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## G = D20.D4order 320 = 26·5

### 9th non-split extension by D20 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — D20.D4
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×D20 — D20⋊8C4 — D20.D4
 Lower central C5 — C10 — C2×C20 — D20.D4
 Upper central C1 — C22 — C2×C4 — D4⋊C4

Generators and relations for D20.D4
G = < a,b,c,d | a20=b2=c4=1, d2=a5, bab=a-1, cac-1=a11, ad=da, bc=cb, dbd-1=a15b, dcd-1=a5c-1 >

Subgroups: 566 in 124 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, 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, C40, Dic10, C4×D5, D20, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×D4, C22×D5, C22×C10, D4.2D4, C40⋊C2, C2×C52C8, C4×Dic5, D10⋊C4, D4⋊D5, C23.D5, C5×C4⋊C4, C2×C40, C2×Dic10, C2×C4×D5, C2×D20, D4×C10, C10.Q16, C20.8Q8, C5×D4⋊C4, D208C4, C2×C40⋊C2, C2×D4⋊D5, C20.17D4, D20.D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C4○D8, C8⋊C22, C22×D5, D4.2D4, C4○D20, D4×D5, D10⋊D4, D8⋊D5, SD163D5, D20.D4

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

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

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

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

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 8 20 20 2 2 4 4 10 10 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 4 4 4 4 4 type + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D5 C4○D4 D10 D10 D10 C4○D8 C4○D20 C8⋊C22 D4×D5 D4×D5 D8⋊D5 SD16⋊3D5 kernel D20.D4 C10.Q16 C20.8Q8 C5×D4⋊C4 D20⋊8C4 C2×C40⋊C2 C2×D4⋊D5 C20.17D4 D20 C2×Dic5 D4⋊C4 C20 C4⋊C4 C2×C8 C2×D4 C10 C4 C10 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 8 1 2 2 4 4

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

 1 9 0 0 18 40 0 0 0 0 6 5 0 0 1 1
,
 40 32 0 0 0 1 0 0 0 0 1 36 0 0 0 40
,
 9 40 0 0 0 32 0 0 0 0 9 0 0 0 0 9
,
 0 29 0 0 17 30 0 0 0 0 2 4 0 0 9 39
`G:=sub<GL(4,GF(41))| [1,18,0,0,9,40,0,0,0,0,6,1,0,0,5,1],[40,0,0,0,32,1,0,0,0,0,1,0,0,0,36,40],[9,0,0,0,40,32,0,0,0,0,9,0,0,0,0,9],[0,17,0,0,29,30,0,0,0,0,2,9,0,0,4,39] >;`

D20.D4 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(320,414);`
`// by ID`

`G=gap.SmallGroup(320,414);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,64,590,555,1684,851,438,102,12550]);`
`// Polycyclic`

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

׿
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