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## G = C22×C4○D20order 320 = 26·5

### Direct product of C22 and C4○D20

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C22×C4○D20
 Chief series C1 — C5 — C10 — D10 — C22×D5 — C23×D5 — D5×C22×C4 — C22×C4○D20
 Lower central C5 — C10 — C22×C4○D20
 Upper central C1 — C22×C4 — C23×C4

Generators and relations for C22×C4○D20
G = < a,b,c,d,e | a2=b2=c4=e2=1, d10=c2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c2d9 >

Subgroups: 2558 in 890 conjugacy classes, 463 normal (17 characteristic)
C1, C2, C2 [×6], C2 [×12], C4 [×8], C4 [×8], C22 [×11], C22 [×44], C5, C2×C4 [×28], C2×C4 [×44], D4 [×48], Q8 [×16], C23, C23 [×6], C23 [×24], D5 [×8], C10, C10 [×6], C10 [×4], C22×C4 [×2], C22×C4 [×12], C22×C4 [×26], C2×D4 [×36], C2×Q8 [×12], C4○D4 [×64], C24, C24 [×2], Dic5 [×8], C20 [×8], D10 [×8], D10 [×24], C2×C10 [×11], C2×C10 [×12], C23×C4, C23×C4 [×2], C22×D4 [×3], C22×Q8, C2×C4○D4 [×24], Dic10 [×16], C4×D5 [×32], D20 [×16], C2×Dic5 [×12], C5⋊D4 [×32], C2×C20 [×28], C22×D5 [×12], C22×D5 [×8], C22×C10, C22×C10 [×6], C22×C10 [×4], C22×C4○D4, C2×Dic10 [×12], C2×C4×D5 [×24], C2×D20 [×12], C4○D20 [×64], C22×Dic5 [×2], C2×C5⋊D4 [×24], C22×C20 [×2], C22×C20 [×12], C23×D5 [×2], C23×C10, C22×Dic10, D5×C22×C4 [×2], C22×D20, C2×C4○D20 [×24], C22×C5⋊D4 [×2], C23×C20, C22×C4○D20
Quotients: C1, C2 [×31], C22 [×155], C23 [×155], D5, C4○D4 [×4], C24 [×31], D10 [×15], C2×C4○D4 [×6], C25, C22×D5 [×35], C22×C4○D4, C4○D20 [×4], C23×D5 [×15], C2×C4○D20 [×6], D5×C24, C22×C4○D20

Smallest permutation representation of C22×C4○D20
On 160 points
Generators in S160
(1 98)(2 99)(3 100)(4 81)(5 82)(6 83)(7 84)(8 85)(9 86)(10 87)(11 88)(12 89)(13 90)(14 91)(15 92)(16 93)(17 94)(18 95)(19 96)(20 97)(21 149)(22 150)(23 151)(24 152)(25 153)(26 154)(27 155)(28 156)(29 157)(30 158)(31 159)(32 160)(33 141)(34 142)(35 143)(36 144)(37 145)(38 146)(39 147)(40 148)(41 117)(42 118)(43 119)(44 120)(45 101)(46 102)(47 103)(48 104)(49 105)(50 106)(51 107)(52 108)(53 109)(54 110)(55 111)(56 112)(57 113)(58 114)(59 115)(60 116)(61 129)(62 130)(63 131)(64 132)(65 133)(66 134)(67 135)(68 136)(69 137)(70 138)(71 139)(72 140)(73 121)(74 122)(75 123)(76 124)(77 125)(78 126)(79 127)(80 128)
(1 32)(2 33)(3 34)(4 35)(5 36)(6 37)(7 38)(8 39)(9 40)(10 21)(11 22)(12 23)(13 24)(14 25)(15 26)(16 27)(17 28)(18 29)(19 30)(20 31)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)(81 143)(82 144)(83 145)(84 146)(85 147)(86 148)(87 149)(88 150)(89 151)(90 152)(91 153)(92 154)(93 155)(94 156)(95 157)(96 158)(97 159)(98 160)(99 141)(100 142)(101 133)(102 134)(103 135)(104 136)(105 137)(106 138)(107 139)(108 140)(109 121)(110 122)(111 123)(112 124)(113 125)(114 126)(115 127)(116 128)(117 129)(118 130)(119 131)(120 132)
(1 72 11 62)(2 73 12 63)(3 74 13 64)(4 75 14 65)(5 76 15 66)(6 77 16 67)(7 78 17 68)(8 79 18 69)(9 80 19 70)(10 61 20 71)(21 41 31 51)(22 42 32 52)(23 43 33 53)(24 44 34 54)(25 45 35 55)(26 46 36 56)(27 47 37 57)(28 48 38 58)(29 49 39 59)(30 50 40 60)(81 123 91 133)(82 124 92 134)(83 125 93 135)(84 126 94 136)(85 127 95 137)(86 128 96 138)(87 129 97 139)(88 130 98 140)(89 131 99 121)(90 132 100 122)(101 143 111 153)(102 144 112 154)(103 145 113 155)(104 146 114 156)(105 147 115 157)(106 148 116 158)(107 149 117 159)(108 150 118 160)(109 151 119 141)(110 152 120 142)
(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)
(1 159)(2 158)(3 157)(4 156)(5 155)(6 154)(7 153)(8 152)(9 151)(10 150)(11 149)(12 148)(13 147)(14 146)(15 145)(16 144)(17 143)(18 142)(19 141)(20 160)(21 88)(22 87)(23 86)(24 85)(25 84)(26 83)(27 82)(28 81)(29 100)(30 99)(31 98)(32 97)(33 96)(34 95)(35 94)(36 93)(37 92)(38 91)(39 90)(40 89)(41 130)(42 129)(43 128)(44 127)(45 126)(46 125)(47 124)(48 123)(49 122)(50 121)(51 140)(52 139)(53 138)(54 137)(55 136)(56 135)(57 134)(58 133)(59 132)(60 131)(61 118)(62 117)(63 116)(64 115)(65 114)(66 113)(67 112)(68 111)(69 110)(70 109)(71 108)(72 107)(73 106)(74 105)(75 104)(76 103)(77 102)(78 101)(79 120)(80 119)

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

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

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

104 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L ··· 2S 4A ··· 4H 4I 4J 4K 4L 4M ··· 4T 5A 5B 10A ··· 10AD 20A ··· 20AF order 1 2 ··· 2 2 2 2 2 2 ··· 2 4 ··· 4 4 4 4 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 2 2 2 2 10 ··· 10 1 ··· 1 2 2 2 2 10 ··· 10 2 2 2 ··· 2 2 ··· 2

104 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 D5 C4○D4 D10 D10 C4○D20 kernel C22×C4○D20 C22×Dic10 D5×C22×C4 C22×D20 C2×C4○D20 C22×C5⋊D4 C23×C20 C23×C4 C2×C10 C22×C4 C24 C22 # reps 1 1 2 1 24 2 1 2 8 28 2 32

Matrix representation of C22×C4○D20 in GL5(𝔽41)

 40 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 40 0 0 0 0 0 40
,
 1 0 0 0 0 0 9 0 0 0 0 0 9 0 0 0 0 0 40 0 0 0 0 0 40
,
 1 0 0 0 0 0 0 9 0 0 0 9 0 0 0 0 0 0 6 1 0 0 0 40 0
,
 1 0 0 0 0 0 0 9 0 0 0 32 0 0 0 0 0 0 40 35 0 0 0 0 1

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,0,9,0,0,0,9,0,0,0,0,0,0,6,40,0,0,0,1,0],[1,0,0,0,0,0,0,32,0,0,0,9,0,0,0,0,0,0,40,0,0,0,0,35,1] >;

C22×C4○D20 in GAP, Magma, Sage, TeX

C_2^2\times C_4\circ D_{20}
% in TeX

G:=Group("C2^2xC4oD20");
// GroupNames label

G:=SmallGroup(320,1611);
// by ID

G=gap.SmallGroup(320,1611);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,136,1684,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^4=e^2=1,d^10=c^2,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,c*d=d*c,c*e=e*c,e*d*e=c^2*d^9>;
// generators/relations

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