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## G = D4.1D20order 320 = 26·5

### 1st non-split extension by D4 of D20 acting via D20/C20=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — D4.1D20
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×D20 — C4.D20 — D4.1D20
 Lower central C5 — C10 — C2×C20 — D4.1D20
 Upper central C1 — C22 — C42 — C4×D4

Generators and relations for D4.1D20
G = < a,b,c,d | a4=b2=c20=1, d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=a2c-1 >

Subgroups: 502 in 124 conjugacy classes, 43 normal (39 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, D8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C2×C10, D4⋊C4, Q8⋊C4, C4⋊C8, C4×D4, C4.4D4, C2×D8, C2×SD16, C52C8, Dic10, D20, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×D4, C22×D5, C22×C10, D4.2D4, C2×C52C8, D10⋊C4, D4⋊D5, D4.D5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×D20, C22×C20, D4×C10, C203C8, D206C4, C10.Q16, C4.D20, C2×D4⋊D5, C2×D4.D5, D4×C20, D4.1D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C4○D8, C8⋊C22, D20, C5⋊D4, C22×D5, D4.2D4, C2×D20, C4○D20, C2×C5⋊D4, C207D4, D4.D10, D4.8D10, D4.1D20

Smallest permutation representation of D4.1D20
On 160 points
Generators in S160
```(1 139 141 104)(2 140 142 105)(3 121 143 106)(4 122 144 107)(5 123 145 108)(6 124 146 109)(7 125 147 110)(8 126 148 111)(9 127 149 112)(10 128 150 113)(11 129 151 114)(12 130 152 115)(13 131 153 116)(14 132 154 117)(15 133 155 118)(16 134 156 119)(17 135 157 120)(18 136 158 101)(19 137 159 102)(20 138 160 103)(21 81 64 45)(22 82 65 46)(23 83 66 47)(24 84 67 48)(25 85 68 49)(26 86 69 50)(27 87 70 51)(28 88 71 52)(29 89 72 53)(30 90 73 54)(31 91 74 55)(32 92 75 56)(33 93 76 57)(34 94 77 58)(35 95 78 59)(36 96 79 60)(37 97 80 41)(38 98 61 42)(39 99 62 43)(40 100 63 44)
(1 60)(2 41)(3 42)(4 43)(5 44)(6 45)(7 46)(8 47)(9 48)(10 49)(11 50)(12 51)(13 52)(14 53)(15 54)(16 55)(17 56)(18 57)(19 58)(20 59)(21 109)(22 110)(23 111)(24 112)(25 113)(26 114)(27 115)(28 116)(29 117)(30 118)(31 119)(32 120)(33 101)(34 102)(35 103)(36 104)(37 105)(38 106)(39 107)(40 108)(61 121)(62 122)(63 123)(64 124)(65 125)(66 126)(67 127)(68 128)(69 129)(70 130)(71 131)(72 132)(73 133)(74 134)(75 135)(76 136)(77 137)(78 138)(79 139)(80 140)(81 146)(82 147)(83 148)(84 149)(85 150)(86 151)(87 152)(88 153)(89 154)(90 155)(91 156)(92 157)(93 158)(94 159)(95 160)(96 141)(97 142)(98 143)(99 144)(100 145)
(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)
(1 160 141 20)(2 19 142 159)(3 158 143 18)(4 17 144 157)(5 156 145 16)(6 15 146 155)(7 154 147 14)(8 13 148 153)(9 152 149 12)(10 11 150 151)(21 90 64 54)(22 53 65 89)(23 88 66 52)(24 51 67 87)(25 86 68 50)(26 49 69 85)(27 84 70 48)(28 47 71 83)(29 82 72 46)(30 45 73 81)(31 100 74 44)(32 43 75 99)(33 98 76 42)(34 41 77 97)(35 96 78 60)(36 59 79 95)(37 94 80 58)(38 57 61 93)(39 92 62 56)(40 55 63 91)(101 121 136 106)(102 105 137 140)(103 139 138 104)(107 135 122 120)(108 119 123 134)(109 133 124 118)(110 117 125 132)(111 131 126 116)(112 115 127 130)(113 129 128 114)```

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

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

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

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 10G ··· 10N 20A ··· 20H 20I ··· 20X order 1 2 2 2 2 2 2 4 4 4 4 4 4 4 4 5 5 8 8 8 8 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 4 4 40 2 2 2 2 4 4 4 40 2 2 20 20 20 20 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

59 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D5 C4○D4 D10 D10 D10 C4○D8 C5⋊D4 D20 C4○D20 C8⋊C22 D4.D10 D4.8D10 kernel D4.1D20 C20⋊3C8 D20⋊6C4 C10.Q16 C4.D20 C2×D4⋊D5 C2×D4.D5 D4×C20 C2×C20 C5×D4 C4×D4 C20 C42 C4⋊C4 C2×D4 C10 C2×C4 D4 C4 C10 C2 C2 # reps 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 8 8 8 1 4 4

Matrix representation of D4.1D20 in GL4(𝔽41) generated by

 1 0 0 0 0 1 0 0 0 0 1 21 0 0 37 40
,
 40 0 0 0 0 40 0 0 0 0 0 35 0 0 34 0
,
 16 30 0 0 27 2 0 0 0 0 32 0 0 0 0 32
,
 39 30 0 0 4 2 0 0 0 0 32 0 0 0 36 9
`G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,1,37,0,0,21,40],[40,0,0,0,0,40,0,0,0,0,0,34,0,0,35,0],[16,27,0,0,30,2,0,0,0,0,32,0,0,0,0,32],[39,4,0,0,30,2,0,0,0,0,32,36,0,0,0,9] >;`

D4.1D20 in GAP, Magma, Sage, TeX

`D_4._1D_{20}`
`% in TeX`

`G:=Group("D4.1D20");`
`// GroupNames label`

`G:=SmallGroup(320,643);`
`// by ID`

`G=gap.SmallGroup(320,643);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,254,1123,297,136,12550]);`
`// Polycyclic`

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

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