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Properties

Label 28T24
28T24 1 2 1->2 13 1->13 16 1->16 14 2->14 15 2->15 3 4 3->4 3->13 24 3->24 4->14 23 4->23 5 6 5->6 11 5->11 12 6->12 7 8 7->8 9 7->9 7->16 10 8->10 8->15 9->10 26 9->26 25 10->25 11->7 11->12 12->8 13->14 18 13->18 17 14->17 15->16 28 15->28 27 16->27 17->10 17->18 17->28 18->9 18->27 19 20 19->20 19->26 20->25 21 21->1 22 21->22 21->24 22->2 22->23 23->12 23->24 24->11 25->22 25->26 26->21 27->3 27->28 28->4
Degree $28$
Order $168$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^2\times F_7$

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Copy content magma:G := TransitiveGroup(28, 24);
 

Group invariants

Abstract group:  $C_2^2\times F_7$
Copy content magma:IdentifyGroup(G);
 
Order:  $168=2^{3} \cdot 3 \cdot 7$
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 $n$:  $28$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $24$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $4$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,13,18,9,26,21)(2,14,17,10,25,22)(3,24,11,7,16,27)(4,23,12,8,15,28)$, $(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)$, $(1,16)(2,15)(3,13)(4,14)(5,11)(6,12)(7,9)(8,10)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 7
$3$:  $C_3$
$4$:  $C_2^2$ x 7
$6$:  $C_6$ x 7
$8$:  $C_2^3$
$12$:  $C_6\times C_2$ x 7
$24$:  24T3
$42$:  $F_7$
$84$:  $F_7 \times C_2$ x 3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 7: $F_7$

Degree 14: $F_7 \times C_2$ x 3

Low degree siblings

28T24 x 3

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{28}$ $1$ $1$ $0$ $()$
2A $2^{14}$ $1$ $2$ $14$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)$
2B $2^{14}$ $1$ $2$ $14$ $( 1,16)( 2,15)( 3,18)( 4,17)( 5,19)( 6,20)( 7,21)( 8,22)( 9,24)(10,23)(11,26)(12,25)(13,27)(14,28)$
2C $2^{14}$ $1$ $2$ $14$ $( 1,15)( 2,16)( 3,17)( 4,18)( 5,20)( 6,19)( 7,22)( 8,21)( 9,23)(10,24)(11,25)(12,26)(13,28)(14,27)$
2D $2^{14}$ $7$ $2$ $14$ $( 1,16)( 2,15)( 3,13)( 4,14)( 5,11)( 6,12)( 7, 9)( 8,10)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)$
2E $2^{14}$ $7$ $2$ $14$ $( 1,28)( 2,27)( 3,25)( 4,26)( 5,23)( 6,24)( 7,22)( 8,21)( 9,20)(10,19)(11,17)(12,18)(13,15)(14,16)$
2F $2^{12},1^{4}$ $7$ $2$ $12$ $( 3,27)( 4,28)( 5,26)( 6,25)( 7,24)( 8,23)( 9,21)(10,22)(11,19)(12,20)(13,18)(14,17)$
2G $2^{14}$ $7$ $2$ $14$ $( 1,14)( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(15,27)(16,28)(17,26)(18,25)(19,23)(20,24)(21,22)$
3A1 $3^{8},1^{4}$ $7$ $3$ $16$ $( 3,19,24)( 4,20,23)( 5, 9,18)( 6,10,17)( 7,27,11)( 8,28,12)(13,26,21)(14,25,22)$
3A-1 $3^{8},1^{4}$ $7$ $3$ $16$ $( 3,24,19)( 4,23,20)( 5,18, 9)( 6,17,10)( 7,11,27)( 8,12,28)(13,21,26)(14,22,25)$
6A1 $6^{4},2^{2}$ $7$ $6$ $22$ $( 1,16)( 2,15)( 3,21,19,13,24,26)( 4,22,20,14,23,25)( 5,27, 9,11,18, 7)( 6,28,10,12,17, 8)$
6A-1 $6^{4},2^{2}$ $7$ $6$ $22$ $( 1,16)( 2,15)( 3,26,24,13,19,21)( 4,25,23,14,20,22)( 5, 7,18,11, 9,27)( 6, 8,17,12,10,28)$
6B1 $6^{4},2^{2}$ $7$ $6$ $22$ $( 1,17,21, 2,18,22)( 3, 8,16, 4, 7,15)( 5,25, 9, 6,26,10)(11,23,19,12,24,20)(13,14)(27,28)$
6B-1 $6^{4},2^{2}$ $7$ $6$ $22$ $( 1,17,26, 2,18,25)( 3,12,16, 4,11,15)( 5, 6)( 7,28,24, 8,27,23)( 9,22,13,10,21,14)(19,20)$
6C1 $6^{4},2^{2}$ $7$ $6$ $22$ $( 1, 4, 9,28,26,20)( 2, 3,10,27,25,19)( 5,15,18,23,13,12)( 6,16,17,24,14,11)( 7,22)( 8,21)$
6C-1 $6^{4},2^{2}$ $7$ $6$ $22$ $( 1, 4,13, 8, 5,23)( 2, 3,14, 7, 6,24)( 9,15,18,28,21,20)(10,16,17,27,22,19)(11,25)(12,26)$
6D1 $6^{4},2^{2}$ $7$ $6$ $22$ $( 1,16)( 2,15)( 3, 5,24,18,19, 9)( 4, 6,23,17,20,10)( 7,13,11,21,27,26)( 8,14,12,22,28,25)$
6D-1 $6^{4},2^{2}$ $7$ $6$ $22$ $( 1,16)( 2,15)( 3, 9,19,18,24, 5)( 4,10,20,17,23, 6)( 7,26,27,21,11,13)( 8,25,28,22,12,14)$
6E1 $6^{4},1^{4}$ $7$ $6$ $20$ $( 3, 7,19,27,24,11)( 4, 8,20,28,23,12)( 5,13, 9,26,18,21)( 6,14,10,25,17,22)$
6E-1 $6^{4},1^{4}$ $7$ $6$ $20$ $( 3,11,24,27,19, 7)( 4,12,23,28,20, 8)( 5,21,18,26, 9,13)( 6,22,17,25,10,14)$
6F1 $6^{4},2^{2}$ $7$ $6$ $22$ $( 1, 4,21,15,18, 8)( 2, 3,22,16,17, 7)( 5,12, 9,20,26,23)( 6,11,10,19,25,24)(13,28)(14,27)$
6F-1 $6^{4},2^{2}$ $7$ $6$ $22$ $( 1, 4,26,15,18,12)( 2, 3,25,16,17,11)( 5,20)( 6,19)( 7,14,24,22,27,10)( 8,13,23,21,28, 9)$
6G1 $6^{4},2^{2}$ $7$ $6$ $22$ $( 1,17, 9,14,26, 6)( 2,18,10,13,25, 5)( 3,23,27,12,19,15)( 4,24,28,11,20,16)( 7, 8)(21,22)$
6G-1 $6^{4},2^{2}$ $7$ $6$ $22$ $( 1,17,13,22, 5,10)( 2,18,14,21, 6, 9)( 3,28, 7,20,24,15)( 4,27, 8,19,23,16)(11,12)(25,26)$
7A $7^{4}$ $6$ $7$ $24$ $( 1, 5, 9,13,18,21,26)( 2, 6,10,14,17,22,25)( 3, 7,11,16,19,24,27)( 4, 8,12,15,20,23,28)$
14A $14^{2}$ $6$ $14$ $26$ $( 1,17, 5,22, 9,25,13, 2,18, 6,21,10,26,14)( 3,20, 7,23,11,28,16, 4,19, 8,24,12,27,15)$
14B $14^{2}$ $6$ $14$ $26$ $( 1,19, 9,27,18, 7,26,16, 5,24,13, 3,21,11)( 2,20,10,28,17, 8,25,15, 6,23,14, 4,22,12)$
14C $14^{2}$ $6$ $14$ $26$ $( 1, 4, 5, 8, 9,12,13,15,18,20,21,23,26,28)( 2, 3, 6, 7,10,11,14,16,17,19,22,24,25,27)$

Malle's constant $a(G)$:     $1/12$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 2E 2F 2G 3A1 3A-1 6A1 6A-1 6B1 6B-1 6C1 6C-1 6D1 6D-1 6E1 6E-1 6F1 6F-1 6G1 6G-1 7A 14A 14B 14C
Size 1 1 1 1 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 6 6 6 6
2 P 1A 1A 1A 1A 1A 1A 1A 1A 3A-1 3A1 3A1 3A-1 3A-1 3A1 3A1 3A-1 3A-1 3A1 3A1 3A-1 3A-1 3A1 3A1 3A-1 7A 7A 7A 7A
3 P 1A 2A 2B 2C 2D 2E 2F 2G 1A 1A 2D 2D 2A 2A 2E 2E 2B 2B 2F 2F 2C 2C 2G 2G 7A 14A 14B 14C
7 P 1A 2A 2B 2C 2D 2E 2F 2G 3A1 3A-1 6A1 6A-1 6B1 6B-1 6C1 6C-1 6D1 6D-1 6E1 6E-1 6F1 6F-1 6G1 6G-1 1A 2A 2B 2C
Type
168.47.1a 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
168.47.1b 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
168.47.1c 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
168.47.1d 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
168.47.1e 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
168.47.1f 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
168.47.1g 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
168.47.1h 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
168.47.1i1 C 1 1 1 1 1 1 1 1 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3 ζ3 ζ3 ζ3−1 ζ3−1 ζ3−1 ζ3 ζ3−1 1 1 1 1
168.47.1i2 C 1 1 1 1 1 1 1 1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3−1 ζ3−1 ζ3−1 ζ3 ζ3 ζ3 ζ3−1 ζ3 1 1 1 1
168.47.1j1 C 1 −1 −1 1 −1 1 1 −1 ζ3−1 ζ3 −ζ3−1 −ζ3−1 −ζ3 ζ3−1 −ζ3 ζ3 ζ3 −ζ3 −ζ3 −ζ3−1 ζ3−1 −ζ3−1 ζ3 ζ3−1 1 −1 1 −1
168.47.1j2 C 1 −1 −1 1 −1 1 1 −1 ζ3 ζ3−1 −ζ3 −ζ3 −ζ3−1 ζ3 −ζ3−1 ζ3−1 ζ3−1 −ζ3−1 −ζ3−1 −ζ3 ζ3 −ζ3 ζ3−1 ζ3 1 −1 1 −1
168.47.1k1 C 1 −1 −1 1 1 −1 −1 1 ζ3−1 ζ3 −ζ3−1 ζ3−1 −ζ3 ζ3−1 −ζ3 −ζ3 ζ3 ζ3 ζ3 ζ3−1 −ζ3−1 −ζ3−1 −ζ3 −ζ3−1 1 −1 1 −1
168.47.1k2 C 1 −1 −1 1 1 −1 −1 1 ζ3 ζ3−1 −ζ3 ζ3 −ζ3−1 ζ3 −ζ3−1 −ζ3−1 ζ3−1 ζ3−1 ζ3−1 ζ3 −ζ3 −ζ3 −ζ3−1 −ζ3 1 −1 1 −1
168.47.1l1 C 1 −1 1 −1 −1 1 −1 1 ζ3−1 ζ3 −ζ3−1 ζ3−1 −ζ3 −ζ3−1 ζ3 −ζ3 −ζ3 ζ3 −ζ3 −ζ3−1 ζ3−1 ζ3−1 ζ3 −ζ3−1 1 1 −1 −1
168.47.1l2 C 1 −1 1 −1 −1 1 −1 1 ζ3 ζ3−1 −ζ3 ζ3 −ζ3−1 −ζ3 ζ3−1 −ζ3−1 −ζ3−1 ζ3−1 −ζ3−1 −ζ3 ζ3 ζ3 ζ3−1 −ζ3 1 1 −1 −1
168.47.1m1 C 1 −1 1 −1 1 −1 1 −1 ζ3−1 ζ3 −ζ3−1 −ζ3−1 −ζ3 −ζ3−1 ζ3 ζ3 −ζ3 −ζ3 ζ3 ζ3−1 −ζ3−1 ζ3−1 −ζ3 ζ3−1 1 1 −1 −1
168.47.1m2 C 1 −1 1 −1 1 −1 1 −1 ζ3 ζ3−1 −ζ3 −ζ3 −ζ3−1 −ζ3 ζ3−1 ζ3−1 −ζ3−1 −ζ3−1 ζ3−1 ζ3 −ζ3 ζ3 −ζ3−1 ζ3 1 1 −1 −1
168.47.1n1 C 1 1 −1 −1 −1 −1 1 1 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 −ζ3−1 −ζ3 ζ3 −ζ3 ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3−1 −ζ3 ζ3−1 1 −1 −1 1
168.47.1n2 C 1 1 −1 −1 −1 −1 1 1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 −ζ3 −ζ3−1 ζ3−1 −ζ3−1 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3 −ζ3−1 ζ3 1 −1 −1 1
168.47.1o1 C 1 1 −1 −1 1 1 −1 −1 ζ3−1 ζ3 ζ3−1 −ζ3−1 ζ3 −ζ3−1 −ζ3 −ζ3 −ζ3 −ζ3 ζ3 ζ3−1 ζ3−1 −ζ3−1 ζ3 −ζ3−1 1 −1 −1 1
168.47.1o2 C 1 1 −1 −1 1 1 −1 −1 ζ3 ζ3−1 ζ3 −ζ3 ζ3−1 −ζ3 −ζ3−1 −ζ3−1 −ζ3−1 −ζ3−1 ζ3−1 ζ3 ζ3 −ζ3 ζ3−1 −ζ3 1 −1 −1 1
168.47.1p1 C 1 1 1 1 −1 −1 −1 −1 ζ3−1 ζ3 ζ3−1 −ζ3−1 ζ3 ζ3−1 ζ3 −ζ3 ζ3 −ζ3 −ζ3 −ζ3−1 −ζ3−1 ζ3−1 −ζ3 −ζ3−1 1 1 1 1
168.47.1p2 C 1 1 1 1 −1 −1 −1 −1 ζ3 ζ3−1 ζ3 −ζ3 ζ3−1 ζ3 ζ3−1 −ζ3−1 ζ3−1 −ζ3−1 −ζ3−1 −ζ3 −ζ3 ζ3 −ζ3−1 −ζ3 1 1 1 1
168.47.6a R 6 6 6 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 −1 −1 −1
168.47.6b R 6 −6 −6 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 1 −1 1
168.47.6c R 6 −6 6 −6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 −1 1 1
168.47.6d R 6 6 −6 −6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 1 1 −1

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed