Properties

Label 16T672
16T672 1 10 1->10 11 1->11 12 1->12 2 9 2->9 2->11 2->12 3 3->10 3->12 14 3->14 4 4->9 4->11 13 4->13 5 5->13 16 5->16 5->16 6 6->14 15 6->15 6->15 7 7->10 7->14 7->15 8 8->9 8->13 8->16 9->1 9->5 9->6 10->2 10->5 10->6 11->3 11->4 11->8 12->3 12->4 12->7 13->1 13->2 13->5 14->1 14->2 14->6 15->4 15->8 15->8 16->3 16->7 16->7
Degree 1616
Order 256256
Cyclic no
Abelian no
Solvable yes
Primitive no
pp-group yes
Group: C42.(C2×D4)C_4^2.(C_2\times D_4)

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Copy content magma:G := TransitiveGroup(16, 672);
 

Group invariants

Abstract group:  C42.(C2×D4)C_4^2.(C_2\times D_4)
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Order:  256=28256=2^{8}
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Cyclic:  no
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Abelian:  no
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Solvable:  yes
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Nilpotency class:  55
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Group action invariants

Degree nn:  1616
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number tt:  672672
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  11
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Primitive:  no
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#Aut(F/K)\card{\Aut(F/K)}:  22
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Generators:  (1,12,3,10,2,11,4,9)(5,16,7,14,6,15,8,13)(1,12,3,10,2,11,4,9)(5,16,7,14,6,15,8,13), (1,10,6,14,2,9,5,13)(3,12,7,15,4,11,8,16)(1,10,6,14,2,9,5,13)(3,12,7,15,4,11,8,16), (1,11,3,14)(2,12,4,13)(5,16,7,10)(6,15,8,9)(1,11,3,14)(2,12,4,13)(5,16,7,10)(6,15,8,9)
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Low degree resolvents

#(G/N)\card{(G/N)}Galois groups for stem field(s)
22C2C_2 x 7
44C22C_2^2 x 7
88D4D_{4} x 6, C23C_2^3
1616D4×C2D_4\times C_2 x 3
3232C22C2C_2^2 \wr C_2
6464(((C4×C2):C2):C2):C2(((C_4 \times C_2): C_2):C_2):C_2
128128C2C2C2C_2 \wr C_2\wr C_2

Resolvents shown for degrees 47\leq 47

Subfields

Degree 2: C2C_2

Degree 4: D4D_{4}

Degree 8: (((C4×C2):C2):C2):C2(((C_4 \times C_2): C_2):C_2):C_2

Low degree siblings

16T662 x 2, 16T672, 32T3230, 32T3231 x 2, 32T3232, 32T3270, 32T3271, 32T6290 x 2, 32T7435, 32T7447

Siblings are shown with degree 47\leq 47

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A 1161^{16} 11 11 00 ()()
2A 282^{8} 11 22 88 (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)
2B 24,182^{4},1^{8} 22 22 44 (1,2)(3,4)(5,6)(7,8)(1,2)(3,4)(5,6)(7,8)
2C 282^{8} 88 22 88 (1,15)(2,16)(3,9)(4,10)(5,12)(6,11)(7,13)(8,14)( 1,15)( 2,16)( 3, 9)( 4,10)( 5,12)( 6,11)( 7,13)( 8,14)
2D 282^{8} 88 22 88 (1,16)(2,15)(3,13)(4,14)(5,12)(6,11)(7,9)(8,10)( 1,16)( 2,15)( 3,13)( 4,14)( 5,12)( 6,11)( 7, 9)( 8,10)
2E 282^{8} 88 22 88 (1,7)(2,8)(3,6)(4,5)(9,12)(10,11)(13,16)(14,15)( 1, 7)( 2, 8)( 3, 6)( 4, 5)( 9,12)(10,11)(13,16)(14,15)
2F 26,142^{6},1^{4} 1616 22 66 (1,2)(3,7)(4,8)(9,13)(10,14)(15,16)( 1, 2)( 3, 7)( 4, 8)( 9,13)(10,14)(15,16)
2G 282^{8} 1616 22 88 (1,8)(2,7)(3,6)(4,5)(9,13)(10,14)(11,16)(12,15)( 1, 8)( 2, 7)( 3, 6)( 4, 5)( 9,13)(10,14)(11,16)(12,15)
4A 444^{4} 44 44 1212 (1,5,2,6)(3,8,4,7)(9,14,10,13)(11,15,12,16)( 1, 5, 2, 6)( 3, 8, 4, 7)( 9,14,10,13)(11,15,12,16)
4B 444^{4} 88 44 1212 (1,12,2,11)(3,14,4,13)(5,16,6,15)(7,9,8,10)( 1,12, 2,11)( 3,14, 4,13)( 5,16, 6,15)( 7, 9, 8,10)
4C 444^{4} 88 44 1212 (1,9,2,10)(3,12,4,11)(5,14,6,13)(7,15,8,16)( 1, 9, 2,10)( 3,12, 4,11)( 5,14, 6,13)( 7,15, 8,16)
4D 42,22,144^{2},2^{2},1^{4} 88 44 88 (1,2)(5,6)(9,13,10,14)(11,15,12,16)( 1, 2)( 5, 6)( 9,13,10,14)(11,15,12,16)
4E 444^{4} 88 44 1212 (1,7,2,8)(3,5,4,6)(9,15,10,16)(11,13,12,14)( 1, 7, 2, 8)( 3, 5, 4, 6)( 9,15,10,16)(11,13,12,14)
4F 42,22,144^{2},2^{2},1^{4} 1616 44 88 (1,4,2,3)(5,7,6,8)(13,14)(15,16)( 1, 4, 2, 3)( 5, 7, 6, 8)(13,14)(15,16)
4G 444^{4} 3232 44 1212 (1,13,7,16)(2,14,8,15)(3,11,6,10)(4,12,5,9)( 1,13, 7,16)( 2,14, 8,15)( 3,11, 6,10)( 4,12, 5, 9)
8A 828^{2} 88 88 1414 (1,8,5,4,2,7,6,3)(9,15,14,12,10,16,13,11)( 1, 8, 5, 4, 2, 7, 6, 3)( 9,15,14,12,10,16,13,11)
8B 828^{2} 88 88 1414 (1,8,5,4,2,7,6,3)(9,16,14,11,10,15,13,12)( 1, 8, 5, 4, 2, 7, 6, 3)( 9,16,14,11,10,15,13,12)
8C 828^{2} 1616 88 1414 (1,14,5,10,2,13,6,9)(3,15,8,12,4,16,7,11)( 1,14, 5,10, 2,13, 6, 9)( 3,15, 8,12, 4,16, 7,11)
8D 828^{2} 1616 88 1414 (1,11,6,16,2,12,5,15)(3,14,7,9,4,13,8,10)( 1,11, 6,16, 2,12, 5,15)( 3,14, 7, 9, 4,13, 8,10)
8E 8,4,148,4,1^{4} 1616 88 1010 (1,5,2,6)(9,11,13,15,10,12,14,16)( 1, 5, 2, 6)( 9,11,13,15,10,12,14,16)
8F 8,4,228,4,2^{2} 1616 88 1212 (1,6,2,5)(3,4)(7,8)(9,12,13,16,10,11,14,15)( 1, 6, 2, 5)( 3, 4)( 7, 8)( 9,12,13,16,10,11,14,15)
8G 828^{2} 3232 88 1414 (1,15,7,10,2,16,8,9)(3,13,5,12,4,14,6,11)( 1,15, 7,10, 2,16, 8, 9)( 3,13, 5,12, 4,14, 6,11)

Malle's constant a(G)a(G):     1/41/4

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Character table

1A 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 8A 8B 8C 8D 8E 8F 8G
Size 1 1 2 8 8 8 16 16 4 8 8 8 8 16 32 8 8 16 16 16 16 32
2 P 1A 1A 1A 1A 1A 1A 1A 1A 2A 2A 2A 2B 2A 2B 2E 4A 4A 4A 4A 4D 4D 4E
Type
256.6665.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
256.6665.1b R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
256.6665.1c R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
256.6665.1d R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
256.6665.1e R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
256.6665.1f R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
256.6665.1g R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
256.6665.1h R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
256.6665.2a R 2 2 2 2 0 2 0 0 2 2 0 2 2 0 0 0 0 0 2 0 0 0
256.6665.2b R 2 2 2 0 2 2 0 0 2 0 2 2 2 0 0 0 0 0 0 0 2 0
256.6665.2c R 2 2 2 0 0 2 2 0 2 0 0 2 2 0 0 2 2 0 0 0 0 0
256.6665.2d R 2 2 2 0 0 2 2 0 2 0 0 2 2 0 0 2 2 0 0 0 0 0
256.6665.2e R 2 2 2 0 2 2 0 0 2 0 2 2 2 0 0 0 0 0 0 0 2 0
256.6665.2f R 2 2 2 2 0 2 0 0 2 2 0 2 2 0 0 0 0 0 2 0 0 0
256.6665.4a R 4 4 4 0 0 0 0 2 4 0 0 0 0 2 0 0 0 0 0 0 0 0
256.6665.4b R 4 4 4 0 0 0 0 2 4 0 0 0 0 2 0 0 0 0 0 0 0 0
256.6665.4c R 4 4 4 2 2 0 0 0 0 2 2 0 0 0 0 2 2 0 0 0 0 0
256.6665.4d R 4 4 4 2 2 0 0 0 0 2 2 0 0 0 0 2 2 0 0 0 0 0
256.6665.4e R 4 4 4 2 2 0 0 0 0 2 2 0 0 0 0 2 2 0 0 0 0 0
256.6665.4f R 4 4 4 2 2 0 0 0 0 2 2 0 0 0 0 2 2 0 0 0 0 0
256.6665.8a R 8 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 2 0 0
256.6665.8b R 8 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 2 0 0

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Regular extensions

Data not computed