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G = D4.D20order 320 = 26·5

2nd non-split extension by D4 of D20 acting via D20/D10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — D4.D20
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×C4×D5 — C2×D4⋊2D5 — D4.D20
 Lower central C5 — C10 — C2×C20 — D4.D20
 Upper central C1 — C22 — C2×C4 — D4⋊C4

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

Subgroups: 638 in 152 conjugacy classes, 43 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, Q16, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C22⋊C8, D4⋊C4, Q8⋊C4, C22⋊Q8, C2×SD16, C2×Q16, C2×C4○D4, C52C8, C40, Dic10, Dic10, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C22×D5, C22×C10, D4.7D4, Dic20, C2×C52C8, C4⋊Dic5, D10⋊C4, D4.D5, C5×C4⋊C4, C2×C40, C2×Dic10, C2×C4×D5, D42D5, C22×Dic5, C2×C5⋊D4, D4×C10, C10.Q16, D101C8, C5×D4⋊C4, D102Q8, C2×Dic20, C2×D4.D5, C2×D42D5, D4.D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C22≀C2, C4○D8, C8.C22, D20, C22×D5, D4.7D4, C2×D20, D4×D5, C22⋊D20, D83D5, SD16⋊D5, D4.D20

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

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

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

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

47 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 10H 10I 10J 20A 20B 20C 20D 20E 20F 20G 20H 40A ··· 40H 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 10 10 20 20 20 20 20 20 20 20 40 ··· 40 size 1 1 1 1 4 4 20 2 2 8 10 10 20 20 40 2 2 4 4 20 20 2 ··· 2 8 8 8 8 4 4 4 4 8 8 8 8 4 ··· 4

47 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + + + - + + - - image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D4 D5 D10 D10 D10 C4○D8 D20 C8.C22 D4×D5 D4×D5 D8⋊3D5 SD16⋊D5 kernel D4.D20 C10.Q16 D10⋊1C8 C5×D4⋊C4 D10⋊2Q8 C2×Dic20 C2×D4.D5 C2×D4⋊2D5 Dic10 C2×Dic5 C5×D4 C22×D5 D4⋊C4 C4⋊C4 C2×C8 C2×D4 C10 D4 C10 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 2 1 2 1 2 2 2 2 4 8 1 2 2 4 4

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

 32 0 0 0 40 9 0 0 0 0 1 0 0 0 0 1
,
 19 27 0 0 14 22 0 0 0 0 40 0 0 0 0 40
,
 20 9 0 0 24 21 0 0 0 0 27 30 0 0 11 32
,
 20 9 0 0 1 21 0 0 0 0 11 14 0 0 9 30
`G:=sub<GL(4,GF(41))| [32,40,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[19,14,0,0,27,22,0,0,0,0,40,0,0,0,0,40],[20,24,0,0,9,21,0,0,0,0,27,11,0,0,30,32],[20,1,0,0,9,21,0,0,0,0,11,9,0,0,14,30] >;`

D4.D20 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(320,410);`
`// by ID`

`G=gap.SmallGroup(320,410);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,254,219,226,851,438,102,12550]);`
`// Polycyclic`

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

׿
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