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G = C2×C23.2F5order 320 = 26·5

Direct product of C2 and C23.2F5

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×C23.2F5, C24.3F5, C232(C5⋊C8), (C22×C10)⋊4C8, C102(C22⋊C8), (C23×C10).6C4, C23.63(C2×F5), C10.24(C22×C8), Dic5.116(C2×D4), (C2×Dic5).263D4, (C2×C10).25M4(2), C10.33(C2×M4(2)), C22.53(C22×F5), (C23×Dic5).12C2, (C22×Dic5).36C4, C22.53(C22⋊F5), Dic5.51(C22⋊C4), (C2×Dic5).359C23, C22.10(C22.F5), (C22×Dic5).279C22, C53(C2×C22⋊C8), C222(C2×C5⋊C8), (C2×C10)⋊8(C2×C8), (C22×C5⋊C8)⋊7C2, (C2×C5⋊C8)⋊8C22, C2.9(C22×C5⋊C8), C2.5(C2×C22⋊F5), C10.38(C2×C22⋊C4), C2.5(C2×C22.F5), (C2×C10).91(C22×C4), (C22×C10).73(C2×C4), (C2×C10).62(C22⋊C4), (C2×Dic5).195(C2×C4), SmallGroup(320,1135)

Series: Derived Chief Lower central Upper central

C1C10 — C2×C23.2F5
C1C5C10Dic5C2×Dic5C2×C5⋊C8C22×C5⋊C8 — C2×C23.2F5
C5C10 — C2×C23.2F5
C1C23C24

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···2G2H2I2J2K4A···4H4I4J4K4L 5 8A···8P10A···10O
order12···222224···4444458···810···10
size11···122225···510101010410···104···4

56 irreducible representations

dim11111112244444
type++++++-+-+
imageC1C2C2C2C4C4C8D4M4(2)F5C5⋊C8C2×F5C22.F5C22⋊F5
kernelC2×C23.2F5C23.2F5C22×C5⋊C8C23×Dic5C22×Dic5C23×C10C22×C10C2×Dic5C2×C10C24C23C23C22C22
# reps142162164414344

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

10000000
01000000
00100000
00010000
000040000
000004000
000000400
000000040
,
10000000
040000000
004000000
003710000
00001000
00000100
00000010
00000001
,
400000000
040000000
004000000
000400000
00001000
00000100
00000010
00000001
,
400000000
040000000
00100000
00010000
000040000
000004000
000000400
000000040
,
10000000
01000000
00100000
00010000
000037000
000001000
000000160
000000018
,
039000000
160000000
009160000
0036320000
00000010
00000001
00000100
000040000

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

׿
×
𝔽