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## G = Q16.F5order 320 = 26·5

### 1st non-split extension by Q16 of F5 acting via F5/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C40 — Q16.F5
 Chief series C1 — C5 — C10 — C20 — C4×D5 — C8×D5 — D10.Q8 — Q16.F5
 Lower central C5 — C10 — C20 — C40 — Q16.F5
 Upper central C1 — C2 — C4 — C8 — Q16

Generators and relations for Q16.F5
G = < a,b,c,d | a8=c5=1, b2=d4=a4, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a-1b, dcd-1=c3 >

Character table of Q16.F5

 class 1 2A 2B 2C 4A 4B 4C 4D 5 8A 8B 8C 8D 8E 8F 10 16A 16B 16C 16D 16E 16F 16G 16H 20A 20B 20C 40A 40B size 1 1 10 40 2 5 5 8 4 2 2 10 10 40 40 4 10 10 10 10 10 10 10 10 8 16 16 8 8 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 i -i 1 -i i i i i -i -i -i 1 -1 -1 1 1 linear of order 4 ρ6 1 1 -1 1 1 -1 -1 -1 1 1 1 -1 -1 -i i 1 i -i -i -i -i i i i 1 -1 -1 1 1 linear of order 4 ρ7 1 1 -1 -1 1 -1 -1 1 1 1 1 -1 -1 i -i 1 i -i -i -i -i i i i 1 1 1 1 1 linear of order 4 ρ8 1 1 -1 -1 1 -1 -1 1 1 1 1 -1 -1 -i i 1 -i i i i i -i -i -i 1 1 1 1 1 linear of order 4 ρ9 2 2 -2 0 2 -2 -2 0 2 -2 -2 2 2 0 0 2 0 0 0 0 0 0 0 0 2 0 0 -2 -2 orthogonal lifted from D4 ρ10 2 2 2 0 2 2 2 0 2 -2 -2 -2 -2 0 0 2 0 0 0 0 0 0 0 0 2 0 0 -2 -2 orthogonal lifted from D4 ρ11 2 2 -2 0 -2 2 2 0 2 0 0 0 0 0 0 2 √2 -√2 -√2 √2 √2 -√2 -√2 √2 -2 0 0 0 0 orthogonal lifted from D8 ρ12 2 2 -2 0 -2 2 2 0 2 0 0 0 0 0 0 2 -√2 √2 √2 -√2 -√2 √2 √2 -√2 -2 0 0 0 0 orthogonal lifted from D8 ρ13 2 2 2 0 -2 -2 -2 0 2 0 0 0 0 0 0 2 -√-2 -√-2 -√-2 √-2 √-2 √-2 √-2 -√-2 -2 0 0 0 0 complex lifted from SD16 ρ14 2 2 2 0 -2 -2 -2 0 2 0 0 0 0 0 0 2 √-2 √-2 √-2 -√-2 -√-2 -√-2 -√-2 √-2 -2 0 0 0 0 complex lifted from SD16 ρ15 2 -2 0 0 0 2i -2i 0 2 -√2 √2 √-2 -√-2 0 0 -2 ζ169+ζ163 ζ163+ζ16 ζ1611+ζ169 ζ1615+ζ165 ζ1613+ζ167 ζ1615+ζ1613 ζ167+ζ165 ζ1611+ζ16 0 0 0 √2 -√2 complex lifted from D8.C4 ρ16 2 -2 0 0 0 -2i 2i 0 2 √2 -√2 √-2 -√-2 0 0 -2 ζ163+ζ16 ζ1611+ζ16 ζ169+ζ163 ζ167+ζ165 ζ1615+ζ1613 ζ1615+ζ165 ζ1613+ζ167 ζ1611+ζ169 0 0 0 -√2 √2 complex lifted from D8.C4 ρ17 2 -2 0 0 0 2i -2i 0 2 √2 -√2 -√-2 √-2 0 0 -2 ζ1615+ζ1613 ζ1615+ζ165 ζ1613+ζ167 ζ1611+ζ169 ζ163+ζ16 ζ1611+ζ16 ζ169+ζ163 ζ167+ζ165 0 0 0 -√2 √2 complex lifted from D8.C4 ρ18 2 -2 0 0 0 2i -2i 0 2 -√2 √2 √-2 -√-2 0 0 -2 ζ1611+ζ16 ζ1611+ζ169 ζ163+ζ16 ζ1613+ζ167 ζ1615+ζ165 ζ167+ζ165 ζ1615+ζ1613 ζ169+ζ163 0 0 0 √2 -√2 complex lifted from D8.C4 ρ19 2 -2 0 0 0 -2i 2i 0 2 -√2 √2 -√-2 √-2 0 0 -2 ζ1615+ζ165 ζ167+ζ165 ζ1615+ζ1613 ζ169+ζ163 ζ1611+ζ16 ζ1611+ζ169 ζ163+ζ16 ζ1613+ζ167 0 0 0 √2 -√2 complex lifted from D8.C4 ρ20 2 -2 0 0 0 2i -2i 0 2 √2 -√2 -√-2 √-2 0 0 -2 ζ167+ζ165 ζ1613+ζ167 ζ1615+ζ165 ζ163+ζ16 ζ1611+ζ169 ζ169+ζ163 ζ1611+ζ16 ζ1615+ζ1613 0 0 0 -√2 √2 complex lifted from D8.C4 ρ21 2 -2 0 0 0 -2i 2i 0 2 -√2 √2 -√-2 √-2 0 0 -2 ζ1613+ζ167 ζ1615+ζ1613 ζ167+ζ165 ζ1611+ζ16 ζ169+ζ163 ζ163+ζ16 ζ1611+ζ169 ζ1615+ζ165 0 0 0 √2 -√2 complex lifted from D8.C4 ρ22 2 -2 0 0 0 -2i 2i 0 2 √2 -√2 √-2 -√-2 0 0 -2 ζ1611+ζ169 ζ169+ζ163 ζ1611+ζ16 ζ1615+ζ1613 ζ167+ζ165 ζ1613+ζ167 ζ1615+ζ165 ζ163+ζ16 0 0 0 -√2 √2 complex lifted from D8.C4 ρ23 4 4 0 0 4 0 0 4 -1 4 4 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ24 4 4 0 0 4 0 0 -4 -1 4 4 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 1 1 -1 -1 orthogonal lifted from C2×F5 ρ25 4 4 0 0 4 0 0 0 -1 -4 -4 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 -√5 √5 1 1 orthogonal lifted from C22⋊F5 ρ26 4 4 0 0 4 0 0 0 -1 -4 -4 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 √5 -√5 1 1 orthogonal lifted from C22⋊F5 ρ27 8 8 0 0 -8 0 0 0 -2 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 0 2 0 0 0 0 orthogonal lifted from D20⋊C4, Schur index 2 ρ28 8 -8 0 0 0 0 0 0 -2 -4√2 4√2 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 -√2 √2 orthogonal faithful, Schur index 2 ρ29 8 -8 0 0 0 0 0 0 -2 4√2 -4√2 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 √2 -√2 orthogonal faithful, Schur index 2

Smallest permutation representation of Q16.F5
On 160 points
Generators in S160
```(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)(129 130 131 132 133 134 135 136)(137 138 139 140 141 142 143 144)(145 146 147 148 149 150 151 152)(153 154 155 156 157 158 159 160)
(1 87 5 83)(2 86 6 82)(3 85 7 81)(4 84 8 88)(9 120 13 116)(10 119 14 115)(11 118 15 114)(12 117 16 113)(17 135 21 131)(18 134 22 130)(19 133 23 129)(20 132 24 136)(25 139 29 143)(26 138 30 142)(27 137 31 141)(28 144 32 140)(33 127 37 123)(34 126 38 122)(35 125 39 121)(36 124 40 128)(41 106 45 110)(42 105 46 109)(43 112 47 108)(44 111 48 107)(49 93 53 89)(50 92 54 96)(51 91 55 95)(52 90 56 94)(57 97 61 101)(58 104 62 100)(59 103 63 99)(60 102 64 98)(65 77 69 73)(66 76 70 80)(67 75 71 79)(68 74 72 78)(145 159 149 155)(146 158 150 154)(147 157 151 153)(148 156 152 160)
(1 49 44 66 63)(2 50 45 67 64)(3 51 46 68 57)(4 52 47 69 58)(5 53 48 70 59)(6 54 41 71 60)(7 55 42 72 61)(8 56 43 65 62)(9 21 27 155 33)(10 22 28 156 34)(11 23 29 157 35)(12 24 30 158 36)(13 17 31 159 37)(14 18 32 160 38)(15 19 25 153 39)(16 20 26 154 40)(73 104 84 90 108)(74 97 85 91 109)(75 98 86 92 110)(76 99 87 93 111)(77 100 88 94 112)(78 101 81 95 105)(79 102 82 96 106)(80 103 83 89 107)(113 132 138 146 128)(114 133 139 147 121)(115 134 140 148 122)(116 135 141 149 123)(117 136 142 150 124)(118 129 143 151 125)(119 130 144 152 126)(120 131 137 145 127)
(1 38 83 127 5 34 87 123)(2 37 84 126 6 33 88 122)(3 36 85 125 7 40 81 121)(4 35 86 124 8 39 82 128)(9 112 148 67 13 108 152 71)(10 111 149 66 14 107 145 70)(11 110 150 65 15 106 146 69)(12 109 151 72 16 105 147 68)(17 104 144 54 21 100 140 50)(18 103 137 53 22 99 141 49)(19 102 138 52 23 98 142 56)(20 101 139 51 24 97 143 55)(25 96 132 58 29 92 136 62)(26 95 133 57 30 91 129 61)(27 94 134 64 31 90 130 60)(28 93 135 63 32 89 131 59)(41 155 77 115 45 159 73 119)(42 154 78 114 46 158 74 118)(43 153 79 113 47 157 75 117)(44 160 80 120 48 156 76 116)```

`G:=sub<Sym(160)| (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)(129,130,131,132,133,134,135,136)(137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152)(153,154,155,156,157,158,159,160), (1,87,5,83)(2,86,6,82)(3,85,7,81)(4,84,8,88)(9,120,13,116)(10,119,14,115)(11,118,15,114)(12,117,16,113)(17,135,21,131)(18,134,22,130)(19,133,23,129)(20,132,24,136)(25,139,29,143)(26,138,30,142)(27,137,31,141)(28,144,32,140)(33,127,37,123)(34,126,38,122)(35,125,39,121)(36,124,40,128)(41,106,45,110)(42,105,46,109)(43,112,47,108)(44,111,48,107)(49,93,53,89)(50,92,54,96)(51,91,55,95)(52,90,56,94)(57,97,61,101)(58,104,62,100)(59,103,63,99)(60,102,64,98)(65,77,69,73)(66,76,70,80)(67,75,71,79)(68,74,72,78)(145,159,149,155)(146,158,150,154)(147,157,151,153)(148,156,152,160), (1,49,44,66,63)(2,50,45,67,64)(3,51,46,68,57)(4,52,47,69,58)(5,53,48,70,59)(6,54,41,71,60)(7,55,42,72,61)(8,56,43,65,62)(9,21,27,155,33)(10,22,28,156,34)(11,23,29,157,35)(12,24,30,158,36)(13,17,31,159,37)(14,18,32,160,38)(15,19,25,153,39)(16,20,26,154,40)(73,104,84,90,108)(74,97,85,91,109)(75,98,86,92,110)(76,99,87,93,111)(77,100,88,94,112)(78,101,81,95,105)(79,102,82,96,106)(80,103,83,89,107)(113,132,138,146,128)(114,133,139,147,121)(115,134,140,148,122)(116,135,141,149,123)(117,136,142,150,124)(118,129,143,151,125)(119,130,144,152,126)(120,131,137,145,127), (1,38,83,127,5,34,87,123)(2,37,84,126,6,33,88,122)(3,36,85,125,7,40,81,121)(4,35,86,124,8,39,82,128)(9,112,148,67,13,108,152,71)(10,111,149,66,14,107,145,70)(11,110,150,65,15,106,146,69)(12,109,151,72,16,105,147,68)(17,104,144,54,21,100,140,50)(18,103,137,53,22,99,141,49)(19,102,138,52,23,98,142,56)(20,101,139,51,24,97,143,55)(25,96,132,58,29,92,136,62)(26,95,133,57,30,91,129,61)(27,94,134,64,31,90,130,60)(28,93,135,63,32,89,131,59)(41,155,77,115,45,159,73,119)(42,154,78,114,46,158,74,118)(43,153,79,113,47,157,75,117)(44,160,80,120,48,156,76,116)>;`

`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)(129,130,131,132,133,134,135,136)(137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152)(153,154,155,156,157,158,159,160), (1,87,5,83)(2,86,6,82)(3,85,7,81)(4,84,8,88)(9,120,13,116)(10,119,14,115)(11,118,15,114)(12,117,16,113)(17,135,21,131)(18,134,22,130)(19,133,23,129)(20,132,24,136)(25,139,29,143)(26,138,30,142)(27,137,31,141)(28,144,32,140)(33,127,37,123)(34,126,38,122)(35,125,39,121)(36,124,40,128)(41,106,45,110)(42,105,46,109)(43,112,47,108)(44,111,48,107)(49,93,53,89)(50,92,54,96)(51,91,55,95)(52,90,56,94)(57,97,61,101)(58,104,62,100)(59,103,63,99)(60,102,64,98)(65,77,69,73)(66,76,70,80)(67,75,71,79)(68,74,72,78)(145,159,149,155)(146,158,150,154)(147,157,151,153)(148,156,152,160), (1,49,44,66,63)(2,50,45,67,64)(3,51,46,68,57)(4,52,47,69,58)(5,53,48,70,59)(6,54,41,71,60)(7,55,42,72,61)(8,56,43,65,62)(9,21,27,155,33)(10,22,28,156,34)(11,23,29,157,35)(12,24,30,158,36)(13,17,31,159,37)(14,18,32,160,38)(15,19,25,153,39)(16,20,26,154,40)(73,104,84,90,108)(74,97,85,91,109)(75,98,86,92,110)(76,99,87,93,111)(77,100,88,94,112)(78,101,81,95,105)(79,102,82,96,106)(80,103,83,89,107)(113,132,138,146,128)(114,133,139,147,121)(115,134,140,148,122)(116,135,141,149,123)(117,136,142,150,124)(118,129,143,151,125)(119,130,144,152,126)(120,131,137,145,127), (1,38,83,127,5,34,87,123)(2,37,84,126,6,33,88,122)(3,36,85,125,7,40,81,121)(4,35,86,124,8,39,82,128)(9,112,148,67,13,108,152,71)(10,111,149,66,14,107,145,70)(11,110,150,65,15,106,146,69)(12,109,151,72,16,105,147,68)(17,104,144,54,21,100,140,50)(18,103,137,53,22,99,141,49)(19,102,138,52,23,98,142,56)(20,101,139,51,24,97,143,55)(25,96,132,58,29,92,136,62)(26,95,133,57,30,91,129,61)(27,94,134,64,31,90,130,60)(28,93,135,63,32,89,131,59)(41,155,77,115,45,159,73,119)(42,154,78,114,46,158,74,118)(43,153,79,113,47,157,75,117)(44,160,80,120,48,156,76,116) );`

`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),(129,130,131,132,133,134,135,136),(137,138,139,140,141,142,143,144),(145,146,147,148,149,150,151,152),(153,154,155,156,157,158,159,160)], [(1,87,5,83),(2,86,6,82),(3,85,7,81),(4,84,8,88),(9,120,13,116),(10,119,14,115),(11,118,15,114),(12,117,16,113),(17,135,21,131),(18,134,22,130),(19,133,23,129),(20,132,24,136),(25,139,29,143),(26,138,30,142),(27,137,31,141),(28,144,32,140),(33,127,37,123),(34,126,38,122),(35,125,39,121),(36,124,40,128),(41,106,45,110),(42,105,46,109),(43,112,47,108),(44,111,48,107),(49,93,53,89),(50,92,54,96),(51,91,55,95),(52,90,56,94),(57,97,61,101),(58,104,62,100),(59,103,63,99),(60,102,64,98),(65,77,69,73),(66,76,70,80),(67,75,71,79),(68,74,72,78),(145,159,149,155),(146,158,150,154),(147,157,151,153),(148,156,152,160)], [(1,49,44,66,63),(2,50,45,67,64),(3,51,46,68,57),(4,52,47,69,58),(5,53,48,70,59),(6,54,41,71,60),(7,55,42,72,61),(8,56,43,65,62),(9,21,27,155,33),(10,22,28,156,34),(11,23,29,157,35),(12,24,30,158,36),(13,17,31,159,37),(14,18,32,160,38),(15,19,25,153,39),(16,20,26,154,40),(73,104,84,90,108),(74,97,85,91,109),(75,98,86,92,110),(76,99,87,93,111),(77,100,88,94,112),(78,101,81,95,105),(79,102,82,96,106),(80,103,83,89,107),(113,132,138,146,128),(114,133,139,147,121),(115,134,140,148,122),(116,135,141,149,123),(117,136,142,150,124),(118,129,143,151,125),(119,130,144,152,126),(120,131,137,145,127)], [(1,38,83,127,5,34,87,123),(2,37,84,126,6,33,88,122),(3,36,85,125,7,40,81,121),(4,35,86,124,8,39,82,128),(9,112,148,67,13,108,152,71),(10,111,149,66,14,107,145,70),(11,110,150,65,15,106,146,69),(12,109,151,72,16,105,147,68),(17,104,144,54,21,100,140,50),(18,103,137,53,22,99,141,49),(19,102,138,52,23,98,142,56),(20,101,139,51,24,97,143,55),(25,96,132,58,29,92,136,62),(26,95,133,57,30,91,129,61),(27,94,134,64,31,90,130,60),(28,93,135,63,32,89,131,59),(41,155,77,115,45,159,73,119),(42,154,78,114,46,158,74,118),(43,153,79,113,47,157,75,117),(44,160,80,120,48,156,76,116)]])`

Matrix representation of Q16.F5 in GL6(𝔽241)

 11 230 0 0 0 0 11 11 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240
,
 64 0 0 0 0 0 0 177 0 0 0 0 0 0 117 0 234 234 0 0 7 124 7 0 0 0 0 7 124 7 0 0 234 234 0 117
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 240 240 240 240 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 87 140 0 0 0 0 140 154 0 0 0 0 0 0 58 116 14 102 0 0 139 227 125 183 0 0 139 197 14 153 0 0 58 197 44 183

`G:=sub<GL(6,GF(241))| [11,11,0,0,0,0,230,11,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[64,0,0,0,0,0,0,177,0,0,0,0,0,0,117,7,0,234,0,0,0,124,7,234,0,0,234,7,124,0,0,0,234,0,7,117],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,1,0,0,0,0,240,0,1,0,0,0,240,0,0,1,0,0,240,0,0,0],[87,140,0,0,0,0,140,154,0,0,0,0,0,0,58,139,139,58,0,0,116,227,197,197,0,0,14,125,14,44,0,0,102,183,153,183] >;`

Q16.F5 in GAP, Magma, Sage, TeX

`Q_{16}.F_5`
`% in TeX`

`G:=Group("Q16.F5");`
`// GroupNames label`

`G:=SmallGroup(320,247);`
`// by ID`

`G=gap.SmallGroup(320,247);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,232,184,675,346,192,1684,851,102,6278,3156]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=c^5=1,b^2=d^4=a^4,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^3>;`
`// generators/relations`

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