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## G = C10.Q16order 160 = 25·5

### 2nd non-split extension by C10 of Q16 acting via Q16/Q8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C10.Q16
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×Dic10 — C10.Q16
 Lower central C5 — C10 — C20 — C10.Q16
 Upper central C1 — C22 — C2×C4 — C4⋊C4

Generators and relations for C10.Q16
G = < a,b,c | a10=b8=1, c2=a5b4, bab-1=a-1, ac=ca, cbc-1=a5b-1 >

Smallest permutation representation of C10.Q16
Regular action 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 30 50 142 11 140 34 157)(2 29 41 141 12 139 35 156)(3 28 42 150 13 138 36 155)(4 27 43 149 14 137 37 154)(5 26 44 148 15 136 38 153)(6 25 45 147 16 135 39 152)(7 24 46 146 17 134 40 151)(8 23 47 145 18 133 31 160)(9 22 48 144 19 132 32 159)(10 21 49 143 20 131 33 158)(51 116 64 99 87 101 79 124)(52 115 65 98 88 110 80 123)(53 114 66 97 89 109 71 122)(54 113 67 96 90 108 72 121)(55 112 68 95 81 107 73 130)(56 111 69 94 82 106 74 129)(57 120 70 93 83 105 75 128)(58 119 61 92 84 104 76 127)(59 118 62 91 85 103 77 126)(60 117 63 100 86 102 78 125)
(1 63 16 73)(2 64 17 74)(3 65 18 75)(4 66 19 76)(5 67 20 77)(6 68 11 78)(7 69 12 79)(8 70 13 80)(9 61 14 71)(10 62 15 72)(21 113 136 103)(22 114 137 104)(23 115 138 105)(24 116 139 106)(25 117 140 107)(26 118 131 108)(27 119 132 109)(28 120 133 110)(29 111 134 101)(30 112 135 102)(31 83 42 52)(32 84 43 53)(33 85 44 54)(34 86 45 55)(35 87 46 56)(36 88 47 57)(37 89 48 58)(38 90 49 59)(39 81 50 60)(40 82 41 51)(91 143 121 153)(92 144 122 154)(93 145 123 155)(94 146 124 156)(95 147 125 157)(96 148 126 158)(97 149 127 159)(98 150 128 160)(99 141 129 151)(100 142 130 152)```

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

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

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

34 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 5A 5B 8A 8B 8C 8D 10A ··· 10F 20A ··· 20L order 1 2 2 2 4 4 4 4 4 4 5 5 8 8 8 8 10 ··· 10 20 ··· 20 size 1 1 1 1 2 2 4 4 20 20 2 2 10 10 10 10 2 ··· 2 4 ··· 4

34 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + - + + - - image C1 C2 C2 C2 C4 D4 D4 D5 SD16 Q16 D10 C4×D5 D20 C5⋊D4 D4.D5 C5⋊Q16 kernel C10.Q16 C2×C5⋊2C8 C5×C4⋊C4 C2×Dic10 Dic10 C20 C2×C10 C4⋊C4 C10 C10 C2×C4 C4 C4 C22 C2 C2 # reps 1 1 1 1 4 1 1 2 2 2 2 4 4 4 2 2

Matrix representation of C10.Q16 in GL5(𝔽41)

 40 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 35 0 0 0 7 34
,
 9 0 0 0 0 0 29 12 0 0 0 29 29 0 0 0 0 0 28 15 0 0 0 16 13
,
 32 0 0 0 0 0 6 39 0 0 0 39 35 0 0 0 0 0 17 6 0 0 0 34 24

`G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,7,0,0,0,35,34],[9,0,0,0,0,0,29,29,0,0,0,12,29,0,0,0,0,0,28,16,0,0,0,15,13],[32,0,0,0,0,0,6,39,0,0,0,39,35,0,0,0,0,0,17,34,0,0,0,6,24] >;`

C10.Q16 in GAP, Magma, Sage, TeX

`C_{10}.Q_{16}`
`% in TeX`

`G:=Group("C10.Q16");`
`// GroupNames label`

`G:=SmallGroup(160,17);`
`// by ID`

`G=gap.SmallGroup(160,17);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,96,121,31,579,297,69,4613]);`
`// Polycyclic`

`G:=Group<a,b,c|a^10=b^8=1,c^2=a^5*b^4,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^5*b^-1>;`
`// generators/relations`

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