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## G = C22×D4⋊D5order 320 = 26·5

### Direct product of C22 and D4⋊D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C22×D4⋊D5
 Chief series C1 — C5 — C10 — C20 — D20 — C2×D20 — C22×D20 — C22×D4⋊D5
 Lower central C5 — C10 — C20 — C22×D4⋊D5
 Upper central C1 — C23 — C22×C4 — C22×D4

Generators and relations for C22×D4⋊D5
G = < a,b,c,d,e,f | a2=b2=c4=d2=e5=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=fcf=c-1, ce=ec, de=ed, fdf=cd, fef=e-1 >

Subgroups: 1342 in 338 conjugacy classes, 127 normal (15 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, D4, D4, C23, C23, D5, C10, C10, C10, C2×C8, D8, C22×C4, C2×D4, C2×D4, C24, C20, C20, D10, C2×C10, C2×C10, C22×C8, C2×D8, C22×D4, C22×D4, C52C8, D20, D20, C2×C20, C5×D4, C5×D4, C22×D5, C22×C10, C22×C10, C22×D8, C2×C52C8, D4⋊D5, C2×D20, C2×D20, C22×C20, D4×C10, D4×C10, C23×D5, C23×C10, C22×C52C8, C2×D4⋊D5, C22×D20, D4×C2×C10, C22×D4⋊D5
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, C24, D10, C2×D8, C22×D4, C5⋊D4, C22×D5, C22×D8, D4⋊D5, C2×C5⋊D4, C23×D5, C2×D4⋊D5, C22×C5⋊D4, C22×D4⋊D5

Smallest permutation representation of C22×D4⋊D5
On 160 points
Generators in S160
(1 86)(2 87)(3 88)(4 89)(5 90)(6 81)(7 82)(8 83)(9 84)(10 85)(11 96)(12 97)(13 98)(14 99)(15 100)(16 91)(17 92)(18 93)(19 94)(20 95)(21 106)(22 107)(23 108)(24 109)(25 110)(26 101)(27 102)(28 103)(29 104)(30 105)(31 116)(32 117)(33 118)(34 119)(35 120)(36 111)(37 112)(38 113)(39 114)(40 115)(41 126)(42 127)(43 128)(44 129)(45 130)(46 121)(47 122)(48 123)(49 124)(50 125)(51 136)(52 137)(53 138)(54 139)(55 140)(56 131)(57 132)(58 133)(59 134)(60 135)(61 146)(62 147)(63 148)(64 149)(65 150)(66 141)(67 142)(68 143)(69 144)(70 145)(71 156)(72 157)(73 158)(74 159)(75 160)(76 151)(77 152)(78 153)(79 154)(80 155)
(1 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)(81 121)(82 122)(83 123)(84 124)(85 125)(86 126)(87 127)(88 128)(89 129)(90 130)(91 131)(92 132)(93 133)(94 134)(95 135)(96 136)(97 137)(98 138)(99 139)(100 140)(101 141)(102 142)(103 143)(104 144)(105 145)(106 146)(107 147)(108 148)(109 149)(110 150)(111 151)(112 152)(113 153)(114 154)(115 155)(116 156)(117 157)(118 158)(119 159)(120 160)
(1 16 6 11)(2 17 7 12)(3 18 8 13)(4 19 9 14)(5 20 10 15)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)(41 56 46 51)(42 57 47 52)(43 58 48 53)(44 59 49 54)(45 60 50 55)(61 71 66 76)(62 72 67 77)(63 73 68 78)(64 74 69 79)(65 75 70 80)(81 96 86 91)(82 97 87 92)(83 98 88 93)(84 99 89 94)(85 100 90 95)(101 111 106 116)(102 112 107 117)(103 113 108 118)(104 114 109 119)(105 115 110 120)(121 136 126 131)(122 137 127 132)(123 138 128 133)(124 139 129 134)(125 140 130 135)(141 151 146 156)(142 152 147 157)(143 153 148 158)(144 154 149 159)(145 155 150 160)
(1 156)(2 157)(3 158)(4 159)(5 160)(6 151)(7 152)(8 153)(9 154)(10 155)(11 141)(12 142)(13 143)(14 144)(15 145)(16 146)(17 147)(18 148)(19 149)(20 150)(21 131)(22 132)(23 133)(24 134)(25 135)(26 136)(27 137)(28 138)(29 139)(30 140)(31 126)(32 127)(33 128)(34 129)(35 130)(36 121)(37 122)(38 123)(39 124)(40 125)(41 116)(42 117)(43 118)(44 119)(45 120)(46 111)(47 112)(48 113)(49 114)(50 115)(51 101)(52 102)(53 103)(54 104)(55 105)(56 106)(57 107)(58 108)(59 109)(60 110)(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(67 97)(68 98)(69 99)(70 100)(71 86)(72 87)(73 88)(74 89)(75 90)(76 81)(77 82)(78 83)(79 84)(80 85)
(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 45)(2 44)(3 43)(4 42)(5 41)(6 50)(7 49)(8 48)(9 47)(10 46)(11 60)(12 59)(13 58)(14 57)(15 56)(16 55)(17 54)(18 53)(19 52)(20 51)(21 75)(22 74)(23 73)(24 72)(25 71)(26 80)(27 79)(28 78)(29 77)(30 76)(31 65)(32 64)(33 63)(34 62)(35 61)(36 70)(37 69)(38 68)(39 67)(40 66)(81 125)(82 124)(83 123)(84 122)(85 121)(86 130)(87 129)(88 128)(89 127)(90 126)(91 140)(92 139)(93 138)(94 137)(95 136)(96 135)(97 134)(98 133)(99 132)(100 131)(101 155)(102 154)(103 153)(104 152)(105 151)(106 160)(107 159)(108 158)(109 157)(110 156)(111 145)(112 144)(113 143)(114 142)(115 141)(116 150)(117 149)(118 148)(119 147)(120 146)

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

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

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

68 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L 2M 2N 2O 4A 4B 4C 4D 5A 5B 8A ··· 8H 10A ··· 10N 10O ··· 10AD 20A ··· 20H order 1 2 ··· 2 2 2 2 2 2 2 2 2 4 4 4 4 5 5 8 ··· 8 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 ··· 1 4 4 4 4 20 20 20 20 2 2 2 2 2 2 10 ··· 10 2 ··· 2 4 ··· 4 4 ··· 4

68 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 D4 D4 D5 D8 D10 D10 C5⋊D4 C5⋊D4 D4⋊D5 kernel C22×D4⋊D5 C22×C5⋊2C8 C2×D4⋊D5 C22×D20 D4×C2×C10 C2×C20 C22×C10 C22×D4 C2×C10 C22×C4 C2×D4 C2×C4 C23 C22 # reps 1 1 12 1 1 3 1 2 8 2 12 12 4 8

Matrix representation of C22×D4⋊D5 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 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 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 40 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 17 35 0 0 0 0 7 24 0 0 0 0 0 0 29 12 0 0 0 0 12 12
,
 34 40 0 0 0 0 1 0 0 0 0 0 0 0 40 40 0 0 0 0 8 7 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 7 1 0 0 0 0 34 34 0 0 0 0 0 0 0 35 0 0 0 0 34 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,17,7,0,0,0,0,35,24,0,0,0,0,0,0,29,12,0,0,0,0,12,12],[34,1,0,0,0,0,40,0,0,0,0,0,0,0,40,8,0,0,0,0,40,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[7,34,0,0,0,0,1,34,0,0,0,0,0,0,0,34,0,0,0,0,35,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C22×D4⋊D5 in GAP, Magma, Sage, TeX

C_2^2\times D_4\rtimes D_5
% in TeX

G:=Group("C2^2xD4:D5");
// GroupNames label

G:=SmallGroup(320,1464);
// by ID

G=gap.SmallGroup(320,1464);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,675,1684,235,102,12550]);
// Polycyclic

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

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