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## G = D20.19D4order 320 = 26·5

### 2nd non-split extension by D20 of D4 acting via D4/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — D20.19D4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×D20 — C4×D20 — D20.19D4
 Lower central C5 — C10 — C2×C20 — D20.19D4
 Upper central C1 — C22 — C42 — C4⋊C8

Generators and relations for D20.19D4
G = < a,b,c,d | a20=b2=c4=1, d2=a15, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a15b, dcd-1=a10c-1 >

Subgroups: 662 in 124 conjugacy classes, 41 normal (39 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C2×C8, D8, SD16, C22×C4, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, D4⋊C4, Q8⋊C4, C4⋊C8, C4×D4, C4.4D4, C2×D8, C2×SD16, C40, Dic10, C4×D5, D20, D20, C2×Dic5, C2×C20, C22×D5, D4.2D4, C40⋊C2, D40, C4⋊Dic5, D10⋊C4, C4×C20, C2×C40, C2×Dic10, C2×C4×D5, C2×D20, C20.44D4, D205C4, C5×C4⋊C8, C4×D20, C4.D20, C2×C40⋊C2, C2×D40, D20.19D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C4○D8, C8⋊C22, D20, C22×D5, D4.2D4, C2×D20, D4×D5, Q82D5, C4⋊D20, D407C2, C8⋊D10, D20.19D4

Smallest permutation representation of D20.19D4
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 38)(2 37)(3 36)(4 35)(5 34)(6 33)(7 32)(8 31)(9 30)(10 29)(11 28)(12 27)(13 26)(14 25)(15 24)(16 23)(17 22)(18 21)(19 40)(20 39)(41 119)(42 118)(43 117)(44 116)(45 115)(46 114)(47 113)(48 112)(49 111)(50 110)(51 109)(52 108)(53 107)(54 106)(55 105)(56 104)(57 103)(58 102)(59 101)(60 120)(61 121)(62 140)(63 139)(64 138)(65 137)(66 136)(67 135)(68 134)(69 133)(70 132)(71 131)(72 130)(73 129)(74 128)(75 127)(76 126)(77 125)(78 124)(79 123)(80 122)(81 153)(82 152)(83 151)(84 150)(85 149)(86 148)(87 147)(88 146)(89 145)(90 144)(91 143)(92 142)(93 141)(94 160)(95 159)(96 158)(97 157)(98 156)(99 155)(100 154)
(1 64 48 86)(2 65 49 87)(3 66 50 88)(4 67 51 89)(5 68 52 90)(6 69 53 91)(7 70 54 92)(8 71 55 93)(9 72 56 94)(10 73 57 95)(11 74 58 96)(12 75 59 97)(13 76 60 98)(14 77 41 99)(15 78 42 100)(16 79 43 81)(17 80 44 82)(18 61 45 83)(19 62 46 84)(20 63 47 85)(21 121 115 151)(22 122 116 152)(23 123 117 153)(24 124 118 154)(25 125 119 155)(26 126 120 156)(27 127 101 157)(28 128 102 158)(29 129 103 159)(30 130 104 160)(31 131 105 141)(32 132 106 142)(33 133 107 143)(34 134 108 144)(35 135 109 145)(36 136 110 146)(37 137 111 147)(38 138 112 148)(39 139 113 149)(40 140 114 150)
(1 108 16 103 11 118 6 113)(2 109 17 104 12 119 7 114)(3 110 18 105 13 120 8 115)(4 111 19 106 14 101 9 116)(5 112 20 107 15 102 10 117)(21 50 36 45 31 60 26 55)(22 51 37 46 32 41 27 56)(23 52 38 47 33 42 28 57)(24 53 39 48 34 43 29 58)(25 54 40 49 35 44 30 59)(61 121 76 136 71 131 66 126)(62 122 77 137 72 132 67 127)(63 123 78 138 73 133 68 128)(64 124 79 139 74 134 69 129)(65 125 80 140 75 135 70 130)(81 149 96 144 91 159 86 154)(82 150 97 145 92 160 87 155)(83 151 98 146 93 141 88 156)(84 152 99 147 94 142 89 157)(85 153 100 148 95 143 90 158)```

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

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

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

59 conjugacy classes

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

59 irreducible representations

 dim 1 1 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 C2 C2 D4 D4 D5 C4○D4 D10 D10 C4○D8 D20 D40⋊7C2 C8⋊C22 D4×D5 Q8⋊2D5 C8⋊D10 kernel D20.19D4 C20.44D4 D20⋊5C4 C5×C4⋊C8 C4×D20 C4.D20 C2×C40⋊C2 C2×D40 D20 C2×C20 C4⋊C8 C20 C42 C2×C8 C10 C2×C4 C2 C10 C4 C4 C2 # reps 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 8 16 1 2 2 4

Matrix representation of D20.19D4 in GL4(𝔽41) generated by

 30 39 0 0 16 14 0 0 0 0 40 0 0 0 0 40
,
 38 8 0 0 40 3 0 0 0 0 32 23 0 0 9 9
,
 32 0 0 0 0 32 0 0 0 0 40 39 0 0 1 1
,
 20 33 0 0 23 38 0 0 0 0 32 23 0 0 0 9
`G:=sub<GL(4,GF(41))| [30,16,0,0,39,14,0,0,0,0,40,0,0,0,0,40],[38,40,0,0,8,3,0,0,0,0,32,9,0,0,23,9],[32,0,0,0,0,32,0,0,0,0,40,1,0,0,39,1],[20,23,0,0,33,38,0,0,0,0,32,0,0,0,23,9] >;`

D20.19D4 in GAP, Magma, Sage, TeX

`D_{20}._{19}D_4`
`% in TeX`

`G:=Group("D20.19D4");`
`// GroupNames label`

`G:=SmallGroup(320,471);`
`// by ID`

`G=gap.SmallGroup(320,471);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,344,254,219,58,1123,136,12550]);`
`// Polycyclic`

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

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