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## G = D20⋊9Q8order 320 = 26·5

### 7th semidirect product of D20 and Q8 acting via Q8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — D20⋊9Q8
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×D20 — C4×D20 — D20⋊9Q8
 Lower central C5 — C2×C10 — D20⋊9Q8
 Upper central C1 — C22 — C4⋊Q8

Generators and relations for D209Q8
G = < a,b,c,d | a20=b2=c4=1, d2=c2, bab=a-1, cac-1=a11, ad=da, cbc-1=dbd-1=a10b, dcd-1=c-1 >

Subgroups: 774 in 228 conjugacy classes, 107 normal (27 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×11], C22, C22 [×8], C5, C2×C4 [×3], C2×C4 [×4], C2×C4 [×14], D4 [×4], Q8 [×4], C23 [×2], D5 [×4], C10 [×3], C42, C42 [×2], C22⋊C4 [×6], C4⋊C4 [×4], C4⋊C4 [×12], C22×C4 [×6], C2×D4, C2×Q8 [×2], C2×Q8, Dic5 [×6], C20 [×4], C20 [×5], D10 [×4], D10 [×4], C2×C10, C2×C4⋊C4 [×2], C4×D4 [×3], C4×Q8, C22⋊Q8 [×6], C42.C2 [×2], C4⋊Q8, Dic10 [×2], C4×D5 [×8], D20 [×4], C2×Dic5 [×6], C2×C20 [×3], C2×C20 [×4], C5×Q8 [×2], C22×D5 [×2], D43Q8, C4×Dic5 [×2], C10.D4 [×6], C4⋊Dic5 [×2], C4⋊Dic5 [×4], D10⋊C4 [×6], C4×C20, C5×C4⋊C4 [×4], C2×Dic10, C2×C4×D5 [×6], C2×D20, Q8×C10 [×2], C4×Dic10, C4×D20, C4.Dic10 [×2], D5×C4⋊C4 [×2], D208C4 [×2], D102Q8 [×2], D103Q8 [×4], C5×C4⋊Q8, D209Q8
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D5, C2×Q8 [×6], C4○D4 [×2], C24, D10 [×7], C22×Q8, C2×C4○D4, 2+ 1+4, C22×D5 [×7], D43Q8, Q8×D5 [×2], Q82D5 [×2], C23×D5, D46D10, C2×Q8×D5, C2×Q82D5, D209Q8

Smallest permutation representation of D209Q8
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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 37)(22 36)(23 35)(24 34)(25 33)(26 32)(27 31)(28 30)(38 40)(41 49)(42 48)(43 47)(44 46)(50 60)(51 59)(52 58)(53 57)(54 56)(61 71)(62 70)(63 69)(64 68)(65 67)(72 80)(73 79)(74 78)(75 77)(81 93)(82 92)(83 91)(84 90)(85 89)(86 88)(94 100)(95 99)(96 98)(102 120)(103 119)(104 118)(105 117)(106 116)(107 115)(108 114)(109 113)(110 112)(121 131)(122 130)(123 129)(124 128)(125 127)(132 140)(133 139)(134 138)(135 137)(141 143)(144 160)(145 159)(146 158)(147 157)(148 156)(149 155)(150 154)(151 153)
(1 150 104 43)(2 141 105 54)(3 152 106 45)(4 143 107 56)(5 154 108 47)(6 145 109 58)(7 156 110 49)(8 147 111 60)(9 158 112 51)(10 149 113 42)(11 160 114 53)(12 151 115 44)(13 142 116 55)(14 153 117 46)(15 144 118 57)(16 155 119 48)(17 146 120 59)(18 157 101 50)(19 148 102 41)(20 159 103 52)(21 84 68 133)(22 95 69 124)(23 86 70 135)(24 97 71 126)(25 88 72 137)(26 99 73 128)(27 90 74 139)(28 81 75 130)(29 92 76 121)(30 83 77 132)(31 94 78 123)(32 85 79 134)(33 96 80 125)(34 87 61 136)(35 98 62 127)(36 89 63 138)(37 100 64 129)(38 91 65 140)(39 82 66 131)(40 93 67 122)
(1 95 104 124)(2 96 105 125)(3 97 106 126)(4 98 107 127)(5 99 108 128)(6 100 109 129)(7 81 110 130)(8 82 111 131)(9 83 112 132)(10 84 113 133)(11 85 114 134)(12 86 115 135)(13 87 116 136)(14 88 117 137)(15 89 118 138)(16 90 119 139)(17 91 120 140)(18 92 101 121)(19 93 102 122)(20 94 103 123)(21 42 68 149)(22 43 69 150)(23 44 70 151)(24 45 71 152)(25 46 72 153)(26 47 73 154)(27 48 74 155)(28 49 75 156)(29 50 76 157)(30 51 77 158)(31 52 78 159)(32 53 79 160)(33 54 80 141)(34 55 61 142)(35 56 62 143)(36 57 63 144)(37 58 64 145)(38 59 65 146)(39 60 66 147)(40 41 67 148)```

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

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

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

53 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E ··· 4I 4J 4K 4L 4M 4N 4O 4P 4Q 5A 5B 10A ··· 10F 20A ··· 20L 20M ··· 20T order 1 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 4 4 4 4 4 4 4 4 5 5 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 10 10 10 10 2 2 2 2 4 ··· 4 10 10 10 10 20 20 20 20 2 2 2 ··· 2 4 ··· 4 8 ··· 8

53 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + - + + + + + - + image C1 C2 C2 C2 C2 C2 C2 C2 C2 Q8 D5 C4○D4 D10 D10 D10 2+ 1+4 Q8×D5 Q8⋊2D5 D4⋊6D10 kernel D20⋊9Q8 C4×Dic10 C4×D20 C4.Dic10 D5×C4⋊C4 D20⋊8C4 D10⋊2Q8 D10⋊3Q8 C5×C4⋊Q8 D20 C4⋊Q8 C20 C42 C4⋊C4 C2×Q8 C10 C4 C4 C2 # reps 1 1 1 2 2 2 2 4 1 4 2 4 2 8 4 1 4 4 4

Matrix representation of D209Q8 in GL6(𝔽41)

 9 23 0 0 0 0 0 32 0 0 0 0 0 0 35 40 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 40 0 0 0 0 0 40 1 0 0 0 0 0 0 1 6 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 39 0 0 0 0 1 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 1 0 0 0 0 40 0
,
 32 18 0 0 0 0 0 9 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 14 7 0 0 0 0 7 27

`G:=sub<GL(6,GF(41))| [9,0,0,0,0,0,23,32,0,0,0,0,0,0,35,1,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,40,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,6,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,1,0,0,0,0,39,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[32,0,0,0,0,0,18,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,14,7,0,0,0,0,7,27] >;`

D209Q8 in GAP, Magma, Sage, TeX

`D_{20}\rtimes_9Q_8`
`% in TeX`

`G:=Group("D20:9Q8");`
`// GroupNames label`

`G:=SmallGroup(320,1402);`
`// by ID`

`G=gap.SmallGroup(320,1402);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,232,100,570,185,192,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^20=b^2=c^4=1,d^2=c^2,b*a*b=a^-1,c*a*c^-1=a^11,a*d=d*a,c*b*c^-1=d*b*d^-1=a^10*b,d*c*d^-1=c^-1>;`
`// generators/relations`

׿
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