Copied to
clipboard

G = C2×C23.21D10order 320 = 26·5

Direct product of C2 and C23.21D10

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

Aliases: C2×C23.21D10, C24.68D10, (C22×C20)⋊27C4, (C23×C4).12D5, C10.64(C23×C4), (C23×C20).17C2, C4⋊Dic583C22, (C22×C4)⋊10Dic5, C105(C42⋊C2), C2.5(C23×Dic5), C20.240(C22×C4), (C2×C20).885C23, (C2×C10).284C24, (C4×Dic5)⋊81C22, (C22×C4).448D10, C23.36(C2×Dic5), C4.39(C22×Dic5), C22.41(C23×D5), C22.80(C4○D20), C23.232(C22×D5), (C22×C20).547C22, (C22×C10).413C23, (C23×C10).106C22, (C2×Dic5).288C23, C23.D5.144C22, C22.31(C22×Dic5), (C22×Dic5).252C22, (C2×C20)⋊50(C2×C4), C56(C2×C42⋊C2), (C2×C4×Dic5)⋊37C2, C2.5(C2×C4○D20), (C2×C4⋊Dic5)⋊50C2, C10.60(C2×C4○D4), (C2×C4)⋊11(C2×Dic5), (C2×C4).829(C22×D5), (C2×C23.D5).26C2, (C2×C10).111(C4○D4), (C2×C10).309(C22×C4), (C22×C10).211(C2×C4), SmallGroup(320,1458)

Series: Derived Chief Lower central Upper central

C1C10 — C2×C23.21D10
C1C5C10C2×C10C2×Dic5C22×Dic5C2×C4×Dic5 — C2×C23.21D10
C5C10 — C2×C23.21D10
C1C22×C4C23×C4

Generators and relations for C2×C23.21D10
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=d, f2=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, ce=ec, cf=fc, de=ed, df=fd, fef-1=e9 >

Subgroups: 734 in 330 conjugacy classes, 207 normal (17 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×8], C4 [×8], C22, C22 [×10], C22 [×12], C5, C2×C4 [×28], C2×C4 [×16], C23, C23 [×6], C23 [×4], C10, C10 [×6], C10 [×4], C42 [×8], C22⋊C4 [×8], C4⋊C4 [×8], C22×C4 [×2], C22×C4 [×12], C22×C4 [×4], C24, Dic5 [×8], C20 [×8], C2×C10, C2×C10 [×10], C2×C10 [×12], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C42⋊C2 [×8], C23×C4, C2×Dic5 [×8], C2×Dic5 [×8], C2×C20 [×28], C22×C10, C22×C10 [×6], C22×C10 [×4], C2×C42⋊C2, C4×Dic5 [×8], C4⋊Dic5 [×8], C23.D5 [×8], C22×Dic5 [×4], C22×C20 [×2], C22×C20 [×12], C23×C10, C2×C4×Dic5 [×2], C2×C4⋊Dic5 [×2], C23.21D10 [×8], C2×C23.D5 [×2], C23×C20, C2×C23.21D10
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D5, C22×C4 [×14], C4○D4 [×4], C24, Dic5 [×8], D10 [×7], C42⋊C2 [×4], C23×C4, C2×C4○D4 [×2], C2×Dic5 [×28], C22×D5 [×7], C2×C42⋊C2, C4○D20 [×4], C22×Dic5 [×14], C23×D5, C23.21D10 [×4], C2×C4○D20 [×2], C23×Dic5, C2×C23.21D10

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

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

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

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

104 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I4J4K4L4M···4AB5A5B10A···10AD20A···20AF
order12···222224···444444···45510···1020···20
size11···122221···1222210···10222···22···2

104 irreducible representations

dim1111111222222
type+++++++-++
imageC1C2C2C2C2C2C4D5C4○D4Dic5D10D10C4○D20
kernelC2×C23.21D10C2×C4×Dic5C2×C4⋊Dic5C23.21D10C2×C23.D5C23×C20C22×C20C23×C4C2×C10C22×C4C22×C4C24C22
# reps12282116281612232

Matrix representation of C2×C23.21D10 in GL4(𝔽41) generated by

40000
04000
0010
0001
,
40000
0100
00400
0001
,
40000
0100
00400
00040
,
1000
0100
00400
00040
,
1000
0100
00210
00039
,
32000
04000
00039
00200
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,1,0,0,0,0,40,0,0,0,0,1],[40,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,21,0,0,0,0,39],[32,0,0,0,0,40,0,0,0,0,0,20,0,0,39,0] >;

C2×C23.21D10 in GAP, Magma, Sage, TeX

C_2\times C_2^3._{21}D_{10}
% in TeX

G:=Group("C2xC2^3.21D10");
// GroupNames label

G:=SmallGroup(320,1458);
// by ID

G=gap.SmallGroup(320,1458);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,184,1123,12550]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^10=d,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,f*b*f^-1=b*d=d*b,b*e=e*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