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## G = D20.2Q8order 320 = 26·5

### 2nd non-split extension by D20 of Q8 acting via Q8/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — D20.2Q8
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×D20 — D20⋊8C4 — D20.2Q8
 Lower central C5 — C10 — C2×C20 — D20.2Q8
 Upper central C1 — C22 — C2×C4 — C2.D8

Generators and relations for D20.2Q8
G = < a,b,c,d | a20=b2=1, c4=a10, d2=a15c2, bab=a-1, ac=ca, dad-1=a11, cbc-1=a15b, bd=db, dcd-1=a10c3 >

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, C20.Q8, D206C4, C20.8Q8, D205C4, C5×C2.D8, C4.Dic10, D208C4, D20.2Q8
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, D8⋊D5, Q8.D10, D20.2Q8

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

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

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

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

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 D8⋊D5 Q8.D10 kernel D20.2Q8 C20.Q8 D20⋊6C4 C20.8Q8 D20⋊5C4 C5×C2.D8 C4.Dic10 D20⋊8C4 D20 C2×Dic5 C2.D8 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 D20.2Q8 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 40 6 0 0 0 0 0 0 1 21 0 0 0 0 37 40
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 40 20 0 0 0 0 0 1
,
 0 1 0 0 0 0 40 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 6 0 0 0 0 34 17
,
 32 0 0 0 0 0 0 9 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 32 16 0 0 0 0 0 9

`G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,6,0,0,0,0,0,0,1,37,0,0,0,0,21,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,20,1],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,34,0,0,0,0,6,17],[32,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,16,9] >;`

D20.2Q8 in GAP, Magma, Sage, TeX

`D_{20}._2Q_8`
`% in TeX`

`G:=Group("D20.2Q8");`
`// GroupNames label`

`G:=SmallGroup(320,518);`
`// by ID`

`G=gap.SmallGroup(320,518);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,64,590,555,100,1684,851,102,12550]);`
`// Polycyclic`

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

׿
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