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G = C2×C23.D10order 320 = 26·5

Direct product of C2 and C23.D10

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

Aliases: C2×C23.D10, C24.22D10, (C2×C10).26C24, C4⋊Dic550C22, C22⋊C4.85D10, C102(C422C2), (C2×C20).126C23, (C4×Dic5)⋊72C22, (C22×C4).168D10, C23.78(C22×D5), C22.68(C23×D5), C22.71(C4○D20), C10.D447C22, (C23×C10).52C22, C23.D5.84C22, C22.65(D42D5), (C22×C10).118C23, (C22×C20).350C22, (C2×Dic5).186C23, (C22×Dic5).226C22, C52(C2×C422C2), (C2×C4×Dic5)⋊29C2, (C2×C4⋊Dic5)⋊18C2, C10.11(C2×C4○D4), C2.13(C2×C4○D20), C2.8(C2×D42D5), (C2×C22⋊C4).17D5, (C2×C10.D4)⋊34C2, (C10×C22⋊C4).20C2, (C2×C4).255(C22×D5), (C2×C23.D5).21C2, (C2×C10).100(C4○D4), (C5×C22⋊C4).108C22, SmallGroup(320,1154)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C2×C23.D10
C1C5C10C2×C10C2×Dic5C22×Dic5C2×C4×Dic5 — C2×C23.D10
C5C2×C10 — C2×C23.D10
C1C23C2×C22⋊C4

Generators and relations for C2×C23.D10
 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, fbf-1=bc=cb, ebe-1=bd=db, ce=ec, cf=fc, de=ed, df=fd, fef-1=e9 >

Subgroups: 686 in 246 conjugacy classes, 111 normal (31 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×12], C22, C22 [×6], C22 [×10], C5, C2×C4 [×4], C2×C4 [×20], C23, C23 [×2], C23 [×6], C10 [×3], C10 [×4], C10 [×2], C42 [×4], C22⋊C4 [×4], C22⋊C4 [×8], C4⋊C4 [×12], C22×C4 [×2], C22×C4 [×4], C24, Dic5 [×8], C20 [×4], C2×C10, C2×C10 [×6], C2×C10 [×10], C2×C42, C2×C22⋊C4, C2×C22⋊C4 [×2], C2×C4⋊C4 [×3], C422C2 [×8], C2×Dic5 [×8], C2×Dic5 [×8], C2×C20 [×4], C2×C20 [×4], C22×C10, C22×C10 [×2], C22×C10 [×6], C2×C422C2, C4×Dic5 [×4], C10.D4 [×8], C4⋊Dic5 [×4], C23.D5 [×8], C5×C22⋊C4 [×4], C22×Dic5 [×4], C22×C20 [×2], C23×C10, C23.D10 [×8], C2×C4×Dic5, C2×C10.D4 [×2], C2×C4⋊Dic5, C2×C23.D5 [×2], C10×C22⋊C4, C2×C23.D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×6], C24, D10 [×7], C422C2 [×4], C2×C4○D4 [×3], C22×D5 [×7], C2×C422C2, C4○D20 [×2], D42D5 [×4], C23×D5, C23.D10 [×4], C2×C4○D20, C2×D42D5 [×2], C2×C23.D10

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

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

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

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

68 conjugacy classes

class 1 2A···2G2H2I4A4B4C4D4E4F4G···4N4O4P4Q4R5A5B10A···10N10O···10V20A···20P
order12···2224444444···444445510···1010···1020···20
size11···14422224410···1020202020222···24···44···4

68 irreducible representations

dim11111112222224
type+++++++++++-
imageC1C2C2C2C2C2C2D5C4○D4D10D10D10C4○D20D42D5
kernelC2×C23.D10C23.D10C2×C4×Dic5C2×C10.D4C2×C4⋊Dic5C2×C23.D5C10×C22⋊C4C2×C22⋊C4C2×C10C22⋊C4C22×C4C24C22C22
# reps1812121212842168

Matrix representation of C2×C23.D10 in GL5(𝔽41)

400000
01000
00100
000400
000040
,
400000
01000
004000
00010
0003940
,
10000
040000
004000
000400
000040
,
10000
01000
00100
000400
000040
,
10000
02000
002000
0003232
00009
,
10000
002000
02000
00011
000040

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],[40,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,1,39,0,0,0,0,40],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[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],[1,0,0,0,0,0,2,0,0,0,0,0,20,0,0,0,0,0,32,0,0,0,0,32,9],[1,0,0,0,0,0,0,2,0,0,0,20,0,0,0,0,0,0,1,0,0,0,0,1,40] >;

C2×C23.D10 in GAP, Magma, Sage, TeX

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

G:=Group("C2xC2^3.D10");
// GroupNames label

G:=SmallGroup(320,1154);
// by ID

G=gap.SmallGroup(320,1154);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,100,1571,297,12550]);
// 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,f*b*f^-1=b*c=c*b,e*b*e^-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