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## G = D4⋊2D5⋊C4order 320 = 26·5

### 2nd semidirect product of D4⋊2D5 and C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — D4⋊2D5⋊C4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C4×D5 — C2×D4⋊2D5 — D4⋊2D5⋊C4
 Lower central C5 — C10 — C20 — D4⋊2D5⋊C4
 Upper central C1 — C22 — C2×C4 — D4⋊C4

Generators and relations for D42D5⋊C4
G = < a,b,c,d,e | a4=b2=c5=d2=e4=1, bab=eae-1=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, ebe-1=ab, dcd=c-1, ce=ec, ede-1=a2d >

Subgroups: 590 in 158 conjugacy classes, 55 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, D4⋊C4, D4⋊C4, Q8⋊C4, C42⋊C2, C22×C8, C2×C4○D4, C52C8, C40, Dic10, Dic10, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C22×D5, C22×C10, C23.24D4, C8×D5, C2×C52C8, C4×Dic5, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C2×C40, C2×Dic10, C2×C4×D5, D42D5, D42D5, C22×Dic5, C2×C5⋊D4, D4×C10, C10.Q16, C20.44D4, D4⋊Dic5, C5×D4⋊C4, C4⋊C47D5, D5×C2×C8, C2×D42D5, D42D5⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, D10, C2×C22⋊C4, C4○D8, C4×D5, C22×D5, C23.24D4, C2×C4×D5, D4×D5, D5×C22⋊C4, D83D5, SD163D5, D42D5⋊C4

Smallest permutation representation of D42D5⋊C4
On 160 points
Generators in S160
(1 16 6 11)(2 17 7 12)(3 18 8 13)(4 19 9 14)(5 20 10 15)(21 36 26 31)(22 37 27 32)(23 38 28 33)(24 39 29 34)(25 40 30 35)(41 56 46 51)(42 57 47 52)(43 58 48 53)(44 59 49 54)(45 60 50 55)(61 76 66 71)(62 77 67 72)(63 78 68 73)(64 79 69 74)(65 80 70 75)(81 91 86 96)(82 92 87 97)(83 93 88 98)(84 94 89 99)(85 95 90 100)(101 111 106 116)(102 112 107 117)(103 113 108 118)(104 114 109 119)(105 115 110 120)(121 131 126 136)(122 132 127 137)(123 133 128 138)(124 134 129 139)(125 135 130 140)(141 151 146 156)(142 152 147 157)(143 153 148 158)(144 154 149 159)(145 155 150 160)
(1 81)(2 82)(3 83)(4 84)(5 85)(6 86)(7 87)(8 88)(9 89)(10 90)(11 91)(12 92)(13 93)(14 94)(15 95)(16 96)(17 97)(18 98)(19 99)(20 100)(21 101)(22 102)(23 103)(24 104)(25 105)(26 106)(27 107)(28 108)(29 109)(30 110)(31 111)(32 112)(33 113)(34 114)(35 115)(36 116)(37 117)(38 118)(39 119)(40 120)(41 121)(42 122)(43 123)(44 124)(45 125)(46 126)(47 127)(48 128)(49 129)(50 130)(51 131)(52 132)(53 133)(54 134)(55 135)(56 136)(57 137)(58 138)(59 139)(60 140)(61 141)(62 142)(63 143)(64 144)(65 145)(66 146)(67 147)(68 148)(69 149)(70 150)(71 151)(72 152)(73 153)(74 154)(75 155)(76 156)(77 157)(78 158)(79 159)(80 160)
(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 35)(2 34)(3 33)(4 32)(5 31)(6 40)(7 39)(8 38)(9 37)(10 36)(11 30)(12 29)(13 28)(14 27)(15 26)(16 25)(17 24)(18 23)(19 22)(20 21)(41 75)(42 74)(43 73)(44 72)(45 71)(46 80)(47 79)(48 78)(49 77)(50 76)(51 70)(52 69)(53 68)(54 67)(55 66)(56 65)(57 64)(58 63)(59 62)(60 61)(81 120)(82 119)(83 118)(84 117)(85 116)(86 115)(87 114)(88 113)(89 112)(90 111)(91 105)(92 104)(93 103)(94 102)(95 101)(96 110)(97 109)(98 108)(99 107)(100 106)(121 160)(122 159)(123 158)(124 157)(125 156)(126 155)(127 154)(128 153)(129 152)(130 151)(131 145)(132 144)(133 143)(134 142)(135 141)(136 150)(137 149)(138 148)(139 147)(140 146)
(1 151 31 121)(2 152 32 122)(3 153 33 123)(4 154 34 124)(5 155 35 125)(6 156 36 126)(7 157 37 127)(8 158 38 128)(9 159 39 129)(10 160 40 130)(11 146 26 131)(12 147 27 132)(13 148 28 133)(14 149 29 134)(15 150 30 135)(16 141 21 136)(17 142 22 137)(18 143 23 138)(19 144 24 139)(20 145 25 140)(41 96 71 101)(42 97 72 102)(43 98 73 103)(44 99 74 104)(45 100 75 105)(46 91 76 106)(47 92 77 107)(48 93 78 108)(49 94 79 109)(50 95 80 110)(51 81 66 111)(52 82 67 112)(53 83 68 113)(54 84 69 114)(55 85 70 115)(56 86 61 116)(57 87 62 117)(58 88 63 118)(59 89 64 119)(60 90 65 120)

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

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

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

56 conjugacy classes

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

56 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 C4 D4 D4 D4 D5 D10 D10 D10 C4○D8 C4×D5 D4×D5 D4×D5 D8⋊3D5 SD16⋊3D5 kernel D4⋊2D5⋊C4 C10.Q16 C20.44D4 D4⋊Dic5 C5×D4⋊C4 C4⋊C4⋊7D5 D5×C2×C8 C2×D4⋊2D5 D4⋊2D5 C4×D5 C2×Dic5 C22×D5 D4⋊C4 C4⋊C4 C2×C8 C2×D4 C10 D4 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 8 2 1 1 2 2 2 2 8 8 2 2 4 4

Matrix representation of D42D5⋊C4 in GL4(𝔽41) generated by

 32 2 0 0 0 9 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 9 40 0 0 0 0 40 0 0 0 0 40
,
 1 0 0 0 0 1 0 0 0 0 0 40 0 0 1 6
,
 40 23 0 0 0 1 0 0 0 0 6 1 0 0 6 35
,
 38 11 0 0 14 3 0 0 0 0 9 0 0 0 0 9
G:=sub<GL(4,GF(41))| [32,0,0,0,2,9,0,0,0,0,1,0,0,0,0,1],[1,9,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,40,6],[40,0,0,0,23,1,0,0,0,0,6,6,0,0,1,35],[38,14,0,0,11,3,0,0,0,0,9,0,0,0,0,9] >;

D42D5⋊C4 in GAP, Magma, Sage, TeX

D_4\rtimes_2D_5\rtimes C_4
% in TeX

G:=Group("D4:2D5:C4");
// GroupNames label

G:=SmallGroup(320,399);
// by ID

G=gap.SmallGroup(320,399);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,219,58,570,136,851,102,12550]);
// Polycyclic

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

׿
×
𝔽