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## G = D4.13D20order 320 = 26·5

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — D4.13D20
 Chief series C1 — C5 — C10 — C20 — D20 — C4○D20 — D4.10D10 — D4.13D20
 Lower central C5 — C10 — C20 — D4.13D20
 Upper central C1 — C2 — C4○D4 — C8○D4

Generators and relations for D4.13D20
G = < a,b,c,d | a4=b2=d2=1, c20=a2, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=a2c19 >

Subgroups: 902 in 248 conjugacy classes, 107 normal (16 characteristic)
C1, C2, C2 [×5], C4, C4 [×3], C4 [×6], C22 [×3], C22 [×2], C5, C8, C8 [×3], C2×C4 [×3], C2×C4 [×12], D4 [×3], D4 [×8], Q8, Q8 [×12], D5 [×2], C10, C10 [×3], C2×C8 [×3], M4(2) [×3], D8, SD16 [×6], Q16 [×9], C2×Q8 [×8], C4○D4, C4○D4 [×12], Dic5 [×6], C20, C20 [×3], D10 [×2], C2×C10 [×3], C8○D4, C2×Q16 [×3], C4○D8 [×3], C8.C22 [×6], 2- 1+4 [×2], C40, C40 [×3], Dic10 [×6], Dic10 [×6], C4×D5 [×6], D20 [×2], C2×Dic5 [×6], C5⋊D4 [×6], C2×C20 [×3], C5×D4 [×3], C5×Q8, Q8○D8, C40⋊C2 [×6], D40, Dic20 [×9], C2×C40 [×3], C5×M4(2) [×3], C2×Dic10 [×6], C4○D20 [×6], D42D5 [×6], Q8×D5 [×2], C5×C4○D4, D407C2 [×3], C2×Dic20 [×3], C8.D10 [×6], C5×C8○D4, D4.10D10 [×2], D4.13D20
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, D20 [×4], C22×D5 [×7], Q8○D8, C2×D20 [×6], C23×D5, C22×D20, D4.13D20

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

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

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

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

62 conjugacy classes

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

62 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 D4 D4 D5 D10 D10 D10 D20 D20 Q8○D8 D4.13D20 kernel D4.13D20 D40⋊7C2 C2×Dic20 C8.D10 C5×C8○D4 D4.10D10 C5×D4 C5×Q8 C8○D4 C2×C8 M4(2) C4○D4 D4 Q8 C5 C1 # reps 1 3 3 6 1 2 3 1 2 6 6 2 12 4 2 8

Matrix representation of D4.13D20 in GL4(𝔽41) generated by

 0 0 1 0 0 0 0 1 40 0 0 0 0 40 0 0
,
 33 14 15 25 27 8 16 26 15 25 8 27 16 26 14 33
,
 35 3 0 0 38 15 0 0 0 0 35 3 0 0 38 15
,
 20 39 0 0 15 21 0 0 0 0 20 39 0 0 15 21
`G:=sub<GL(4,GF(41))| [0,0,40,0,0,0,0,40,1,0,0,0,0,1,0,0],[33,27,15,16,14,8,25,26,15,16,8,14,25,26,27,33],[35,38,0,0,3,15,0,0,0,0,35,38,0,0,3,15],[20,15,0,0,39,21,0,0,0,0,20,15,0,0,39,21] >;`

D4.13D20 in GAP, Magma, Sage, TeX

`D_4._{13}D_{20}`
`% in TeX`

`G:=Group("D4.13D20");`
`// GroupNames label`

`G:=SmallGroup(320,1425);`
`// by ID`

`G=gap.SmallGroup(320,1425);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,387,184,675,192,1684,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^2=d^2=1,c^20=a^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^2*b,d*c*d=a^2*c^19>;`
`// generators/relations`

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