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## G = D4.(C5⋊C8)  order 320 = 26·5

### The non-split extension by D4 of C5⋊C8 acting via C5⋊C8/Dic5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — D4.(C5⋊C8)
 Chief series C1 — C5 — C10 — C20 — C5⋊2C8 — C2×C5⋊2C8 — C20.C8 — D4.(C5⋊C8)
 Lower central C5 — C10 — C20 — D4.(C5⋊C8)
 Upper central C1 — C4 — C2×C4 — C4○D4

Generators and relations for D4.(C5⋊C8)
G = < a,b,c,d | a4=b2=c5=1, d8=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c3 >

Smallest permutation representation of D4.(C5⋊C8)
On 160 points
Generators in S160
```(1 26 9 18)(2 19 10 27)(3 28 11 20)(4 21 12 29)(5 30 13 22)(6 23 14 31)(7 32 15 24)(8 25 16 17)(33 125 41 117)(34 118 42 126)(35 127 43 119)(36 120 44 128)(37 113 45 121)(38 122 46 114)(39 115 47 123)(40 124 48 116)(49 103 57 111)(50 112 58 104)(51 105 59 97)(52 98 60 106)(53 107 61 99)(54 100 62 108)(55 109 63 101)(56 102 64 110)(65 130 73 138)(66 139 74 131)(67 132 75 140)(68 141 76 133)(69 134 77 142)(70 143 78 135)(71 136 79 144)(72 129 80 137)(81 159 89 151)(82 152 90 160)(83 145 91 153)(84 154 92 146)(85 147 93 155)(86 156 94 148)(87 149 95 157)(88 158 96 150)
(1 18)(2 10)(3 20)(4 12)(5 22)(6 14)(7 24)(8 16)(9 26)(11 28)(13 30)(15 32)(33 125)(35 127)(37 113)(39 115)(41 117)(43 119)(45 121)(47 123)(49 103)(51 105)(53 107)(55 109)(57 111)(59 97)(61 99)(63 101)(65 130)(67 132)(69 134)(71 136)(73 138)(75 140)(77 142)(79 144)(81 151)(82 90)(83 153)(84 92)(85 155)(86 94)(87 157)(88 96)(89 159)(91 145)(93 147)(95 149)(98 106)(100 108)(102 110)(104 112)(114 122)(116 124)(118 126)(120 128)(129 137)(131 139)(133 141)(135 143)
(1 101 87 113 136)(2 114 102 137 88)(3 138 115 89 103)(4 90 139 104 116)(5 105 91 117 140)(6 118 106 141 92)(7 142 119 93 107)(8 94 143 108 120)(9 109 95 121 144)(10 122 110 129 96)(11 130 123 81 111)(12 82 131 112 124)(13 97 83 125 132)(14 126 98 133 84)(15 134 127 85 99)(16 86 135 100 128)(17 156 70 62 36)(18 63 157 37 71)(19 38 64 72 158)(20 73 39 159 49)(21 160 74 50 40)(22 51 145 41 75)(23 42 52 76 146)(24 77 43 147 53)(25 148 78 54 44)(26 55 149 45 79)(27 46 56 80 150)(28 65 47 151 57)(29 152 66 58 48)(30 59 153 33 67)(31 34 60 68 154)(32 69 35 155 61)
(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)```

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

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

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

38 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 5 8A 8B 8C 8D 8E 8F 8G 8H 10A 10B 10C 10D 16A ··· 16H 16I 16J 16K 16L 20A 20B 20C 20D 20E order 1 2 2 2 4 4 4 4 5 8 8 8 8 8 8 8 8 10 10 10 10 16 ··· 16 16 16 16 16 20 20 20 20 20 size 1 1 2 4 1 1 2 4 4 5 5 5 5 10 10 20 20 4 8 8 8 10 ··· 10 20 20 20 20 4 4 8 8 8

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 4 4 4 4 4 4 8 type + + + + + + + - - + - image C1 C2 C2 C2 C4 C4 C8 C8 D4 M4(2) D4.C8 F5 C2×F5 C5⋊C8 C5⋊C8 C22⋊F5 C22.F5 D4.(C5⋊C8) kernel D4.(C5⋊C8) C2×C5⋊C16 C20.C8 D4.Dic5 C4.Dic5 C5×C4○D4 C5×D4 C5×Q8 C5⋊2C8 C2×C10 C5 C4○D4 C2×C4 D4 Q8 C4 C22 C1 # reps 1 1 1 1 2 2 4 4 2 2 8 1 1 1 1 2 2 2

Matrix representation of D4.(C5⋊C8) in GL6(𝔽241)

 0 1 0 0 0 0 240 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 240 0 0 1 0 0 240 0 0 0 1 0 240 0 0 0 0 1 240
,
 181 60 0 0 0 0 60 60 0 0 0 0 0 0 220 82 26 67 0 0 5 149 113 46 0 0 92 128 195 72 0 0 174 154 21 159

`G:=sub<GL(6,GF(241))| [0,240,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,240,240,240,240],[181,60,0,0,0,0,60,60,0,0,0,0,0,0,220,5,92,174,0,0,82,149,128,154,0,0,26,113,195,21,0,0,67,46,72,159] >;`

D4.(C5⋊C8) in GAP, Magma, Sage, TeX

`D_4.(C_5\rtimes C_8)`
`% in TeX`

`G:=Group("D4.(C5:C8)");`
`// GroupNames label`

`G:=SmallGroup(320,270);`
`// by ID`

`G=gap.SmallGroup(320,270);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,100,1123,570,136,102,6278,3156]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^2=c^5=1,d^8=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=c^3>;`
`// generators/relations`

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