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## G = D20.Q8order 320 = 26·5

### 1st non-split extension by D20 of Q8 acting via Q8/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — D20.Q8
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×D20 — D20⋊8C4 — D20.Q8
 Lower central C5 — C10 — C2×C20 — D20.Q8
 Upper central C1 — C22 — C2×C4 — C4.Q8

Generators and relations for D20.Q8
G = < a,b,c,d | a20=b2=1, c4=a10, d2=a5c2, bab=a-1, ac=ca, dad-1=a11, cbc-1=a15b, bd=db, dcd-1=c3 >

Subgroups: 454 in 102 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, Dic5, C20, C20, D10, C2×C10, D4⋊C4, C4⋊C8, C4.Q8, C2.D8, C4×D4, C42.C2, C52C8, C40, C4×D5, D20, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, D4.Q8, C2×C52C8, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C2×C40, C2×C4×D5, C2×D20, C10.D8, D206C4, C20.8Q8, D205C4, C5×C4.Q8, C4.Dic10, D208C4, D20.Q8
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, C4○D8, C8⋊C22, C22×D5, D4.Q8, C4○D20, D4×D5, Q8×D5, D10⋊Q8, D40⋊C2, SD163D5, D20.Q8

Smallest permutation representation of D20.Q8
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)
(1 155)(2 154)(3 153)(4 152)(5 151)(6 150)(7 149)(8 148)(9 147)(10 146)(11 145)(12 144)(13 143)(14 142)(15 141)(16 160)(17 159)(18 158)(19 157)(20 156)(21 102)(22 101)(23 120)(24 119)(25 118)(26 117)(27 116)(28 115)(29 114)(30 113)(31 112)(32 111)(33 110)(34 109)(35 108)(36 107)(37 106)(38 105)(39 104)(40 103)(41 68)(42 67)(43 66)(44 65)(45 64)(46 63)(47 62)(48 61)(49 80)(50 79)(51 78)(52 77)(53 76)(54 75)(55 74)(56 73)(57 72)(58 71)(59 70)(60 69)(81 129)(82 128)(83 127)(84 126)(85 125)(86 124)(87 123)(88 122)(89 121)(90 140)(91 139)(92 138)(93 137)(94 136)(95 135)(96 134)(97 133)(98 132)(99 131)(100 130)
(1 89 146 127 11 99 156 137)(2 90 147 128 12 100 157 138)(3 91 148 129 13 81 158 139)(4 92 149 130 14 82 159 140)(5 93 150 131 15 83 160 121)(6 94 151 132 16 84 141 122)(7 95 152 133 17 85 142 123)(8 96 153 134 18 86 143 124)(9 97 154 135 19 87 144 125)(10 98 155 136 20 88 145 126)(21 78 115 59 31 68 105 49)(22 79 116 60 32 69 106 50)(23 80 117 41 33 70 107 51)(24 61 118 42 34 71 108 52)(25 62 119 43 35 72 109 53)(26 63 120 44 36 73 110 54)(27 64 101 45 37 74 111 55)(28 65 102 46 38 75 112 56)(29 66 103 47 39 76 113 57)(30 67 104 48 40 77 114 58)
(1 40 151 119)(2 31 152 110)(3 22 153 101)(4 33 154 112)(5 24 155 103)(6 35 156 114)(7 26 157 105)(8 37 158 116)(9 28 159 107)(10 39 160 118)(11 30 141 109)(12 21 142 120)(13 32 143 111)(14 23 144 102)(15 34 145 113)(16 25 146 104)(17 36 147 115)(18 27 148 106)(19 38 149 117)(20 29 150 108)(41 125 75 82)(42 136 76 93)(43 127 77 84)(44 138 78 95)(45 129 79 86)(46 140 80 97)(47 131 61 88)(48 122 62 99)(49 133 63 90)(50 124 64 81)(51 135 65 92)(52 126 66 83)(53 137 67 94)(54 128 68 85)(55 139 69 96)(56 130 70 87)(57 121 71 98)(58 132 72 89)(59 123 73 100)(60 134 74 91)```

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

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

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

47 conjugacy classes

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

47 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + - + + + + + - + + image C1 C2 C2 C2 C2 C2 C2 C2 Q8 D4 D5 C4○D4 D10 D10 C4○D8 C4○D20 C8⋊C22 Q8×D5 D4×D5 D40⋊C2 SD16⋊3D5 kernel D20.Q8 C10.D8 D20⋊6C4 C20.8Q8 D20⋊5C4 C5×C4.Q8 C4.Dic10 D20⋊8C4 D20 C2×Dic5 C4.Q8 C20 C4⋊C4 C2×C8 C10 C4 C10 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 2 2 2 2 4 2 4 8 1 2 2 4 4

Matrix representation of D20.Q8 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 1 0 0 0 0 40 34 0 0 0 0 0 0 21 32 0 0 0 0 40 20
,
 40 0 0 0 0 0 40 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 40 0 0 0 0 0 9 1
,
 1 39 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 28 12 0 0 0 0 15 2
,
 9 0 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 9 0 0 0 0 0 1 32

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,1,34,0,0,0,0,0,0,21,40,0,0,0,0,32,20],[40,40,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,40,9,0,0,0,0,0,1],[1,0,0,0,0,0,39,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,28,15,0,0,0,0,12,2],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,1,0,0,0,0,0,32] >;`

D20.Q8 in GAP, Magma, Sage, TeX

`D_{20}.Q_8`
`% in TeX`

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

`G:=SmallGroup(320,498);`
`// by ID`

`G=gap.SmallGroup(320,498);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,64,590,555,268,1684,851,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^20=b^2=1,c^4=a^10,d^2=a^5*c^2,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^11,c*b*c^-1=a^15*b,b*d=d*b,d*c*d^-1=c^3>;`
`// generators/relations`

׿
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