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## G = C2×C23.11D10order 320 = 26·5

### Direct product of C2 and C23.11D10

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C2×C23.11D10
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — C23×Dic5 — C2×C23.11D10
 Lower central C5 — C10 — C2×C23.11D10
 Upper central C1 — C23 — C2×C22⋊C4

Generators and relations for C2×C23.11D10
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=c, f2=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe-1=fbf-1=bd=db, ce=ec, cf=fc, de=ed, df=fd, fef-1=e9 >

Subgroups: 830 in 330 conjugacy classes, 167 normal (17 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×16], C22, C22 [×10], C22 [×12], C5, C2×C4 [×4], C2×C4 [×40], C23, C23 [×6], C23 [×4], C10, C10 [×6], C10 [×4], C42 [×8], C22⋊C4 [×4], C22⋊C4 [×4], C4⋊C4 [×8], C22×C4 [×2], C22×C4 [×16], C24, Dic5 [×8], Dic5 [×4], C20 [×4], C2×C10, C2×C10 [×10], C2×C10 [×12], C2×C42 [×2], C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4 [×2], C42⋊C2 [×8], C23×C4, C2×Dic5 [×32], C2×Dic5 [×4], C2×C20 [×4], C2×C20 [×4], C22×C10, C22×C10 [×6], C22×C10 [×4], C2×C42⋊C2, C4×Dic5 [×8], C10.D4 [×8], C23.D5 [×4], C5×C22⋊C4 [×4], C22×Dic5 [×2], C22×Dic5 [×14], C22×C20 [×2], C23×C10, C23.11D10 [×8], C2×C4×Dic5 [×2], C2×C10.D4 [×2], C2×C23.D5, C10×C22⋊C4, C23×Dic5, C2×C23.11D10
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D5, C22×C4 [×14], C4○D4 [×4], C24, D10 [×7], C42⋊C2 [×4], C23×C4, C2×C4○D4 [×2], C4×D5 [×4], C22×D5 [×7], C2×C42⋊C2, C2×C4×D5 [×6], D42D5 [×4], C23×D5, C23.11D10 [×4], D5×C22×C4, C2×D42D5 [×2], C2×C23.11D10

Smallest permutation representation of C2×C23.11D10
On 160 points
Generators in S160
(1 124)(2 125)(3 126)(4 127)(5 128)(6 129)(7 130)(8 131)(9 132)(10 133)(11 134)(12 135)(13 136)(14 137)(15 138)(16 139)(17 140)(18 121)(19 122)(20 123)(21 114)(22 115)(23 116)(24 117)(25 118)(26 119)(27 120)(28 101)(29 102)(30 103)(31 104)(32 105)(33 106)(34 107)(35 108)(36 109)(37 110)(38 111)(39 112)(40 113)(41 96)(42 97)(43 98)(44 99)(45 100)(46 81)(47 82)(48 83)(49 84)(50 85)(51 86)(52 87)(53 88)(54 89)(55 90)(56 91)(57 92)(58 93)(59 94)(60 95)(61 146)(62 147)(63 148)(64 149)(65 150)(66 151)(67 152)(68 153)(69 154)(70 155)(71 156)(72 157)(73 158)(74 159)(75 160)(76 141)(77 142)(78 143)(79 144)(80 145)
(1 134)(2 42)(3 136)(4 44)(5 138)(6 46)(7 140)(8 48)(9 122)(10 50)(11 124)(12 52)(13 126)(14 54)(15 128)(16 56)(17 130)(18 58)(19 132)(20 60)(21 104)(22 72)(23 106)(24 74)(25 108)(26 76)(27 110)(28 78)(29 112)(30 80)(31 114)(32 62)(33 116)(34 64)(35 118)(36 66)(37 120)(38 68)(39 102)(40 70)(41 86)(43 88)(45 90)(47 92)(49 94)(51 96)(53 98)(55 100)(57 82)(59 84)(61 156)(63 158)(65 160)(67 142)(69 144)(71 146)(73 148)(75 150)(77 152)(79 154)(81 129)(83 131)(85 133)(87 135)(89 137)(91 139)(93 121)(95 123)(97 125)(99 127)(101 143)(103 145)(105 147)(107 149)(109 151)(111 153)(113 155)(115 157)(117 159)(119 141)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)(81 91)(82 92)(83 93)(84 94)(85 95)(86 96)(87 97)(88 98)(89 99)(90 100)(101 111)(102 112)(103 113)(104 114)(105 115)(106 116)(107 117)(108 118)(109 119)(110 120)(121 131)(122 132)(123 133)(124 134)(125 135)(126 136)(127 137)(128 138)(129 139)(130 140)(141 151)(142 152)(143 153)(144 154)(145 155)(146 156)(147 157)(148 158)(149 159)(150 160)
(1 86)(2 87)(3 88)(4 89)(5 90)(6 91)(7 92)(8 93)(9 94)(10 95)(11 96)(12 97)(13 98)(14 99)(15 100)(16 81)(17 82)(18 83)(19 84)(20 85)(21 146)(22 147)(23 148)(24 149)(25 150)(26 151)(27 152)(28 153)(29 154)(30 155)(31 156)(32 157)(33 158)(34 159)(35 160)(36 141)(37 142)(38 143)(39 144)(40 145)(41 134)(42 135)(43 136)(44 137)(45 138)(46 139)(47 140)(48 121)(49 122)(50 123)(51 124)(52 125)(53 126)(54 127)(55 128)(56 129)(57 130)(58 131)(59 132)(60 133)(61 114)(62 115)(63 116)(64 117)(65 118)(66 119)(67 120)(68 101)(69 102)(70 103)(71 104)(72 105)(73 106)(74 107)(75 108)(76 109)(77 110)(78 111)(79 112)(80 113)
(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)
(1 109 96 66)(2 118 97 75)(3 107 98 64)(4 116 99 73)(5 105 100 62)(6 114 81 71)(7 103 82 80)(8 112 83 69)(9 101 84 78)(10 110 85 67)(11 119 86 76)(12 108 87 65)(13 117 88 74)(14 106 89 63)(15 115 90 72)(16 104 91 61)(17 113 92 70)(18 102 93 79)(19 111 94 68)(20 120 95 77)(21 46 156 129)(22 55 157 138)(23 44 158 127)(24 53 159 136)(25 42 160 125)(26 51 141 134)(27 60 142 123)(28 49 143 132)(29 58 144 121)(30 47 145 130)(31 56 146 139)(32 45 147 128)(33 54 148 137)(34 43 149 126)(35 52 150 135)(36 41 151 124)(37 50 152 133)(38 59 153 122)(39 48 154 131)(40 57 155 140)

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

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

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

80 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4H 4I ··· 4P 4Q ··· 4AB 5A 5B 10A ··· 10N 10O ··· 10V 20A ··· 20P order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 ··· 4 4 ··· 4 5 5 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 ··· 1 2 2 2 2 2 ··· 2 5 ··· 5 10 ··· 10 2 2 2 ··· 2 4 ··· 4 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C4 D5 C4○D4 D10 D10 D10 C4×D5 D4⋊2D5 kernel C2×C23.11D10 C23.11D10 C2×C4×Dic5 C2×C10.D4 C2×C23.D5 C10×C22⋊C4 C23×Dic5 C22×Dic5 C2×C22⋊C4 C2×C10 C22⋊C4 C22×C4 C24 C23 C22 # reps 1 8 2 2 1 1 1 16 2 8 8 4 2 16 8

Matrix representation of C2×C23.11D10 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 40 0 0 0 0 0 40 0 0 0 0 0 1 0 0 0 0 1 40
,
 1 0 0 0 0 0 40 0 0 0 0 0 40 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 0 19 0 0 0 13 22 0 0 0 0 0 9 23 0 0 0 9 32
,
 1 0 0 0 0 0 3 20 0 0 0 20 38 0 0 0 0 0 40 2 0 0 0 40 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],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,1,0,0,0,0,40],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[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],[1,0,0,0,0,0,0,13,0,0,0,19,22,0,0,0,0,0,9,9,0,0,0,23,32],[1,0,0,0,0,0,3,20,0,0,0,20,38,0,0,0,0,0,40,40,0,0,0,2,1] >;

C2×C23.11D10 in GAP, Magma, Sage, TeX

C_2\times C_2^3._{11}D_{10}
% in TeX

G:=Group("C2xC2^3.11D10");
// GroupNames label

G:=SmallGroup(320,1152);
// by ID

G=gap.SmallGroup(320,1152);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,297,80,12550]);
// Polycyclic

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

׿
×
𝔽