 6,17)( 7,20)( 8,19)( 9,14)(10,13)(11,16)(12,15)$
4L $4^{2},2^{8}$ $48$ $4$ $14$ $( 1,12)( 2,11)( 3,10)( 4, 9)( 5,16, 6,15)( 7,14, 8,13)(17,20)(18,19)(21,24)(22,23)$
4M $4^{4},2^{4}$ $48$ $4$ $16$ $( 1, 5, 2, 6)( 3, 8, 4, 7)( 9,21,10,22)(11,24,12,23)(13,18)(14,17)(15,19)(16,20)$
6A $6^{4}$ $128$ $6$ $20$ $( 1,24,10, 7,18,15)( 2,23, 9, 8,17,16)( 3,21,11, 6,20,13)( 4,22,12, 5,19,14)$

Malle's constant $a(G)$:     $1/4$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 2N 3A 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 6A
Size 1 3 3 3 4 6 6 6 6 6 12 12 12 24 24 128 12 12 24 24 24 24 24 24 24 48 48 48 48 128
2 P 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 3A 2A 2A 2B 2B 2A 2A 2C 2A 2C 2F 2G 2H 2I 3A
3 P 1A 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 2N 1A 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 2D
Type
768.1090070.1a 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 1 1
768.1090070.1b 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 1 −1
768.1090070.1c 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 −1 −1
768.1090070.1d 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 −1 1
768.1090070.2a R 2 2 2 2 2 2 2 2 2 2 2 2 2 0 0 −1 2 2 0 0 2 0 0 0 0 0 0 0 0 −1
768.1090070.2b R 2 2 2 2 −2 2 2 2 2 2 2 −2 2 0 0 −1 −2 −2 0 0 −2 0 0 0 0 0 0 0 0 1
768.1090070.3a R 3 3 −1 −1 3 −1 −1 −1 3 3 −1 −1 −1 1 1 0 3 −1 −1 −1 −1 −1 1 −1 1 −1 −1 1 1 0
768.1090070.3b R 3 3 −1 −1 3 −1 3 3 −1 −1 −1 −1 −1 1 1 0 −1 3 −1 −1 −1 −1 1 −1 1 1 1 −1 −1 0
768.1090070.3c R 3 3 3 3 3 3 −1 −1 −1 −1 −1 3 −1 1 1 0 −1 −1 1 1 −1 1 1 1 1 −1 −1 −1 −1 0
768.1090070.3d R 3 3 −1 −1 3 −1 −1 −1 3 3 −1 −1 −1 −1 −1 0 3 −1 1 1 −1 1 −1 1 −1 1 1 −1 −1 0
768.1090070.3e R 3 3 −1 −1 3 −1 3 3 −1 −1 −1 −1 −1 −1 −1 0 −1 3 1 1 −1 1 −1 1 −1 −1 −1 1 1 0
768.1090070.3f R 3 3 3 3 3 3 −1 −1 −1 −1 −1 3 −1 −1 −1 0 −1 −1 −1 −1 −1 −1 −1 −1 −1 1 1 1 1 0
768.1090070.3g R 3 3 −1 −1 −3 −1 −1 −1 3 3 −1 1 −1 −1 1 0 −3 1 −1 1 1 −1 −1 1 1 1 −1 −1 1 0
768.1090070.3h R 3 3 −1 −1 −3 −1 −1 −1 3 3 −1 1 −1 1 −1 0 −3 1 1 −1 1 1 1 −1 −1 −1 1 1 −1 0
768.1090070.3i R 3 3 −1 −1 −3 −1 3 3 −1 −1 −1 1 −1 −1 1 0 1 −3 −1 1 1 −1 −1 1 1 −1 1 1 −1 0
768.1090070.3j R 3 3 −1 −1 −3 −1 3 3 −1 −1 −1 1 −1 1 −1 0 1 −3 1 −1 1 1 1 −1 −1 1 −1 −1 1 0
768.1090070.3k R 3 3 3 3 −3 3 −1 −1 −1 −1 −1 −3 −1 −1 1 0 1 1 1 −1 1 1 −1 −1 1 1 −1 1 −1 0
768.1090070.3l R 3 3 3 3 −3 3 −1 −1 −1 −1 −1 −3 −1 1 −1 0 1 1 −1 1 1 −1 1 1 −1 −1 1 −1 1 0
768.1090070.6a R 6 6 −2 −2 6 −2 −2 −2 −2 −2 2 −2 2 0 0 0 −2 −2 0 0 2 0 0 0 0 0 0 0 0 0
768.1090070.6b R 6 −2 −2 6 0 −2 −2 2 −2 2 2 0 −2 0 2 0 0 0 2 0 0 −2 0 0 −2 0 0 0 0 0
768.1090070.6c R 6 −2 −2 6 0 −2 2 −2 2 −2 −2 0 2 2 0 0 0 0 0 2 0 0 −2 −2 0 0 0 0 0 0
768.1090070.6d R 6 −2 6 −2 0 −2 −2 2 −2 2 −2 0 2 0 2 0 0 0 −2 0 0 2 0 0 −2 0 0 0 0 0
768.1090070.6e R 6 −2 6 −2 0 −2 2 −2 2 −2 2 0 −2 2 0 0 0 0 0 −2 0 0 −2 2 0 0 0 0 0 0
768.1090070.6f R 6 6 −2 −2 −6 −2 −2 −2 −2 −2 2 2 2 0 0 0 2 2 0 0 −2 0 0 0 0 0 0 0 0 0
768.1090070.6g R 6 −2 −2 6 0 −2 −2 2 −2 2 2 0 −2 0 −2 0 0 0 −2 0 0 2 0 0 2 0 0 0 0 0
768.1090070.6h R 6 −2 −2 6 0 −2 2 −2 2 −2 −2 0 2 −2 0 0 0 0 0 −2 0 0 2 2 0 0 0 0 0 0
768.1090070.6i R 6 −2 6 −2 0 −2 −2 2 −2 2 −2 0 2 0 −2 0 0 0 2 0 0 −2 0 0 2 0 0 0 0 0
768.1090070.6j R 6 −2 6 −2 0 −2 2 −2 2 −2 2 0 −2 −2 0 0 0 0 0 2 0 0 2 −2 0 0 0 0 0 0
768.1090070.12a R 12 −4 −4 −4 0 4 −4 4 4 −4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
768.1090070.12b R 12 −4 −4 −4 0 4 4 −4 −4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed