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## G = C22×D10⋊C4order 320 = 26·5

### Direct product of C22 and D10⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C22×D10⋊C4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — D5×C24 — C22×D10⋊C4
 Lower central C5 — C10 — C22×D10⋊C4
 Upper central C1 — C24 — C23×C4

Generators and relations for C22×D10⋊C4
G = < a,b,c,d,e | a2=b2=c10=d2=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=c-1, ce=ec, ede-1=c5d >

Subgroups: 2558 in 674 conjugacy classes, 247 normal (17 characteristic)
C1, C2 [×3], C2 [×12], C2 [×8], C4 [×8], C22, C22 [×34], C22 [×64], C5, C2×C4 [×4], C2×C4 [×28], C23 [×15], C23 [×84], D5 [×8], C10 [×3], C10 [×12], C22⋊C4 [×16], C22×C4 [×6], C22×C4 [×14], C24, C24 [×22], Dic5 [×4], C20 [×4], D10 [×8], D10 [×56], C2×C10, C2×C10 [×34], C2×C22⋊C4 [×12], C23×C4, C23×C4, C25, C2×Dic5 [×4], C2×Dic5 [×12], C2×C20 [×4], C2×C20 [×12], C22×D5 [×28], C22×D5 [×56], C22×C10 [×15], C22×C22⋊C4, D10⋊C4 [×16], C22×Dic5 [×6], C22×Dic5 [×4], C22×C20 [×6], C22×C20 [×4], C23×D5 [×14], C23×D5 [×8], C23×C10, C2×D10⋊C4 [×12], C23×Dic5, C23×C20, D5×C24, C22×D10⋊C4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×8], C23 [×15], D5, C22⋊C4 [×16], C22×C4 [×14], C2×D4 [×12], C24, D10 [×7], C2×C22⋊C4 [×12], C23×C4, C22×D4 [×2], C4×D5 [×4], D20 [×4], C5⋊D4 [×4], C22×D5 [×7], C22×C22⋊C4, D10⋊C4 [×16], C2×C4×D5 [×6], C2×D20 [×6], C2×C5⋊D4 [×6], C23×D5, C2×D10⋊C4 [×12], D5×C22×C4, C22×D20, C22×C5⋊D4, C22×D10⋊C4

Smallest permutation representation of C22×D10⋊C4
On 160 points
Generators in S160
(1 92)(2 93)(3 94)(4 95)(5 96)(6 97)(7 98)(8 99)(9 100)(10 91)(11 86)(12 87)(13 88)(14 89)(15 90)(16 81)(17 82)(18 83)(19 84)(20 85)(21 116)(22 117)(23 118)(24 119)(25 120)(26 111)(27 112)(28 113)(29 114)(30 115)(31 106)(32 107)(33 108)(34 109)(35 110)(36 101)(37 102)(38 103)(39 104)(40 105)(41 136)(42 137)(43 138)(44 139)(45 140)(46 131)(47 132)(48 133)(49 134)(50 135)(51 126)(52 127)(53 128)(54 129)(55 130)(56 121)(57 122)(58 123)(59 124)(60 125)(61 156)(62 157)(63 158)(64 159)(65 160)(66 151)(67 152)(68 153)(69 154)(70 155)(71 146)(72 147)(73 148)(74 149)(75 150)(76 141)(77 142)(78 143)(79 144)(80 145)
(1 57)(2 58)(3 59)(4 60)(5 51)(6 52)(7 53)(8 54)(9 55)(10 56)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 71)(22 72)(23 73)(24 74)(25 75)(26 76)(27 77)(28 78)(29 79)(30 80)(31 61)(32 62)(33 63)(34 64)(35 65)(36 66)(37 67)(38 68)(39 69)(40 70)(81 131)(82 132)(83 133)(84 134)(85 135)(86 136)(87 137)(88 138)(89 139)(90 140)(91 121)(92 122)(93 123)(94 124)(95 125)(96 126)(97 127)(98 128)(99 129)(100 130)(101 151)(102 152)(103 153)(104 154)(105 155)(106 156)(107 157)(108 158)(109 159)(110 160)(111 141)(112 142)(113 143)(114 144)(115 145)(116 146)(117 147)(118 148)(119 149)(120 150)
(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 56)(2 55)(3 54)(4 53)(5 52)(6 51)(7 60)(8 59)(9 58)(10 57)(11 42)(12 41)(13 50)(14 49)(15 48)(16 47)(17 46)(18 45)(19 44)(20 43)(21 77)(22 76)(23 75)(24 74)(25 73)(26 72)(27 71)(28 80)(29 79)(30 78)(31 67)(32 66)(33 65)(34 64)(35 63)(36 62)(37 61)(38 70)(39 69)(40 68)(81 132)(82 131)(83 140)(84 139)(85 138)(86 137)(87 136)(88 135)(89 134)(90 133)(91 122)(92 121)(93 130)(94 129)(95 128)(96 127)(97 126)(98 125)(99 124)(100 123)(101 157)(102 156)(103 155)(104 154)(105 153)(106 152)(107 151)(108 160)(109 159)(110 158)(111 147)(112 146)(113 145)(114 144)(115 143)(116 142)(117 141)(118 150)(119 149)(120 148)
(1 157 17 147)(2 158 18 148)(3 159 19 149)(4 160 20 150)(5 151 11 141)(6 152 12 142)(7 153 13 143)(8 154 14 144)(9 155 15 145)(10 156 16 146)(21 121 31 131)(22 122 32 132)(23 123 33 133)(24 124 34 134)(25 125 35 135)(26 126 36 136)(27 127 37 137)(28 128 38 138)(29 129 39 139)(30 130 40 140)(41 111 51 101)(42 112 52 102)(43 113 53 103)(44 114 54 104)(45 115 55 105)(46 116 56 106)(47 117 57 107)(48 118 58 108)(49 119 59 109)(50 120 60 110)(61 81 71 91)(62 82 72 92)(63 83 73 93)(64 84 74 94)(65 85 75 95)(66 86 76 96)(67 87 77 97)(68 88 78 98)(69 89 79 99)(70 90 80 100)

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

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

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

104 conjugacy classes

 class 1 2A ··· 2O 2P ··· 2W 4A ··· 4H 4I ··· 4P 5A 5B 10A ··· 10AD 20A ··· 20AF order 1 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 10 ··· 10 2 ··· 2 10 ··· 10 2 2 2 ··· 2 2 ··· 2

104 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C2 C4 D4 D5 D10 D10 C4×D5 D20 C5⋊D4 kernel C22×D10⋊C4 C2×D10⋊C4 C23×Dic5 C23×C20 D5×C24 C23×D5 C22×C10 C23×C4 C22×C4 C24 C23 C23 C23 # reps 1 12 1 1 1 16 8 2 12 2 16 16 16

Matrix representation of C22×D10⋊C4 in GL5(𝔽41)

 1 0 0 0 0 0 1 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 40
,
 40 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 40
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 6 35 0 0 0 6 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0
,
 40 0 0 0 0 0 32 0 0 0 0 0 40 0 0 0 0 0 23 35 0 0 0 6 18

G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,6,6,0,0,0,35,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0],[40,0,0,0,0,0,32,0,0,0,0,0,40,0,0,0,0,0,23,6,0,0,0,35,18] >;

C22×D10⋊C4 in GAP, Magma, Sage, TeX

C_2^2\times D_{10}\rtimes C_4
% in TeX

G:=Group("C2^2xD10:C4");
// GroupNames label

G:=SmallGroup(320,1459);
// by ID

G=gap.SmallGroup(320,1459);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,1123,80,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^10=d^2=e^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d=c^-1,c*e=e*c,e*d*e^-1=c^5*d>;
// generators/relations

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