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## G = C10.D16order 320 = 26·5

### 7th non-split extension by C10 of D16 acting via D16/D8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C40 — C10.D16
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C40 — C40⋊5C4 — C10.D16
 Lower central C5 — C10 — C20 — C40 — C10.D16
 Upper central C1 — C22 — C2×C4 — C2×C8 — C2×D8

Generators and relations for C10.D16
G = < a,b,c | a10=b16=1, c2=a5, bab-1=cac-1=a-1, cbc-1=a5b-1 >

Subgroups: 254 in 66 conjugacy classes, 31 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, C10, C10, C16, C4⋊C4, C2×C8, D8, D8, C2×D4, Dic5, C20, C2×C10, C2×C10, C2.D8, C2×C16, C2×D8, C40, C2×Dic5, C2×C20, C5×D4, C22×C10, C2.D16, C52C16, C4⋊Dic5, C2×C40, C5×D8, C5×D8, D4×C10, C2×C52C16, C405C4, C10×D8, C10.D16
Quotients: C1, C2, C4, C22, C2×C4, D4, D5, C22⋊C4, D8, SD16, Dic5, D10, D4⋊C4, D16, SD32, C2×Dic5, C5⋊D4, C2.D16, D4⋊D5, D4.D5, C23.D5, C5⋊D16, D8.D5, D4⋊Dic5, C10.D16

Smallest permutation representation of C10.D16
On 160 points
Generators in S160
```(1 29 128 148 88 101 129 42 79 58)(2 59 80 43 130 102 89 149 113 30)(3 31 114 150 90 103 131 44 65 60)(4 61 66 45 132 104 91 151 115 32)(5 17 116 152 92 105 133 46 67 62)(6 63 68 47 134 106 93 153 117 18)(7 19 118 154 94 107 135 48 69 64)(8 49 70 33 136 108 95 155 119 20)(9 21 120 156 96 109 137 34 71 50)(10 51 72 35 138 110 81 157 121 22)(11 23 122 158 82 111 139 36 73 52)(12 53 74 37 140 112 83 159 123 24)(13 25 124 160 84 97 141 38 75 54)(14 55 76 39 142 98 85 145 125 26)(15 27 126 146 86 99 143 40 77 56)(16 57 78 41 144 100 87 147 127 28)
(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 16 101 100)(2 99 102 15)(3 14 103 98)(4 97 104 13)(5 12 105 112)(6 111 106 11)(7 10 107 110)(8 109 108 9)(17 24 133 140)(18 139 134 23)(19 22 135 138)(20 137 136 21)(25 32 141 132)(26 131 142 31)(27 30 143 130)(28 129 144 29)(33 120 119 34)(35 118 121 48)(36 47 122 117)(37 116 123 46)(38 45 124 115)(39 114 125 44)(40 43 126 113)(41 128 127 42)(49 96 95 50)(51 94 81 64)(52 63 82 93)(53 92 83 62)(54 61 84 91)(55 90 85 60)(56 59 86 89)(57 88 87 58)(65 76 150 145)(66 160 151 75)(67 74 152 159)(68 158 153 73)(69 72 154 157)(70 156 155 71)(77 80 146 149)(78 148 147 79)```

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 5A 5B 8A 8B 8C 8D 10A ··· 10F 10G ··· 10N 16A ··· 16H 20A 20B 20C 20D 40A ··· 40H order 1 2 2 2 2 2 4 4 4 4 5 5 8 8 8 8 10 ··· 10 10 ··· 10 16 ··· 16 20 20 20 20 40 ··· 40 size 1 1 1 1 8 8 2 2 40 40 2 2 2 2 2 2 2 ··· 2 8 ··· 8 10 ··· 10 4 4 4 4 4 ··· 4

50 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + - + - + + - image C1 C2 C2 C2 C4 D4 D4 D5 SD16 D8 D10 Dic5 D16 SD32 C5⋊D4 C5⋊D4 D4.D5 D4⋊D5 C5⋊D16 D8.D5 kernel C10.D16 C2×C5⋊2C16 C40⋊5C4 C10×D8 C5×D8 C40 C2×C20 C2×D8 C20 C2×C10 C2×C8 D8 C10 C10 C8 C2×C4 C4 C22 C2 C2 # reps 1 1 1 1 4 1 1 2 2 2 2 4 4 4 4 4 2 2 4 4

Matrix representation of C10.D16 in GL4(𝔽241) generated by

 143 0 0 0 102 150 0 0 0 0 240 0 0 0 0 240
,
 185 171 0 0 124 56 0 0 0 0 179 82 0 0 200 97
,
 56 70 0 0 117 185 0 0 0 0 97 35 0 0 41 144
`G:=sub<GL(4,GF(241))| [143,102,0,0,0,150,0,0,0,0,240,0,0,0,0,240],[185,124,0,0,171,56,0,0,0,0,179,200,0,0,82,97],[56,117,0,0,70,185,0,0,0,0,97,41,0,0,35,144] >;`

C10.D16 in GAP, Magma, Sage, TeX

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

`G:=Group("C10.D16");`
`// GroupNames label`

`G:=SmallGroup(320,120);`
`// by ID`

`G=gap.SmallGroup(320,120);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,675,346,192,1684,851,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c|a^10=b^16=1,c^2=a^5,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^5*b^-1>;`
`// generators/relations`

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