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## G = C5×(C22×C8)⋊C2order 320 = 26·5

### Direct product of C5 and (C22×C8)⋊C2

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C5×(C22×C8)⋊C2
 Chief series C1 — C2 — C4 — C2×C4 — C2×C20 — C2×C40 — C5×C22⋊C8 — C5×(C22×C8)⋊C2
 Lower central C1 — C22 — C5×(C22×C8)⋊C2
 Upper central C1 — C2×C20 — C5×(C22×C8)⋊C2

Generators and relations for C5×(C22×C8)⋊C2
G = < a,b,c,d,e | a5=b2=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe=bd4, ede=cd=dc, ce=ec >

Subgroups: 242 in 158 conjugacy classes, 82 normal (30 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C10, C10, C10, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C20, C20, C20, C2×C10, C2×C10, C2×C10, C22⋊C8, C22×C8, C2×M4(2), C2×C4○D4, C40, C2×C20, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C22×C10, (C22×C8)⋊C2, C2×C40, C2×C40, C5×M4(2), C22×C20, C22×C20, D4×C10, D4×C10, Q8×C10, C5×C4○D4, C5×C22⋊C8, C22×C40, C10×M4(2), C10×C4○D4, C5×(C22×C8)⋊C2
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, C23, C10, C22⋊C4, C22×C4, C2×D4, C20, C2×C10, C2×C22⋊C4, C8○D4, C2×C20, C5×D4, C22×C10, (C22×C8)⋊C2, C5×C22⋊C4, C22×C20, D4×C10, C10×C22⋊C4, C5×C8○D4, C5×(C22×C8)⋊C2

Smallest permutation representation of C5×(C22×C8)⋊C2
On 160 points
Generators in S160
(1 111 31 103 23)(2 112 32 104 24)(3 105 25 97 17)(4 106 26 98 18)(5 107 27 99 19)(6 108 28 100 20)(7 109 29 101 21)(8 110 30 102 22)(9 90 42 114 34)(10 91 43 115 35)(11 92 44 116 36)(12 93 45 117 37)(13 94 46 118 38)(14 95 47 119 39)(15 96 48 120 40)(16 89 41 113 33)(49 121 137 57 129)(50 122 138 58 130)(51 123 139 59 131)(52 124 140 60 132)(53 125 141 61 133)(54 126 142 62 134)(55 127 143 63 135)(56 128 144 64 136)(65 85 153 73 145)(66 86 154 74 146)(67 87 155 75 147)(68 88 156 76 148)(69 81 157 77 149)(70 82 158 78 150)(71 83 159 79 151)(72 84 160 80 152)
(1 39)(2 40)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 106)(10 107)(11 108)(12 109)(13 110)(14 111)(15 112)(16 105)(17 113)(18 114)(19 115)(20 116)(21 117)(22 118)(23 119)(24 120)(25 89)(26 90)(27 91)(28 92)(29 93)(30 94)(31 95)(32 96)(41 97)(42 98)(43 99)(44 100)(45 101)(46 102)(47 103)(48 104)(49 145)(50 146)(51 147)(52 148)(53 149)(54 150)(55 151)(56 152)(57 153)(58 154)(59 155)(60 156)(61 157)(62 158)(63 159)(64 160)(65 121)(66 122)(67 123)(68 124)(69 125)(70 126)(71 127)(72 128)(73 129)(74 130)(75 131)(76 132)(77 133)(78 134)(79 135)(80 136)(81 141)(82 142)(83 143)(84 144)(85 137)(86 138)(87 139)(88 140)
(1 127)(2 128)(3 121)(4 122)(5 123)(6 124)(7 125)(8 126)(9 86)(10 87)(11 88)(12 81)(13 82)(14 83)(15 84)(16 85)(17 49)(18 50)(19 51)(20 52)(21 53)(22 54)(23 55)(24 56)(25 57)(26 58)(27 59)(28 60)(29 61)(30 62)(31 63)(32 64)(33 65)(34 66)(35 67)(36 68)(37 69)(38 70)(39 71)(40 72)(41 73)(42 74)(43 75)(44 76)(45 77)(46 78)(47 79)(48 80)(89 153)(90 154)(91 155)(92 156)(93 157)(94 158)(95 159)(96 160)(97 129)(98 130)(99 131)(100 132)(101 133)(102 134)(103 135)(104 136)(105 137)(106 138)(107 139)(108 140)(109 141)(110 142)(111 143)(112 144)(113 145)(114 146)(115 147)(116 148)(117 149)(118 150)(119 151)(120 152)
(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 37)(2 70)(3 39)(4 72)(5 33)(6 66)(7 35)(8 68)(9 140)(10 109)(11 142)(12 111)(13 144)(14 105)(15 138)(16 107)(17 119)(18 152)(19 113)(20 146)(21 115)(22 148)(23 117)(24 150)(25 95)(26 160)(27 89)(28 154)(29 91)(30 156)(31 93)(32 158)(34 124)(36 126)(38 128)(40 122)(41 99)(42 132)(43 101)(44 134)(45 103)(46 136)(47 97)(48 130)(49 151)(50 120)(51 145)(52 114)(53 147)(54 116)(55 149)(56 118)(57 159)(58 96)(59 153)(60 90)(61 155)(62 92)(63 157)(64 94)(65 123)(67 125)(69 127)(71 121)(73 131)(74 100)(75 133)(76 102)(77 135)(78 104)(79 129)(80 98)(81 143)(82 112)(83 137)(84 106)(85 139)(86 108)(87 141)(88 110)

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

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

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

140 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 5C 5D 8A ··· 8H 8I 8J 8K 8L 10A ··· 10L 10M ··· 10T 10U ··· 10AB 20A ··· 20P 20Q ··· 20X 20Y ··· 20AF 40A ··· 40AF 40AG ··· 40AV order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 5 5 5 5 8 ··· 8 8 8 8 8 10 ··· 10 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 20 ··· 20 40 ··· 40 40 ··· 40 size 1 1 1 1 2 2 4 4 1 1 1 1 2 2 4 4 1 1 1 1 2 ··· 2 4 4 4 4 1 ··· 1 2 ··· 2 4 ··· 4 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

140 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C2 C4 C4 C5 C10 C10 C10 C10 C20 C20 D4 C8○D4 C5×D4 C5×C8○D4 kernel C5×(C22×C8)⋊C2 C5×C22⋊C8 C22×C40 C10×M4(2) C10×C4○D4 D4×C10 Q8×C10 (C22×C8)⋊C2 C22⋊C8 C22×C8 C2×M4(2) C2×C4○D4 C2×D4 C2×Q8 C2×C20 C10 C2×C4 C2 # reps 1 4 1 1 1 6 2 4 16 4 4 4 24 8 4 8 16 32

Matrix representation of C5×(C22×C8)⋊C2 in GL4(𝔽41) generated by

 1 0 0 0 0 1 0 0 0 0 18 0 0 0 0 18
,
 1 0 0 0 0 1 0 0 0 0 1 39 0 0 0 40
,
 40 0 0 0 0 40 0 0 0 0 1 0 0 0 0 1
,
 32 2 0 0 1 9 0 0 0 0 3 0 0 0 0 3
,
 1 0 0 0 9 40 0 0 0 0 9 23 0 0 9 32
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,18,0,0,0,0,18],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,39,40],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[32,1,0,0,2,9,0,0,0,0,3,0,0,0,0,3],[1,9,0,0,0,40,0,0,0,0,9,9,0,0,23,32] >;

C5×(C22×C8)⋊C2 in GAP, Magma, Sage, TeX

C_5\times (C_2^2\times C_8)\rtimes C_2
% in TeX

G:=Group("C5x(C2^2xC8):C2");
// GroupNames label

G:=SmallGroup(320,909);
// by ID

G=gap.SmallGroup(320,909);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,1731,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^2=c^2=d^8=e^2=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,e*b*e=b*d^4,e*d*e=c*d=d*c,c*e=e*c>;
// generators/relations

׿
×
𝔽