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## G = C22⋊C4×C2×C10order 320 = 26·5

### Direct product of C2×C10 and C22⋊C4

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

Series: Derived Chief Lower central Upper central

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

Generators and relations for C22⋊C4×C2×C10
G = < a,b,c,d,e | a2=b10=c2=d2=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, de=ed >

Subgroups: 1010 in 674 conjugacy classes, 338 normal (12 characteristic)
C1, C2, C2 [×14], C2 [×8], C4 [×8], C22, C22 [×42], C22 [×56], C5, C2×C4 [×8], C2×C4 [×24], C23 [×43], C23 [×56], C10, C10 [×14], C10 [×8], C22⋊C4 [×16], C22×C4 [×12], C22×C4 [×8], C24, C24 [×14], C24 [×8], C20 [×8], C2×C10, C2×C10 [×42], C2×C10 [×56], C2×C22⋊C4 [×12], C23×C4 [×2], C25, C2×C20 [×8], C2×C20 [×24], C22×C10 [×43], C22×C10 [×56], C22×C22⋊C4, C5×C22⋊C4 [×16], C22×C20 [×12], C22×C20 [×8], C23×C10, C23×C10 [×14], C23×C10 [×8], C10×C22⋊C4 [×12], C23×C20 [×2], C24×C10, C22⋊C4×C2×C10
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C5, C2×C4 [×28], D4 [×8], C23 [×15], C10 [×15], C22⋊C4 [×16], C22×C4 [×14], C2×D4 [×12], C24, C20 [×8], C2×C10 [×35], C2×C22⋊C4 [×12], C23×C4, C22×D4 [×2], C2×C20 [×28], C5×D4 [×8], C22×C10 [×15], C22×C22⋊C4, C5×C22⋊C4 [×16], C22×C20 [×14], D4×C10 [×12], C23×C10, C10×C22⋊C4 [×12], C23×C20, D4×C2×C10 [×2], C22⋊C4×C2×C10

Smallest permutation representation of C22⋊C4×C2×C10
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)(11 118)(12 119)(13 120)(14 111)(15 112)(16 113)(17 114)(18 115)(19 116)(20 117)(21 139)(22 140)(23 131)(24 132)(25 133)(26 134)(27 135)(28 136)(29 137)(30 138)(31 125)(32 126)(33 127)(34 128)(35 129)(36 130)(37 121)(38 122)(39 123)(40 124)(41 100)(42 91)(43 92)(44 93)(45 94)(46 95)(47 96)(48 97)(49 98)(50 99)(51 109)(52 110)(53 101)(54 102)(55 103)(56 104)(57 105)(58 106)(59 107)(60 108)(61 77)(62 78)(63 79)(64 80)(65 71)(66 72)(67 73)(68 74)(69 75)(70 76)(141 157)(142 158)(143 159)(144 160)(145 151)(146 152)(147 153)(148 154)(149 155)(150 156)
(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 62)(2 63)(3 64)(4 65)(5 66)(6 67)(7 68)(8 69)(9 70)(10 61)(11 158)(12 159)(13 160)(14 151)(15 152)(16 153)(17 154)(18 155)(19 156)(20 157)(21 36)(22 37)(23 38)(24 39)(25 40)(26 31)(27 32)(28 33)(29 34)(30 35)(41 59)(42 60)(43 51)(44 52)(45 53)(46 54)(47 55)(48 56)(49 57)(50 58)(71 89)(72 90)(73 81)(74 82)(75 83)(76 84)(77 85)(78 86)(79 87)(80 88)(91 108)(92 109)(93 110)(94 101)(95 102)(96 103)(97 104)(98 105)(99 106)(100 107)(111 145)(112 146)(113 147)(114 148)(115 149)(116 150)(117 141)(118 142)(119 143)(120 144)(121 140)(122 131)(123 132)(124 133)(125 134)(126 135)(127 136)(128 137)(129 138)(130 139)
(1 54)(2 55)(3 56)(4 57)(5 58)(6 59)(7 60)(8 51)(9 52)(10 53)(11 27)(12 28)(13 29)(14 30)(15 21)(16 22)(17 23)(18 24)(19 25)(20 26)(31 157)(32 158)(33 159)(34 160)(35 151)(36 152)(37 153)(38 154)(39 155)(40 156)(41 67)(42 68)(43 69)(44 70)(45 61)(46 62)(47 63)(48 64)(49 65)(50 66)(71 98)(72 99)(73 100)(74 91)(75 92)(76 93)(77 94)(78 95)(79 96)(80 97)(81 107)(82 108)(83 109)(84 110)(85 101)(86 102)(87 103)(88 104)(89 105)(90 106)(111 138)(112 139)(113 140)(114 131)(115 132)(116 133)(117 134)(118 135)(119 136)(120 137)(121 147)(122 148)(123 149)(124 150)(125 141)(126 142)(127 143)(128 144)(129 145)(130 146)
(1 16 67 32)(2 17 68 33)(3 18 69 34)(4 19 70 35)(5 20 61 36)(6 11 62 37)(7 12 63 38)(8 13 64 39)(9 14 65 40)(10 15 66 31)(21 50 157 53)(22 41 158 54)(23 42 159 55)(24 43 160 56)(25 44 151 57)(26 45 152 58)(27 46 153 59)(28 47 154 60)(29 48 155 51)(30 49 156 52)(71 124 84 111)(72 125 85 112)(73 126 86 113)(74 127 87 114)(75 128 88 115)(76 129 89 116)(77 130 90 117)(78 121 81 118)(79 122 82 119)(80 123 83 120)(91 143 103 131)(92 144 104 132)(93 145 105 133)(94 146 106 134)(95 147 107 135)(96 148 108 136)(97 149 109 137)(98 150 110 138)(99 141 101 139)(100 142 102 140)

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

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

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

200 conjugacy classes

 class 1 2A ··· 2O 2P ··· 2W 4A ··· 4P 5A 5B 5C 5D 10A ··· 10BH 10BI ··· 10CN 20A ··· 20BL order 1 2 ··· 2 2 ··· 2 4 ··· 4 5 5 5 5 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 ··· 1 2 ··· 2 2 ··· 2 1 1 1 1 1 ··· 1 2 ··· 2 2 ··· 2

200 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 type + + + + + image C1 C2 C2 C2 C4 C5 C10 C10 C10 C20 D4 C5×D4 kernel C22⋊C4×C2×C10 C10×C22⋊C4 C23×C20 C24×C10 C23×C10 C22×C22⋊C4 C2×C22⋊C4 C23×C4 C25 C24 C22×C10 C23 # reps 1 12 2 1 16 4 48 8 4 64 8 32

Matrix representation of C22⋊C4×C2×C10 in GL5(𝔽41)

 40 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 1 0 0 0 0 0 40 0 0 0 0 0 37 0 0 0 0 0 37
,
 40 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 40
,
 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 32 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,37,0,0,0,0,0,37],[40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40],[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],[1,0,0,0,0,0,32,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;

C22⋊C4×C2×C10 in GAP, Magma, Sage, TeX

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

G:=Group("C2^2:C4xC2xC10");
// GroupNames label

G:=SmallGroup(320,1514);
// by ID

G=gap.SmallGroup(320,1514);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1120,1149]);
// Polycyclic

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

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