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## G = C8.16Q16order 128 = 27

### 4th non-split extension by C8 of Q16 acting via Q16/Q8=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C8 — C8.16Q16
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C4×C8 — C8.5Q8 — C8.16Q16
 Lower central C1 — C2 — C2×C4 — C2×C8 — C8.16Q16
 Upper central C1 — C22 — C42 — C4×C8 — C8.16Q16
 Jennings C1 — C2 — C2 — C2 — C2 — C2×C4 — C2×C4 — C4×C8 — C8.16Q16

Generators and relations for C8.16Q16
G = < a,b,c | a8=b8=1, c2=a4b4, bab-1=cac-1=a3, cbc-1=ab-1 >

Character table of C8.16Q16

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 16A 16B 16C 16D 16E 16F 16G 16H size 1 1 1 1 2 2 2 2 4 16 16 2 2 2 2 4 4 8 8 8 8 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 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 -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 -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 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 i i -i -i i -i -i i i -i i -i linear of order 4 ρ6 1 1 1 1 -1 -1 1 1 -1 -1 1 -1 -1 -1 -1 1 1 i i -i -i -i i i -i -i i -i i linear of order 4 ρ7 1 1 1 1 -1 -1 1 1 -1 -1 1 -1 -1 -1 -1 1 1 -i -i i i i -i -i i i -i i -i linear of order 4 ρ8 1 1 1 1 -1 -1 1 1 -1 1 -1 -1 -1 -1 -1 1 1 -i -i i i -i i i -i -i i -i i linear of order 4 ρ9 2 2 2 2 -2 -2 2 2 -2 0 0 2 2 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 2 2 2 2 2 0 0 -2 -2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 2 2 -2 -2 -2 0 0 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 -2 0 0 0 0 0 0 0 0 0 0 0 0 -√2 √2 -√2 √2 -√2 -√2 √2 √2 orthogonal lifted from D8 ρ13 2 -2 -2 2 0 0 2 -2 0 0 0 2 2 -2 -2 0 0 √2 -√2 √2 -√2 0 0 0 0 0 0 0 0 symplectic lifted from Q16, Schur index 2 ρ14 2 -2 -2 2 0 0 2 -2 0 0 0 2 2 -2 -2 0 0 -√2 √2 -√2 √2 0 0 0 0 0 0 0 0 symplectic lifted from Q16, Schur index 2 ρ15 2 2 2 2 -2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 -√-2 -√-2 √-2 √-2 -√-2 √-2 √-2 -√-2 complex lifted from SD16 ρ16 2 -2 -2 2 0 0 2 -2 0 0 0 -2 -2 2 2 0 0 -√-2 √-2 √-2 -√-2 0 0 0 0 0 0 0 0 complex lifted from SD16 ρ17 2 -2 -2 2 0 0 2 -2 0 0 0 -2 -2 2 2 0 0 √-2 -√-2 -√-2 √-2 0 0 0 0 0 0 0 0 complex lifted from SD16 ρ18 2 2 2 2 -2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 √-2 √-2 -√-2 -√-2 √-2 -√-2 -√-2 √-2 complex lifted from SD16 ρ19 2 2 -2 -2 2i -2i 0 0 0 0 0 -√-2 √-2 -√-2 √-2 √2 -√2 0 0 0 0 ζ167+ζ165 ζ1613+ζ167 ζ1611+ζ169 ζ1611+ζ16 ζ1615+ζ1613 ζ163+ζ16 ζ169+ζ163 ζ1615+ζ165 complex lifted from D8.C4 ρ20 2 2 -2 -2 -2i 2i 0 0 0 0 0 √-2 -√-2 √-2 -√-2 √2 -√2 0 0 0 0 ζ163+ζ16 ζ1611+ζ16 ζ1615+ζ1613 ζ1613+ζ167 ζ1611+ζ169 ζ167+ζ165 ζ1615+ζ165 ζ169+ζ163 complex lifted from D8.C4 ρ21 2 2 -2 -2 2i -2i 0 0 0 0 0 √-2 -√-2 √-2 -√-2 -√2 √2 0 0 0 0 ζ1611+ζ16 ζ1611+ζ169 ζ1615+ζ165 ζ1615+ζ1613 ζ169+ζ163 ζ1613+ζ167 ζ167+ζ165 ζ163+ζ16 complex lifted from D8.C4 ρ22 2 2 -2 -2 2i -2i 0 0 0 0 0 -√-2 √-2 -√-2 √-2 √2 -√2 0 0 0 0 ζ1615+ζ1613 ζ1615+ζ165 ζ163+ζ16 ζ169+ζ163 ζ167+ζ165 ζ1611+ζ169 ζ1611+ζ16 ζ1613+ζ167 complex lifted from D8.C4 ρ23 2 2 -2 -2 -2i 2i 0 0 0 0 0 -√-2 √-2 -√-2 √-2 -√2 √2 0 0 0 0 ζ1615+ζ165 ζ167+ζ165 ζ1611+ζ16 ζ163+ζ16 ζ1613+ζ167 ζ169+ζ163 ζ1611+ζ169 ζ1615+ζ1613 complex lifted from D8.C4 ρ24 2 2 -2 -2 2i -2i 0 0 0 0 0 √-2 -√-2 √-2 -√-2 -√2 √2 0 0 0 0 ζ169+ζ163 ζ163+ζ16 ζ1613+ζ167 ζ167+ζ165 ζ1611+ζ16 ζ1615+ζ165 ζ1615+ζ1613 ζ1611+ζ169 complex lifted from D8.C4 ρ25 2 2 -2 -2 -2i 2i 0 0 0 0 0 √-2 -√-2 √-2 -√-2 √2 -√2 0 0 0 0 ζ1611+ζ169 ζ169+ζ163 ζ167+ζ165 ζ1615+ζ165 ζ163+ζ16 ζ1615+ζ1613 ζ1613+ζ167 ζ1611+ζ16 complex lifted from D8.C4 ρ26 2 2 -2 -2 -2i 2i 0 0 0 0 0 -√-2 √-2 -√-2 √-2 -√2 √2 0 0 0 0 ζ1613+ζ167 ζ1615+ζ1613 ζ169+ζ163 ζ1611+ζ169 ζ1615+ζ165 ζ1611+ζ16 ζ163+ζ16 ζ167+ζ165 complex lifted from D8.C4 ρ27 4 -4 -4 4 0 0 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C4.10D4, Schur index 2 ρ28 4 -4 4 -4 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 0 0 0 0 complex lifted from D8⋊2C4 ρ29 4 -4 4 -4 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 0 0 0 0 complex lifted from D8⋊2C4

Smallest permutation representation of C8.16Q16
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)
(1 127 29 119 23 10 40 110)(2 122 30 114 24 13 33 105)(3 125 31 117 17 16 34 108)(4 128 32 120 18 11 35 111)(5 123 25 115 19 14 36 106)(6 126 26 118 20 9 37 109)(7 121 27 113 21 12 38 112)(8 124 28 116 22 15 39 107)(41 94 65 79 50 97 62 84)(42 89 66 74 51 100 63 87)(43 92 67 77 52 103 64 82)(44 95 68 80 53 98 57 85)(45 90 69 75 54 101 58 88)(46 93 70 78 55 104 59 83)(47 96 71 73 56 99 60 86)(48 91 72 76 49 102 61 81)
(1 51 19 46)(2 54 20 41)(3 49 21 44)(4 52 22 47)(5 55 23 42)(6 50 24 45)(7 53 17 48)(8 56 18 43)(9 87 122 78)(10 82 123 73)(11 85 124 76)(12 88 125 79)(13 83 126 74)(14 86 127 77)(15 81 128 80)(16 84 121 75)(25 66 40 59)(26 69 33 62)(27 72 34 57)(28 67 35 60)(29 70 36 63)(30 65 37 58)(31 68 38 61)(32 71 39 64)(89 109 104 114)(90 112 97 117)(91 107 98 120)(92 110 99 115)(93 105 100 118)(94 108 101 113)(95 111 102 116)(96 106 103 119)```

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

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

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

Matrix representation of C8.16Q16 in GL4(𝔽17) generated by

 12 12 0 0 5 12 0 0 0 0 1 0 0 0 0 1
,
 0 15 0 0 15 0 0 0 0 0 8 0 0 0 5 15
,
 13 11 0 0 11 4 0 0 0 0 1 15 0 0 1 16
`G:=sub<GL(4,GF(17))| [12,5,0,0,12,12,0,0,0,0,1,0,0,0,0,1],[0,15,0,0,15,0,0,0,0,0,8,5,0,0,0,15],[13,11,0,0,11,4,0,0,0,0,1,1,0,0,15,16] >;`

C8.16Q16 in GAP, Magma, Sage, TeX

`C_8._{16}Q_{16}`
`% in TeX`

`G:=Group("C8.16Q16");`
`// GroupNames label`

`G:=SmallGroup(128,95);`
`// by ID`

`G=gap.SmallGroup(128,95);`
`# by ID`

`G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,120,422,387,520,794,192,2028,124]);`
`// Polycyclic`

`G:=Group<a,b,c|a^8=b^8=1,c^2=a^4*b^4,b*a*b^-1=c*a*c^-1=a^3,c*b*c^-1=a*b^-1>;`
`// generators/relations`

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