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G = D20.47D4order 320 = 26·5

3rd non-split extension by D20 of D4 acting through Inn(D20)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — D20.47D4
 Chief series C1 — C5 — C10 — C20 — C4×D5 — C4○D20 — D4.10D10 — D20.47D4
 Lower central C5 — C10 — C20 — D20.47D4
 Upper central C1 — C2 — C2×C4 — C4○D8

Generators and relations for D20.47D4
G = < a,b,c,d | a8=b2=c10=1, d2=a4, bab=a-1, ac=ca, ad=da, cbc-1=a4b, bd=db, dcd-1=a4c-1 >

Subgroups: 854 in 248 conjugacy classes, 99 normal (31 characteristic)
C1, C2, C2 [×5], C4 [×2], C4 [×8], C22, C22 [×4], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×14], D4 [×2], D4 [×9], Q8 [×2], Q8 [×11], D5 [×2], C10, C10 [×3], C2×C8, C2×C8 [×2], M4(2) [×3], D8, SD16 [×2], SD16 [×4], Q16, Q16 [×8], C2×Q8 [×8], C4○D4 [×2], C4○D4 [×11], Dic5 [×2], Dic5 [×4], C20 [×2], C20 [×2], D10 [×2], C2×C10, C2×C10 [×2], C8○D4, C2×Q16 [×3], C4○D8, C4○D8 [×2], C8.C22 [×6], 2- 1+4 [×2], C52C8 [×2], C40 [×2], Dic10, Dic10 [×4], Dic10 [×6], C4×D5 [×2], C4×D5 [×4], D20, C2×Dic5 [×6], C5⋊D4 [×2], C5⋊D4 [×4], C2×C20, C2×C20 [×2], C5×D4 [×2], C5×D4 [×2], C5×Q8 [×2], Q8○D8, C8×D5 [×2], C8⋊D5 [×2], Dic20 [×4], C4.Dic5, D4.D5 [×4], C5⋊Q16 [×4], C2×C40, C5×D8, C5×SD16 [×2], C5×Q16, C2×Dic10 [×2], C2×Dic10 [×2], C4○D20, C4○D20 [×2], D42D5 [×4], D42D5 [×4], Q8×D5 [×4], C5×C4○D4 [×2], D20.3C4, C2×Dic20, D83D5 [×2], SD16⋊D5 [×4], D5×Q16 [×2], D4.9D10 [×2], C5×C4○D8, D4.10D10 [×2], D20.47D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, C22×D5 [×7], Q8○D8, D4×D5 [×2], C23×D5, C2×D4×D5, D20.47D4

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

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

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

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

50 conjugacy classes

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

50 irreducible representations

 dim 1 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 C2 D4 D4 D4 D5 D10 D10 D10 D10 D10 Q8○D8 D4×D5 D4×D5 D20.47D4 kernel D20.47D4 D20.3C4 C2×Dic20 D8⋊3D5 SD16⋊D5 D5×Q16 D4.9D10 C5×C4○D8 D4.10D10 Dic10 D20 C5⋊D4 C4○D8 C2×C8 D8 SD16 Q16 C4○D4 C5 C4 C22 C1 # reps 1 1 1 2 4 2 2 1 2 1 1 2 2 2 2 4 2 4 2 2 2 8

Matrix representation of D20.47D4 in GL4(𝔽41) generated by

 0 0 33 8 0 0 33 0 0 36 24 0 5 36 0 24
,
 5 29 31 28 12 36 10 18 6 18 34 12 24 24 27 7
,
 39 16 23 15 25 25 6 38 7 7 32 25 1 14 5 27
,
 39 13 21 15 25 2 18 38 7 35 14 25 1 33 25 27
`G:=sub<GL(4,GF(41))| [0,0,0,5,0,0,36,36,33,33,24,0,8,0,0,24],[5,12,6,24,29,36,18,24,31,10,34,27,28,18,12,7],[39,25,7,1,16,25,7,14,23,6,32,5,15,38,25,27],[39,25,7,1,13,2,35,33,21,18,14,25,15,38,25,27] >;`

D20.47D4 in GAP, Magma, Sage, TeX

`D_{20}._{47}D_4`
`% in TeX`

`G:=Group("D20.47D4");`
`// GroupNames label`

`G:=SmallGroup(320,1443);`
`// by ID`

`G=gap.SmallGroup(320,1443);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,387,184,570,185,136,438,235,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=b^2=c^10=1,d^2=a^4,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^4*b,b*d=d*b,d*c*d^-1=a^4*c^-1>;`
`// generators/relations`

׿
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