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## G = C2×C23.2F5order 320 = 26·5

### Direct product of C2 and C23.2F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C2×C23.2F5
 Chief series C1 — C5 — C10 — Dic5 — C2×Dic5 — C2×C5⋊C8 — C22×C5⋊C8 — C2×C23.2F5
 Lower central C5 — C10 — C2×C23.2F5
 Upper central C1 — C23 — C24

Generators and relations for C2×C23.2F5
G = < a,b,c,d,e,f | a2=b2=c2=d2=e5=1, f4=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf-1=bc=cb, bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e3 >

Subgroups: 602 in 202 conjugacy classes, 84 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×6], C22, C22 [×10], C22 [×12], C5, C8 [×4], C2×C4 [×18], C23, C23 [×6], C23 [×4], C10 [×3], C10 [×4], C10 [×4], C2×C8 [×8], C22×C4 [×10], C24, Dic5 [×4], Dic5 [×2], C2×C10, C2×C10 [×10], C2×C10 [×12], C22⋊C8 [×4], C22×C8 [×2], C23×C4, C5⋊C8 [×4], C2×Dic5 [×2], C2×Dic5 [×6], C2×Dic5 [×10], C22×C10, C22×C10 [×6], C22×C10 [×4], C2×C22⋊C8, C2×C5⋊C8 [×4], C2×C5⋊C8 [×4], C22×Dic5 [×2], C22×Dic5 [×4], C22×Dic5 [×4], C23×C10, C23.2F5 [×4], C22×C5⋊C8 [×2], C23×Dic5, C2×C23.2F5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C2×C8 [×6], M4(2) [×2], C22×C4, C2×D4 [×2], F5, C22⋊C8 [×4], C2×C22⋊C4, C22×C8, C2×M4(2), C5⋊C8 [×4], C2×F5 [×3], C2×C22⋊C8, C2×C5⋊C8 [×6], C22.F5 [×2], C22⋊F5 [×2], C22×F5, C23.2F5 [×4], C22×C5⋊C8, C2×C22.F5, C2×C22⋊F5, C2×C23.2F5

Smallest permutation representation of C2×C23.2F5
On 160 points
Generators in S160
(1 112)(2 105)(3 106)(4 107)(5 108)(6 109)(7 110)(8 111)(9 28)(10 29)(11 30)(12 31)(13 32)(14 25)(15 26)(16 27)(17 40)(18 33)(19 34)(20 35)(21 36)(22 37)(23 38)(24 39)(41 52)(42 53)(43 54)(44 55)(45 56)(46 49)(47 50)(48 51)(57 87)(58 88)(59 81)(60 82)(61 83)(62 84)(63 85)(64 86)(65 155)(66 156)(67 157)(68 158)(69 159)(70 160)(71 153)(72 154)(73 142)(74 143)(75 144)(76 137)(77 138)(78 139)(79 140)(80 141)(89 129)(90 130)(91 131)(92 132)(93 133)(94 134)(95 135)(96 136)(97 119)(98 120)(99 113)(100 114)(101 115)(102 116)(103 117)(104 118)(121 150)(122 151)(123 152)(124 145)(125 146)(126 147)(127 148)(128 149)
(1 48)(2 109)(3 42)(4 111)(5 44)(6 105)(7 46)(8 107)(9 151)(10 25)(11 145)(12 27)(13 147)(14 29)(15 149)(16 31)(17 36)(18 138)(19 38)(20 140)(21 40)(22 142)(23 34)(24 144)(26 128)(28 122)(30 124)(32 126)(33 77)(35 79)(37 73)(39 75)(41 56)(43 50)(45 52)(47 54)(49 110)(51 112)(53 106)(55 108)(57 91)(58 84)(59 93)(60 86)(61 95)(62 88)(63 89)(64 82)(65 97)(66 160)(67 99)(68 154)(69 101)(70 156)(71 103)(72 158)(74 139)(76 141)(78 143)(80 137)(81 133)(83 135)(85 129)(87 131)(90 134)(92 136)(94 130)(96 132)(98 116)(100 118)(102 120)(104 114)(113 157)(115 159)(117 153)(119 155)(121 146)(123 148)(125 150)(127 152)
(1 55)(2 56)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 126)(10 127)(11 128)(12 121)(13 122)(14 123)(15 124)(16 125)(17 80)(18 73)(19 74)(20 75)(21 76)(22 77)(23 78)(24 79)(25 152)(26 145)(27 146)(28 147)(29 148)(30 149)(31 150)(32 151)(33 142)(34 143)(35 144)(36 137)(37 138)(38 139)(39 140)(40 141)(41 109)(42 110)(43 111)(44 112)(45 105)(46 106)(47 107)(48 108)(57 135)(58 136)(59 129)(60 130)(61 131)(62 132)(63 133)(64 134)(65 115)(66 116)(67 117)(68 118)(69 119)(70 120)(71 113)(72 114)(81 89)(82 90)(83 91)(84 92)(85 93)(86 94)(87 95)(88 96)(97 159)(98 160)(99 153)(100 154)(101 155)(102 156)(103 157)(104 158)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)(65 69)(66 70)(67 71)(68 72)(73 77)(74 78)(75 79)(76 80)(81 85)(82 86)(83 87)(84 88)(89 93)(90 94)(91 95)(92 96)(97 101)(98 102)(99 103)(100 104)(105 109)(106 110)(107 111)(108 112)(113 117)(114 118)(115 119)(116 120)(121 125)(122 126)(123 127)(124 128)(129 133)(130 134)(131 135)(132 136)(137 141)(138 142)(139 143)(140 144)(145 149)(146 150)(147 151)(148 152)(153 157)(154 158)(155 159)(156 160)
(1 159 63 122 75)(2 123 160 76 64)(3 77 124 57 153)(4 58 78 154 125)(5 155 59 126 79)(6 127 156 80 60)(7 73 128 61 157)(8 62 74 158 121)(9 24 51 101 129)(10 102 17 130 52)(11 131 103 53 18)(12 54 132 19 104)(13 20 55 97 133)(14 98 21 134 56)(15 135 99 49 22)(16 50 136 23 100)(25 120 36 94 45)(26 95 113 46 37)(27 47 96 38 114)(28 39 48 115 89)(29 116 40 90 41)(30 91 117 42 33)(31 43 92 34 118)(32 35 44 119 93)(65 81 147 140 108)(66 141 82 109 148)(67 110 142 149 83)(68 150 111 84 143)(69 85 151 144 112)(70 137 86 105 152)(71 106 138 145 87)(72 146 107 88 139)
(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)

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

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

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

56 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4H 4I 4J 4K 4L 5 8A ··· 8P 10A ··· 10O order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 4 4 4 5 8 ··· 8 10 ··· 10 size 1 1 ··· 1 2 2 2 2 5 ··· 5 10 10 10 10 4 10 ··· 10 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 4 4 4 4 4 type + + + + + + - + - + image C1 C2 C2 C2 C4 C4 C8 D4 M4(2) F5 C5⋊C8 C2×F5 C22.F5 C22⋊F5 kernel C2×C23.2F5 C23.2F5 C22×C5⋊C8 C23×Dic5 C22×Dic5 C23×C10 C22×C10 C2×Dic5 C2×C10 C24 C23 C23 C22 C22 # reps 1 4 2 1 6 2 16 4 4 1 4 3 4 4

Matrix representation of C2×C23.2F5 in GL8(𝔽41)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40
,
 1 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 37 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 37 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 18
,
 0 39 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 9 16 0 0 0 0 0 0 36 32 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0

G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,37,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,37,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,18],[0,16,0,0,0,0,0,0,39,0,0,0,0,0,0,0,0,0,9,36,0,0,0,0,0,0,16,32,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;

C2×C23.2F5 in GAP, Magma, Sage, TeX

C_2\times C_2^3._2F_5
% in TeX

G:=Group("C2xC2^3.2F5");
// GroupNames label

G:=SmallGroup(320,1135);
// by ID

G=gap.SmallGroup(320,1135);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,136,6278,1595]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^5=1,f^4=d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,f*b*f^-1=b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^3>;
// generators/relations

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