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G = C16⋊Q8order 128 = 27

The semidirect product of C16 and Q8 acting via Q8/C2=C22

p-group, metabelian, nilpotent (class 4), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C8 — C16⋊Q8
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C2×C16 — C16⋊5C4 — C16⋊Q8
 Lower central C1 — C2 — C4 — C2×C8 — C16⋊Q8
 Upper central C1 — C22 — C42 — C4×C8 — C16⋊Q8
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — C16⋊Q8

Generators and relations for C16⋊Q8
G = < a,b,c | a16=b4=1, c2=b2, bab-1=a9, cac-1=a7, cbc-1=b-1 >

Character table of C16⋊Q8

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 8A 8B 8C 8D 8E 8F 16A 16B 16C 16D 16E 16F 16G 16H size 1 1 1 1 2 2 4 4 16 16 16 16 2 2 2 2 4 4 4 4 4 4 4 4 4 4 ρ1 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 trivial ρ2 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 linear of order 2 ρ3 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 linear of order 2 ρ4 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 linear of order 2 ρ5 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 linear of order 2 ρ6 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 linear of order 2 ρ7 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 linear of order 2 ρ8 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 linear of order 2 ρ9 2 2 2 2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 -√2 √2 -√2 -√2 -√2 √2 √2 √2 orthogonal lifted from D8 ρ10 2 2 2 2 2 2 2 2 0 0 0 0 -2 -2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 √2 -√2 √2 √2 √2 -√2 -√2 -√2 orthogonal lifted from D8 ρ12 2 2 2 2 2 2 -2 -2 0 0 0 0 -2 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 2 -2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 √2 -√2 -√2 -√2 √2 √2 √2 -√2 orthogonal lifted from D8 ρ14 2 2 2 2 -2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 -√2 √2 √2 √2 -√2 -√2 -√2 √2 orthogonal lifted from D8 ρ15 2 -2 2 -2 -2 2 0 0 0 0 0 0 -2 2 2 -2 0 0 2 -2 0 0 -2 0 0 2 symplectic lifted from Q8, Schur index 2 ρ16 2 -2 2 -2 -2 2 0 0 0 0 0 0 -2 2 2 -2 0 0 -2 2 0 0 2 0 0 -2 symplectic lifted from Q8, Schur index 2 ρ17 2 -2 2 -2 -2 2 0 0 0 0 0 0 2 -2 -2 2 0 0 0 0 -2 2 0 2 -2 0 symplectic lifted from Q8, Schur index 2 ρ18 2 -2 2 -2 -2 2 0 0 0 0 0 0 2 -2 -2 2 0 0 0 0 2 -2 0 -2 2 0 symplectic lifted from Q8, Schur index 2 ρ19 2 -2 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 -2 2 -√2 -√2 √2 -√2 √2 √2 -√2 √2 symplectic lifted from Q16, Schur index 2 ρ20 2 -2 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 2 -2 -√2 -√2 -√2 √2 √2 -√2 √2 √2 symplectic lifted from Q16, Schur index 2 ρ21 2 -2 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 -2 2 √2 √2 -√2 √2 -√2 -√2 √2 -√2 symplectic lifted from Q16, Schur index 2 ρ22 2 -2 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 2 -2 √2 √2 √2 -√2 -√2 √2 -√2 -√2 symplectic lifted from Q16, Schur index 2 ρ23 4 -4 -4 4 0 0 0 0 0 0 0 0 2√2 2√2 -2√2 -2√2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C16⋊C22 ρ24 4 -4 -4 4 0 0 0 0 0 0 0 0 -2√2 -2√2 2√2 2√2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C16⋊C22 ρ25 4 4 -4 -4 0 0 0 0 0 0 0 0 2√2 -2√2 2√2 -2√2 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q32⋊C2, Schur index 2 ρ26 4 4 -4 -4 0 0 0 0 0 0 0 0 -2√2 2√2 -2√2 2√2 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q32⋊C2, Schur index 2

Smallest permutation representation of C16⋊Q8
Regular action on 128 points
Generators in S128
```(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 29 30 31 32)(33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96)(97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128)
(1 93 20 66)(2 86 21 75)(3 95 22 68)(4 88 23 77)(5 81 24 70)(6 90 25 79)(7 83 26 72)(8 92 27 65)(9 85 28 74)(10 94 29 67)(11 87 30 76)(12 96 31 69)(13 89 32 78)(14 82 17 71)(15 91 18 80)(16 84 19 73)(33 112 113 57)(34 105 114 50)(35 98 115 59)(36 107 116 52)(37 100 117 61)(38 109 118 54)(39 102 119 63)(40 111 120 56)(41 104 121 49)(42 97 122 58)(43 106 123 51)(44 99 124 60)(45 108 125 53)(46 101 126 62)(47 110 127 55)(48 103 128 64)
(1 60 20 99)(2 51 21 106)(3 58 22 97)(4 49 23 104)(5 56 24 111)(6 63 25 102)(7 54 26 109)(8 61 27 100)(9 52 28 107)(10 59 29 98)(11 50 30 105)(12 57 31 112)(13 64 32 103)(14 55 17 110)(15 62 18 101)(16 53 19 108)(33 96 113 69)(34 87 114 76)(35 94 115 67)(36 85 116 74)(37 92 117 65)(38 83 118 72)(39 90 119 79)(40 81 120 70)(41 88 121 77)(42 95 122 68)(43 86 123 75)(44 93 124 66)(45 84 125 73)(46 91 126 80)(47 82 127 71)(48 89 128 78)```

`G:=sub<Sym(128)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128), (1,93,20,66)(2,86,21,75)(3,95,22,68)(4,88,23,77)(5,81,24,70)(6,90,25,79)(7,83,26,72)(8,92,27,65)(9,85,28,74)(10,94,29,67)(11,87,30,76)(12,96,31,69)(13,89,32,78)(14,82,17,71)(15,91,18,80)(16,84,19,73)(33,112,113,57)(34,105,114,50)(35,98,115,59)(36,107,116,52)(37,100,117,61)(38,109,118,54)(39,102,119,63)(40,111,120,56)(41,104,121,49)(42,97,122,58)(43,106,123,51)(44,99,124,60)(45,108,125,53)(46,101,126,62)(47,110,127,55)(48,103,128,64), (1,60,20,99)(2,51,21,106)(3,58,22,97)(4,49,23,104)(5,56,24,111)(6,63,25,102)(7,54,26,109)(8,61,27,100)(9,52,28,107)(10,59,29,98)(11,50,30,105)(12,57,31,112)(13,64,32,103)(14,55,17,110)(15,62,18,101)(16,53,19,108)(33,96,113,69)(34,87,114,76)(35,94,115,67)(36,85,116,74)(37,92,117,65)(38,83,118,72)(39,90,119,79)(40,81,120,70)(41,88,121,77)(42,95,122,68)(43,86,123,75)(44,93,124,66)(45,84,125,73)(46,91,126,80)(47,82,127,71)(48,89,128,78)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128), (1,93,20,66)(2,86,21,75)(3,95,22,68)(4,88,23,77)(5,81,24,70)(6,90,25,79)(7,83,26,72)(8,92,27,65)(9,85,28,74)(10,94,29,67)(11,87,30,76)(12,96,31,69)(13,89,32,78)(14,82,17,71)(15,91,18,80)(16,84,19,73)(33,112,113,57)(34,105,114,50)(35,98,115,59)(36,107,116,52)(37,100,117,61)(38,109,118,54)(39,102,119,63)(40,111,120,56)(41,104,121,49)(42,97,122,58)(43,106,123,51)(44,99,124,60)(45,108,125,53)(46,101,126,62)(47,110,127,55)(48,103,128,64), (1,60,20,99)(2,51,21,106)(3,58,22,97)(4,49,23,104)(5,56,24,111)(6,63,25,102)(7,54,26,109)(8,61,27,100)(9,52,28,107)(10,59,29,98)(11,50,30,105)(12,57,31,112)(13,64,32,103)(14,55,17,110)(15,62,18,101)(16,53,19,108)(33,96,113,69)(34,87,114,76)(35,94,115,67)(36,85,116,74)(37,92,117,65)(38,83,118,72)(39,90,119,79)(40,81,120,70)(41,88,121,77)(42,95,122,68)(43,86,123,75)(44,93,124,66)(45,84,125,73)(46,91,126,80)(47,82,127,71)(48,89,128,78) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96),(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128)], [(1,93,20,66),(2,86,21,75),(3,95,22,68),(4,88,23,77),(5,81,24,70),(6,90,25,79),(7,83,26,72),(8,92,27,65),(9,85,28,74),(10,94,29,67),(11,87,30,76),(12,96,31,69),(13,89,32,78),(14,82,17,71),(15,91,18,80),(16,84,19,73),(33,112,113,57),(34,105,114,50),(35,98,115,59),(36,107,116,52),(37,100,117,61),(38,109,118,54),(39,102,119,63),(40,111,120,56),(41,104,121,49),(42,97,122,58),(43,106,123,51),(44,99,124,60),(45,108,125,53),(46,101,126,62),(47,110,127,55),(48,103,128,64)], [(1,60,20,99),(2,51,21,106),(3,58,22,97),(4,49,23,104),(5,56,24,111),(6,63,25,102),(7,54,26,109),(8,61,27,100),(9,52,28,107),(10,59,29,98),(11,50,30,105),(12,57,31,112),(13,64,32,103),(14,55,17,110),(15,62,18,101),(16,53,19,108),(33,96,113,69),(34,87,114,76),(35,94,115,67),(36,85,116,74),(37,92,117,65),(38,83,118,72),(39,90,119,79),(40,81,120,70),(41,88,121,77),(42,95,122,68),(43,86,123,75),(44,93,124,66),(45,84,125,73),(46,91,126,80),(47,82,127,71),(48,89,128,78)]])`

Matrix representation of C16⋊Q8 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 1 15 14 0 0 15 13 6 8 0 0 16 7 14 2 0 0 6 15 9 6
,
 0 16 0 0 0 0 1 0 0 0 0 0 0 0 1 2 8 16 0 0 7 8 12 4 0 0 0 14 6 10 0 0 14 11 12 2
,
 1 7 0 0 0 0 7 16 0 0 0 0 0 0 15 14 3 6 0 0 2 15 7 13 0 0 3 7 1 11 0 0 16 3 4 3

`G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,15,16,6,0,0,1,13,7,15,0,0,15,6,14,9,0,0,14,8,2,6],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,1,7,0,14,0,0,2,8,14,11,0,0,8,12,6,12,0,0,16,4,10,2],[1,7,0,0,0,0,7,16,0,0,0,0,0,0,15,2,3,16,0,0,14,15,7,3,0,0,3,7,1,4,0,0,6,13,11,3] >;`

C16⋊Q8 in GAP, Magma, Sage, TeX

`C_{16}\rtimes Q_8`
`% in TeX`

`G:=Group("C16:Q8");`
`// GroupNames label`

`G:=SmallGroup(128,987);`
`// by ID`

`G=gap.SmallGroup(128,987);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,-2,56,141,64,422,723,268,1684,242,4037,124]);`
`// Polycyclic`

`G:=Group<a,b,c|a^16=b^4=1,c^2=b^2,b*a*b^-1=a^9,c*a*c^-1=a^7,c*b*c^-1=b^-1>;`
`// generators/relations`

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