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## G = D20⋊7Q8order 320 = 26·5

### 5th semidirect product of D20 and Q8 acting via Q8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — D20⋊7Q8
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D5×C4⋊C4 — D20⋊7Q8
 Lower central C5 — C2×C10 — D20⋊7Q8
 Upper central C1 — C22 — C42.C2

Generators and relations for D207Q8
G = < a,b,c,d | a20=b2=c4=1, d2=c2, bab=a-1, ac=ca, dad-1=a9, cbc-1=a10b, dbd-1=a18b, dcd-1=c-1 >

Subgroups: 806 in 228 conjugacy classes, 105 normal (43 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×13], 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 [×2], C4⋊C4 [×4], C4⋊C4 [×10], C22×C4 [×6], C2×D4, C2×Q8 [×3], Dic5 [×2], Dic5 [×5], C20 [×2], C20 [×6], D10 [×4], D10 [×4], C2×C10, C2×C4⋊C4 [×2], C4×D4 [×3], C4×Q8, C22⋊Q8 [×6], C42.C2, C42.C2, C4⋊Q8, Dic10 [×4], C4×D5 [×8], D20 [×4], C2×Dic5 [×4], C2×Dic5 [×2], C2×C20 [×3], C2×C20 [×4], C22×D5 [×2], D43Q8, C4×Dic5 [×2], C10.D4 [×2], C10.D4 [×4], C4⋊Dic5 [×2], C4⋊Dic5 [×2], D10⋊C4 [×6], C4×C20, C5×C4⋊C4 [×2], C5×C4⋊C4 [×4], C2×Dic10, C2×Dic10 [×2], C2×C4×D5 [×6], C2×D20, C4×Dic10, C4×D20, C20⋊Q8, C4.Dic10, D5×C4⋊C4 [×2], D208C4 [×2], D10⋊Q8 [×4], D102Q8 [×2], C5×C42.C2, D207Q8
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], C23×D5, C2×Q8×D5, D5×C4○D4, D48D10, D207Q8

Smallest permutation representation of D207Q8
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)
(1 123)(2 122)(3 121)(4 140)(5 139)(6 138)(7 137)(8 136)(9 135)(10 134)(11 133)(12 132)(13 131)(14 130)(15 129)(16 128)(17 127)(18 126)(19 125)(20 124)(21 61)(22 80)(23 79)(24 78)(25 77)(26 76)(27 75)(28 74)(29 73)(30 72)(31 71)(32 70)(33 69)(34 68)(35 67)(36 66)(37 65)(38 64)(39 63)(40 62)(41 153)(42 152)(43 151)(44 150)(45 149)(46 148)(47 147)(48 146)(49 145)(50 144)(51 143)(52 142)(53 141)(54 160)(55 159)(56 158)(57 157)(58 156)(59 155)(60 154)(81 111)(82 110)(83 109)(84 108)(85 107)(86 106)(87 105)(88 104)(89 103)(90 102)(91 101)(92 120)(93 119)(94 118)(95 117)(96 116)(97 115)(98 114)(99 113)(100 112)
(1 101 139 97)(2 102 140 98)(3 103 121 99)(4 104 122 100)(5 105 123 81)(6 106 124 82)(7 107 125 83)(8 108 126 84)(9 109 127 85)(10 110 128 86)(11 111 129 87)(12 112 130 88)(13 113 131 89)(14 114 132 90)(15 115 133 91)(16 116 134 92)(17 117 135 93)(18 118 136 94)(19 119 137 95)(20 120 138 96)(21 152 77 48)(22 153 78 49)(23 154 79 50)(24 155 80 51)(25 156 61 52)(26 157 62 53)(27 158 63 54)(28 159 64 55)(29 160 65 56)(30 141 66 57)(31 142 67 58)(32 143 68 59)(33 144 69 60)(34 145 70 41)(35 146 71 42)(36 147 72 43)(37 148 73 44)(38 149 74 45)(39 150 75 46)(40 151 76 47)
(1 58 139 142)(2 47 140 151)(3 56 121 160)(4 45 122 149)(5 54 123 158)(6 43 124 147)(7 52 125 156)(8 41 126 145)(9 50 127 154)(10 59 128 143)(11 48 129 152)(12 57 130 141)(13 46 131 150)(14 55 132 159)(15 44 133 148)(16 53 134 157)(17 42 135 146)(18 51 136 155)(19 60 137 144)(20 49 138 153)(21 111 77 87)(22 120 78 96)(23 109 79 85)(24 118 80 94)(25 107 61 83)(26 116 62 92)(27 105 63 81)(28 114 64 90)(29 103 65 99)(30 112 66 88)(31 101 67 97)(32 110 68 86)(33 119 69 95)(34 108 70 84)(35 117 71 93)(36 106 72 82)(37 115 73 91)(38 104 74 100)(39 113 75 89)(40 102 76 98)```

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

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

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

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 1 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + - + + + + - + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 Q8 D5 C4○D4 D10 D10 2+ 1+4 Q8×D5 D5×C4○D4 D4⋊8D10 kernel D20⋊7Q8 C4×Dic10 C4×D20 C20⋊Q8 C4.Dic10 D5×C4⋊C4 D20⋊8C4 D10⋊Q8 D10⋊2Q8 C5×C42.C2 D20 C42.C2 Dic5 C42 C4⋊C4 C10 C4 C2 C2 # reps 1 1 1 1 1 2 2 4 2 1 4 2 4 2 12 1 4 4 4

Matrix representation of D207Q8 in GL6(𝔽41)

 34 1 0 0 0 0 40 0 0 0 0 0 0 0 0 32 0 0 0 0 32 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 34 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 40 0 0 0 0 40 0 0 0 0 0 0 0 1 39 0 0 0 0 1 40
,
 34 7 0 0 0 0 40 7 0 0 0 0 0 0 0 40 0 0 0 0 40 0 0 0 0 0 0 0 7 13 0 0 0 0 34 34

`G:=sub<GL(6,GF(41))| [34,40,0,0,0,0,1,0,0,0,0,0,0,0,0,32,0,0,0,0,32,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,34,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,0,0,0,1,1,0,0,0,0,39,40],[34,40,0,0,0,0,7,7,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,0,0,0,7,34,0,0,0,0,13,34] >;`

D207Q8 in GAP, Magma, Sage, TeX

`D_{20}\rtimes_7Q_8`
`% in TeX`

`G:=Group("D20:7Q8");`
`// GroupNames label`

`G:=SmallGroup(320,1362);`
`// by ID`

`G=gap.SmallGroup(320,1362);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,219,184,1571,297,80,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^20=b^2=c^4=1,d^2=c^2,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^9,c*b*c^-1=a^10*b,d*b*d^-1=a^18*b,d*c*d^-1=c^-1>;`
`// generators/relations`

׿
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