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G = C16.D10order 320 = 26·5

1st non-split extension by C16 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C40 — C16.D10
 Chief series C1 — C5 — C10 — C20 — C40 — D40 — D40⋊7C2 — C16.D10
 Lower central C5 — C10 — C20 — C40 — C16.D10
 Upper central C1 — C2 — C2×C4 — C2×C8 — M5(2)

Generators and relations for C16.D10
G = < a,b,c | a16=1, b10=c2=a8, bab-1=a9, cac-1=a-1, cbc-1=b9 >

Subgroups: 430 in 82 conjugacy classes, 35 normal (25 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, D5, C10, C10, C16, C2×C8, D8, SD16, Q16, C2×Q8, C4○D4, Dic5, C20, D10, C2×C10, M5(2), SD32, Q32, C2×Q16, C4○D8, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, Q32⋊C2, C80, C40⋊C2, D40, Dic20, Dic20, Dic20, C2×C40, C2×Dic10, C4○D20, C16⋊D5, Dic40, C5×M5(2), D407C2, C2×Dic20, C16.D10
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, D10, C2×D8, D20, C22×D5, Q32⋊C2, D40, C2×D20, C2×D40, C16.D10

Smallest permutation representation of C16.D10
On 160 points
Generators in S160
```(1 126 104 31 41 63 99 142 11 136 114 21 51 73 89 152)(2 137 105 22 42 74 100 153 12 127 115 32 52 64 90 143)(3 128 106 33 43 65 81 144 13 138 116 23 53 75 91 154)(4 139 107 24 44 76 82 155 14 129 117 34 54 66 92 145)(5 130 108 35 45 67 83 146 15 140 118 25 55 77 93 156)(6 121 109 26 46 78 84 157 16 131 119 36 56 68 94 147)(7 132 110 37 47 69 85 148 17 122 120 27 57 79 95 158)(8 123 111 28 48 80 86 159 18 133 101 38 58 70 96 149)(9 134 112 39 49 71 87 150 19 124 102 29 59 61 97 160)(10 125 113 30 50 62 88 141 20 135 103 40 60 72 98 151)
(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 10 11 20)(2 19 12 9)(3 8 13 18)(4 17 14 7)(5 6 15 16)(21 62 31 72)(22 71 32 61)(23 80 33 70)(24 69 34 79)(25 78 35 68)(26 67 36 77)(27 76 37 66)(28 65 38 75)(29 74 39 64)(30 63 40 73)(41 60 51 50)(42 49 52 59)(43 58 53 48)(44 47 54 57)(45 56 55 46)(81 101 91 111)(82 110 92 120)(83 119 93 109)(84 108 94 118)(85 117 95 107)(86 106 96 116)(87 115 97 105)(88 104 98 114)(89 113 99 103)(90 102 100 112)(121 146 131 156)(122 155 132 145)(123 144 133 154)(124 153 134 143)(125 142 135 152)(126 151 136 141)(127 160 137 150)(128 149 138 159)(129 158 139 148)(130 147 140 157)```

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 5A 5B 8A 8B 8C 10A 10B 10C 10D 16A 16B 16C 16D 20A 20B 20C 20D 20E 20F 40A ··· 40H 40I 40J 40K 40L 80A ··· 80P order 1 2 2 2 4 4 4 4 4 5 5 8 8 8 10 10 10 10 16 16 16 16 20 20 20 20 20 20 40 ··· 40 40 40 40 40 80 ··· 80 size 1 1 2 40 2 2 40 40 40 2 2 2 2 4 2 2 4 4 4 4 4 4 2 2 2 2 4 4 2 ··· 2 4 4 4 4 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 D4 D4 D5 D8 D8 D10 D10 D20 D20 D40 D40 Q32⋊C2 C16.D10 kernel C16.D10 C16⋊D5 Dic40 C5×M5(2) D40⋊7C2 C2×Dic20 C40 C2×C20 M5(2) C20 C2×C10 C16 C2×C8 C8 C2×C4 C4 C22 C5 C1 # reps 1 2 2 1 1 1 1 1 2 2 2 4 2 4 4 8 8 2 8

Matrix representation of C16.D10 in GL4(𝔽241) generated by

 204 96 239 0 8 108 0 239 5 171 37 145 12 181 233 133
,
 226 91 175 66 129 65 117 51 112 56 65 17 76 239 53 126
,
 226 147 116 233 129 9 176 125 228 153 11 129 11 120 43 236
`G:=sub<GL(4,GF(241))| [204,8,5,12,96,108,171,181,239,0,37,233,0,239,145,133],[226,129,112,76,91,65,56,239,175,117,65,53,66,51,17,126],[226,129,228,11,147,9,153,120,116,176,11,43,233,125,129,236] >;`

C16.D10 in GAP, Magma, Sage, TeX

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

`G:=Group("C16.D10");`
`// GroupNames label`

`G:=SmallGroup(320,536);`
`// by ID`

`G=gap.SmallGroup(320,536);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,254,387,142,1571,80,1684,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c|a^16=1,b^10=c^2=a^8,b*a*b^-1=a^9,c*a*c^-1=a^-1,c*b*c^-1=b^9>;`
`// generators/relations`

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