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## G = C41⋊D4order 328 = 23·41

### The semidirect product of C41 and D4 acting via D4/C22=C2

Aliases: C412D4, C22⋊D41, D822C2, Dic41⋊C2, C2.5D82, C82.5C22, (C2×C82)⋊2C2, SmallGroup(328,8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C82 — C41⋊D4
 Chief series C1 — C41 — C82 — D82 — C41⋊D4
 Lower central C41 — C82 — C41⋊D4
 Upper central C1 — C2 — C22

Generators and relations for C41⋊D4
G = < a,b,c | a41=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >

Smallest permutation representation of C41⋊D4
On 164 points
Generators in S164
```(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)(124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164)
(1 114 69 152)(2 113 70 151)(3 112 71 150)(4 111 72 149)(5 110 73 148)(6 109 74 147)(7 108 75 146)(8 107 76 145)(9 106 77 144)(10 105 78 143)(11 104 79 142)(12 103 80 141)(13 102 81 140)(14 101 82 139)(15 100 42 138)(16 99 43 137)(17 98 44 136)(18 97 45 135)(19 96 46 134)(20 95 47 133)(21 94 48 132)(22 93 49 131)(23 92 50 130)(24 91 51 129)(25 90 52 128)(26 89 53 127)(27 88 54 126)(28 87 55 125)(29 86 56 124)(30 85 57 164)(31 84 58 163)(32 83 59 162)(33 123 60 161)(34 122 61 160)(35 121 62 159)(36 120 63 158)(37 119 64 157)(38 118 65 156)(39 117 66 155)(40 116 67 154)(41 115 68 153)
(2 41)(3 40)(4 39)(5 38)(6 37)(7 36)(8 35)(9 34)(10 33)(11 32)(12 31)(13 30)(14 29)(15 28)(16 27)(17 26)(18 25)(19 24)(20 23)(21 22)(42 55)(43 54)(44 53)(45 52)(46 51)(47 50)(48 49)(56 82)(57 81)(58 80)(59 79)(60 78)(61 77)(62 76)(63 75)(64 74)(65 73)(66 72)(67 71)(68 70)(83 142)(84 141)(85 140)(86 139)(87 138)(88 137)(89 136)(90 135)(91 134)(92 133)(93 132)(94 131)(95 130)(96 129)(97 128)(98 127)(99 126)(100 125)(101 124)(102 164)(103 163)(104 162)(105 161)(106 160)(107 159)(108 158)(109 157)(110 156)(111 155)(112 154)(113 153)(114 152)(115 151)(116 150)(117 149)(118 148)(119 147)(120 146)(121 145)(122 144)(123 143)```

`G:=sub<Sym(164)| (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)(124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164), (1,114,69,152)(2,113,70,151)(3,112,71,150)(4,111,72,149)(5,110,73,148)(6,109,74,147)(7,108,75,146)(8,107,76,145)(9,106,77,144)(10,105,78,143)(11,104,79,142)(12,103,80,141)(13,102,81,140)(14,101,82,139)(15,100,42,138)(16,99,43,137)(17,98,44,136)(18,97,45,135)(19,96,46,134)(20,95,47,133)(21,94,48,132)(22,93,49,131)(23,92,50,130)(24,91,51,129)(25,90,52,128)(26,89,53,127)(27,88,54,126)(28,87,55,125)(29,86,56,124)(30,85,57,164)(31,84,58,163)(32,83,59,162)(33,123,60,161)(34,122,61,160)(35,121,62,159)(36,120,63,158)(37,119,64,157)(38,118,65,156)(39,117,66,155)(40,116,67,154)(41,115,68,153), (2,41)(3,40)(4,39)(5,38)(6,37)(7,36)(8,35)(9,34)(10,33)(11,32)(12,31)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(42,55)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)(56,82)(57,81)(58,80)(59,79)(60,78)(61,77)(62,76)(63,75)(64,74)(65,73)(66,72)(67,71)(68,70)(83,142)(84,141)(85,140)(86,139)(87,138)(88,137)(89,136)(90,135)(91,134)(92,133)(93,132)(94,131)(95,130)(96,129)(97,128)(98,127)(99,126)(100,125)(101,124)(102,164)(103,163)(104,162)(105,161)(106,160)(107,159)(108,158)(109,157)(110,156)(111,155)(112,154)(113,153)(114,152)(115,151)(116,150)(117,149)(118,148)(119,147)(120,146)(121,145)(122,144)(123,143)>;`

`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)(124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164), (1,114,69,152)(2,113,70,151)(3,112,71,150)(4,111,72,149)(5,110,73,148)(6,109,74,147)(7,108,75,146)(8,107,76,145)(9,106,77,144)(10,105,78,143)(11,104,79,142)(12,103,80,141)(13,102,81,140)(14,101,82,139)(15,100,42,138)(16,99,43,137)(17,98,44,136)(18,97,45,135)(19,96,46,134)(20,95,47,133)(21,94,48,132)(22,93,49,131)(23,92,50,130)(24,91,51,129)(25,90,52,128)(26,89,53,127)(27,88,54,126)(28,87,55,125)(29,86,56,124)(30,85,57,164)(31,84,58,163)(32,83,59,162)(33,123,60,161)(34,122,61,160)(35,121,62,159)(36,120,63,158)(37,119,64,157)(38,118,65,156)(39,117,66,155)(40,116,67,154)(41,115,68,153), (2,41)(3,40)(4,39)(5,38)(6,37)(7,36)(8,35)(9,34)(10,33)(11,32)(12,31)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(42,55)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)(56,82)(57,81)(58,80)(59,79)(60,78)(61,77)(62,76)(63,75)(64,74)(65,73)(66,72)(67,71)(68,70)(83,142)(84,141)(85,140)(86,139)(87,138)(88,137)(89,136)(90,135)(91,134)(92,133)(93,132)(94,131)(95,130)(96,129)(97,128)(98,127)(99,126)(100,125)(101,124)(102,164)(103,163)(104,162)(105,161)(106,160)(107,159)(108,158)(109,157)(110,156)(111,155)(112,154)(113,153)(114,152)(115,151)(116,150)(117,149)(118,148)(119,147)(120,146)(121,145)(122,144)(123,143) );`

`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),(124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164)], [(1,114,69,152),(2,113,70,151),(3,112,71,150),(4,111,72,149),(5,110,73,148),(6,109,74,147),(7,108,75,146),(8,107,76,145),(9,106,77,144),(10,105,78,143),(11,104,79,142),(12,103,80,141),(13,102,81,140),(14,101,82,139),(15,100,42,138),(16,99,43,137),(17,98,44,136),(18,97,45,135),(19,96,46,134),(20,95,47,133),(21,94,48,132),(22,93,49,131),(23,92,50,130),(24,91,51,129),(25,90,52,128),(26,89,53,127),(27,88,54,126),(28,87,55,125),(29,86,56,124),(30,85,57,164),(31,84,58,163),(32,83,59,162),(33,123,60,161),(34,122,61,160),(35,121,62,159),(36,120,63,158),(37,119,64,157),(38,118,65,156),(39,117,66,155),(40,116,67,154),(41,115,68,153)], [(2,41),(3,40),(4,39),(5,38),(6,37),(7,36),(8,35),(9,34),(10,33),(11,32),(12,31),(13,30),(14,29),(15,28),(16,27),(17,26),(18,25),(19,24),(20,23),(21,22),(42,55),(43,54),(44,53),(45,52),(46,51),(47,50),(48,49),(56,82),(57,81),(58,80),(59,79),(60,78),(61,77),(62,76),(63,75),(64,74),(65,73),(66,72),(67,71),(68,70),(83,142),(84,141),(85,140),(86,139),(87,138),(88,137),(89,136),(90,135),(91,134),(92,133),(93,132),(94,131),(95,130),(96,129),(97,128),(98,127),(99,126),(100,125),(101,124),(102,164),(103,163),(104,162),(105,161),(106,160),(107,159),(108,158),(109,157),(110,156),(111,155),(112,154),(113,153),(114,152),(115,151),(116,150),(117,149),(118,148),(119,147),(120,146),(121,145),(122,144),(123,143)])`

85 conjugacy classes

 class 1 2A 2B 2C 4 41A ··· 41T 82A ··· 82BH order 1 2 2 2 4 41 ··· 41 82 ··· 82 size 1 1 2 82 82 2 ··· 2 2 ··· 2

85 irreducible representations

 dim 1 1 1 1 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 D4 D41 D82 C41⋊D4 kernel C41⋊D4 Dic41 D82 C2×C82 C41 C22 C2 C1 # reps 1 1 1 1 1 20 20 40

Matrix representation of C41⋊D4 in GL2(𝔽821) generated by

 635 1 476 302
,
 71 62 210 750
,
 461 534 340 360
`G:=sub<GL(2,GF(821))| [635,476,1,302],[71,210,62,750],[461,340,534,360] >;`

C41⋊D4 in GAP, Magma, Sage, TeX

`C_{41}\rtimes D_4`
`% in TeX`

`G:=Group("C41:D4");`
`// GroupNames label`

`G:=SmallGroup(328,8);`
`// by ID`

`G=gap.SmallGroup(328,8);`
`# by ID`

`G:=PCGroup([4,-2,-2,-2,-41,49,5123]);`
`// Polycyclic`

`G:=Group<a,b,c|a^41=b^4=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;`
`// generators/relations`

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