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## G = Q16.D10order 320 = 26·5

### 1st non-split extension by Q16 of D10 acting via D10/C10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C40 — Q16.D10
 Chief series C1 — C5 — C10 — C20 — C40 — D40 — D40⋊7C2 — Q16.D10
 Lower central C5 — C10 — C20 — C40 — Q16.D10
 Upper central C1 — C2 — C2×C4 — C2×C8 — C2×Q16

Generators and relations for Q16.D10
G = < a,b,c,d | a8=1, b2=c10=d2=a4, bab-1=dad-1=a-1, ac=ca, cbc-1=a4b, dbd-1=a3b, dcd-1=c9 >

Subgroups: 334 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, Q16, C2×Q8, C4○D4, Dic5, C20, C20, D10, C2×C10, M5(2), SD32, Q32, C2×Q16, C4○D8, C40, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C2×C20, C5×Q8, Q32⋊C2, C52C16, C40⋊C2, D40, Dic20, C2×C40, C5×Q16, C5×Q16, C4○D20, Q8×C10, C20.4C8, C5⋊SD32, C5⋊Q32, D407C2, C10×Q16, Q16.D10
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, D10, C2×D8, C5⋊D4, C22×D5, Q32⋊C2, D4⋊D5, C2×C5⋊D4, C2×D4⋊D5, Q16.D10

Smallest permutation representation of Q16.D10
On 160 points
Generators in S160
```(1 98 122 23 11 88 132 33)(2 99 123 24 12 89 133 34)(3 100 124 25 13 90 134 35)(4 81 125 26 14 91 135 36)(5 82 126 27 15 92 136 37)(6 83 127 28 16 93 137 38)(7 84 128 29 17 94 138 39)(8 85 129 30 18 95 139 40)(9 86 130 31 19 96 140 21)(10 87 131 32 20 97 121 22)(41 72 109 143 51 62 119 153)(42 73 110 144 52 63 120 154)(43 74 111 145 53 64 101 155)(44 75 112 146 54 65 102 156)(45 76 113 147 55 66 103 157)(46 77 114 148 56 67 104 158)(47 78 115 149 57 68 105 159)(48 79 116 150 58 69 106 160)(49 80 117 151 59 70 107 141)(50 61 118 152 60 71 108 142)
(1 73 11 63)(2 64 12 74)(3 75 13 65)(4 66 14 76)(5 77 15 67)(6 68 16 78)(7 79 17 69)(8 70 18 80)(9 61 19 71)(10 72 20 62)(21 118 31 108)(22 109 32 119)(23 120 33 110)(24 111 34 101)(25 102 35 112)(26 113 36 103)(27 104 37 114)(28 115 38 105)(29 106 39 116)(30 117 40 107)(41 97 51 87)(42 88 52 98)(43 99 53 89)(44 90 54 100)(45 81 55 91)(46 92 56 82)(47 83 57 93)(48 94 58 84)(49 85 59 95)(50 96 60 86)(121 143 131 153)(122 154 132 144)(123 145 133 155)(124 156 134 146)(125 147 135 157)(126 158 136 148)(127 149 137 159)(128 160 138 150)(129 151 139 141)(130 142 140 152)
(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 99 31 89)(22 88 32 98)(23 97 33 87)(24 86 34 96)(25 95 35 85)(26 84 36 94)(27 93 37 83)(28 82 38 92)(29 91 39 81)(30 100 40 90)(41 144 51 154)(42 153 52 143)(43 142 53 152)(44 151 54 141)(45 160 55 150)(46 149 56 159)(47 158 57 148)(48 147 58 157)(49 156 59 146)(50 145 60 155)(61 111 71 101)(62 120 72 110)(63 109 73 119)(64 118 74 108)(65 107 75 117)(66 116 76 106)(67 105 77 115)(68 114 78 104)(69 103 79 113)(70 112 80 102)(121 132 131 122)(123 130 133 140)(124 139 134 129)(125 128 135 138)(126 137 136 127)```

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + - + + image C1 C2 C2 C2 C2 C2 D4 D4 D5 D8 D8 D10 D10 C5⋊D4 C5⋊D4 Q32⋊C2 D4⋊D5 D4⋊D5 Q16.D10 kernel Q16.D10 C20.4C8 C5⋊SD32 C5⋊Q32 D40⋊7C2 C10×Q16 C40 C2×C20 C2×Q16 C20 C2×C10 C2×C8 Q16 C8 C2×C4 C5 C4 C22 C1 # reps 1 1 2 2 1 1 1 1 2 2 2 2 4 4 4 2 2 2 8

Matrix representation of Q16.D10 in GL6(𝔽241)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 230 11 0 0 0 0 230 230 0 0 0 0 0 0 230 230 0 0 0 0 11 230
,
 240 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 240 0 0 0 0 0 0 240 0 0
,
 0 52 0 0 0 0 190 51 0 0 0 0 0 0 94 0 75 38 0 0 0 94 38 166 0 0 75 38 147 0 0 0 38 166 0 147
,
 190 52 0 0 0 0 191 51 0 0 0 0 0 0 94 0 75 38 0 0 0 147 203 75 0 0 75 203 70 171 0 0 38 75 171 171

`G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,230,230,0,0,0,0,11,230,0,0,0,0,0,0,230,11,0,0,0,0,230,230],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,1,0,0,0,0,0,0,1,0,0],[0,190,0,0,0,0,52,51,0,0,0,0,0,0,94,0,75,38,0,0,0,94,38,166,0,0,75,38,147,0,0,0,38,166,0,147],[190,191,0,0,0,0,52,51,0,0,0,0,0,0,94,0,75,38,0,0,0,147,203,75,0,0,75,203,70,171,0,0,38,75,171,171] >;`

Q16.D10 in GAP, Magma, Sage, TeX

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

`G:=Group("Q16.D10");`
`// GroupNames label`

`G:=SmallGroup(320,806);`
`// by ID`

`G=gap.SmallGroup(320,806);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,254,387,184,675,185,192,1684,438,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=1,b^2=c^10=d^2=a^4,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^4*b,d*b*d^-1=a^3*b,d*c*d^-1=c^9>;`
`// generators/relations`

׿
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