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G = C5×C23.46D4order 320 = 26·5

Direct product of C5 and C23.46D4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C5×C23.46D4, C4.Q89C10, C22⋊C810C10, C4⋊D4.6C10, (C2×C20).337D4, D4⋊C412C10, C23.46(C5×D4), C2.12(C10×SD16), C10.92(C2×SD16), (C2×C10).36SD16, C20.318(C4○D4), C22.8(C5×SD16), (C2×C20).937C23, (C2×C40).305C22, (C22×C10).168D4, C22.102(D4×C10), C10.142(C8⋊C22), (D4×C10).196C22, (C22×C20).429C22, C10.96(C22.D4), (C10×C4⋊C4)⋊39C2, (C2×C4⋊C4)⋊12C10, (C5×C4.Q8)⋊24C2, C4.30(C5×C4○D4), (C2×C4).38(C5×D4), C4⋊C4.58(C2×C10), (C5×C22⋊C8)⋊27C2, (C2×C8).42(C2×C10), C2.17(C5×C8⋊C22), (C5×D4⋊C4)⋊36C2, (C2×D4).19(C2×C10), (C5×C4⋊D4).16C2, (C2×C10).658(C2×D4), (C5×C4⋊C4).381C22, (C22×C4).47(C2×C10), (C2×C4).112(C22×C10), C2.12(C5×C22.D4), SmallGroup(320,982)

Series: Derived Chief Lower central Upper central

C1C2×C4 — C5×C23.46D4
C1C2C4C2×C4C2×C20D4×C10C5×C4⋊D4 — C5×C23.46D4
C1C2C2×C4 — C5×C23.46D4
C1C2×C10C22×C20 — C5×C23.46D4

Generators and relations for C5×C23.46D4
 G = < a,b,c,d,e,f | a5=b2=c2=d2=f2=1, e4=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ebe-1=fbf=bc=cb, bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=ce3 >

Subgroups: 226 in 114 conjugacy classes, 54 normal (30 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×5], C5, C8 [×2], C2×C4 [×2], C2×C4 [×7], D4 [×4], C23, C23, C10 [×3], C10 [×3], C22⋊C4, C4⋊C4, C4⋊C4 [×2], C4⋊C4, C2×C8 [×2], C22×C4, C22×C4, C2×D4, C2×D4, C20 [×2], C20 [×4], C2×C10, C2×C10 [×2], C2×C10 [×5], C22⋊C8, D4⋊C4 [×2], C4.Q8 [×2], C2×C4⋊C4, C4⋊D4, C40 [×2], C2×C20 [×2], C2×C20 [×7], C5×D4 [×4], C22×C10, C22×C10, C23.46D4, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4 [×2], C5×C4⋊C4, C2×C40 [×2], C22×C20, C22×C20, D4×C10, D4×C10, C5×C22⋊C8, C5×D4⋊C4 [×2], C5×C4.Q8 [×2], C10×C4⋊C4, C5×C4⋊D4, C5×C23.46D4
Quotients: C1, C2 [×7], C22 [×7], C5, D4 [×2], C23, C10 [×7], SD16 [×2], C2×D4, C4○D4 [×2], C2×C10 [×7], C22.D4, C2×SD16, C8⋊C22, C5×D4 [×2], C22×C10, C23.46D4, C5×SD16 [×2], D4×C10, C5×C4○D4 [×2], C5×C22.D4, C10×SD16, C5×C8⋊C22, C5×C23.46D4

Smallest permutation representation of C5×C23.46D4
On 160 points
Generators in S160
(1 62 20 54 12)(2 63 21 55 13)(3 64 22 56 14)(4 57 23 49 15)(5 58 24 50 16)(6 59 17 51 9)(7 60 18 52 10)(8 61 19 53 11)(25 85 75 33 67)(26 86 76 34 68)(27 87 77 35 69)(28 88 78 36 70)(29 81 79 37 71)(30 82 80 38 72)(31 83 73 39 65)(32 84 74 40 66)(41 155 117 147 109)(42 156 118 148 110)(43 157 119 149 111)(44 158 120 150 112)(45 159 113 151 105)(46 160 114 152 106)(47 153 115 145 107)(48 154 116 146 108)(89 124 138 100 130)(90 125 139 101 131)(91 126 140 102 132)(92 127 141 103 133)(93 128 142 104 134)(94 121 143 97 135)(95 122 144 98 136)(96 123 137 99 129)
(1 111)(2 132)(3 105)(4 134)(5 107)(6 136)(7 109)(8 130)(9 98)(10 147)(11 100)(12 149)(13 102)(14 151)(15 104)(16 145)(17 122)(18 155)(19 124)(20 157)(21 126)(22 159)(23 128)(24 153)(25 129)(26 110)(27 131)(28 112)(29 133)(30 106)(31 135)(32 108)(33 137)(34 118)(35 139)(36 120)(37 141)(38 114)(39 143)(40 116)(41 60)(42 86)(43 62)(44 88)(45 64)(46 82)(47 58)(48 84)(49 142)(50 115)(51 144)(52 117)(53 138)(54 119)(55 140)(56 113)(57 93)(59 95)(61 89)(63 91)(65 97)(66 146)(67 99)(68 148)(69 101)(70 150)(71 103)(72 152)(73 121)(74 154)(75 123)(76 156)(77 125)(78 158)(79 127)(80 160)(81 92)(83 94)(85 96)(87 90)
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 25)(8 26)(9 66)(10 67)(11 68)(12 69)(13 70)(14 71)(15 72)(16 65)(17 74)(18 75)(19 76)(20 77)(21 78)(22 79)(23 80)(24 73)(33 52)(34 53)(35 54)(36 55)(37 56)(38 49)(39 50)(40 51)(41 96)(42 89)(43 90)(44 91)(45 92)(46 93)(47 94)(48 95)(57 82)(58 83)(59 84)(60 85)(61 86)(62 87)(63 88)(64 81)(97 145)(98 146)(99 147)(100 148)(101 149)(102 150)(103 151)(104 152)(105 133)(106 134)(107 135)(108 136)(109 129)(110 130)(111 131)(112 132)(113 141)(114 142)(115 143)(116 144)(117 137)(118 138)(119 139)(120 140)(121 153)(122 154)(123 155)(124 156)(125 157)(126 158)(127 159)(128 160)
(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 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)
(2 30)(3 7)(4 28)(6 26)(8 32)(9 68)(10 14)(11 66)(13 72)(15 70)(17 76)(18 22)(19 74)(21 80)(23 78)(25 29)(33 37)(34 51)(36 49)(38 55)(40 53)(41 92)(42 48)(43 90)(44 46)(45 96)(47 94)(52 56)(57 88)(59 86)(60 64)(61 84)(63 82)(67 71)(75 79)(81 85)(89 95)(91 93)(97 145)(98 100)(99 151)(101 149)(102 104)(103 147)(105 129)(106 112)(107 135)(108 110)(109 133)(111 131)(113 137)(114 120)(115 143)(116 118)(117 141)(119 139)(121 153)(122 124)(123 159)(125 157)(126 128)(127 155)(130 136)(132 134)(138 144)(140 142)(146 148)(150 152)(154 156)(158 160)

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

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

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

95 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C···4G4H5A5B5C5D8A8B8C8D10A···10L10M···10T10U10V10W10X20A···20H20I···20AB20AC20AD20AE20AF40A···40P
order1222222444···445555888810···1010···101010101020···2020···202020202040···40
size1111228224···48111144441···12···288882···24···488884···4

95 irreducible representations

dim1111111111112222222244
type+++++++++
imageC1C2C2C2C2C2C5C10C10C10C10C10D4D4C4○D4SD16C5×D4C5×D4C5×C4○D4C5×SD16C8⋊C22C5×C8⋊C22
kernelC5×C23.46D4C5×C22⋊C8C5×D4⋊C4C5×C4.Q8C10×C4⋊C4C5×C4⋊D4C23.46D4C22⋊C8D4⋊C4C4.Q8C2×C4⋊C4C4⋊D4C2×C20C22×C10C20C2×C10C2×C4C23C4C22C10C2
# reps112211448844114444161614

Matrix representation of C5×C23.46D4 in GL4(𝔽41) generated by

16000
01600
00160
00016
,
0100
1000
00400
00040
,
40000
04000
0010
0001
,
1000
0100
00400
00040
,
32000
0900
001526
001515
,
1000
04000
0010
00040
G:=sub<GL(4,GF(41))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[0,1,0,0,1,0,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[32,0,0,0,0,9,0,0,0,0,15,15,0,0,26,15],[1,0,0,0,0,40,0,0,0,0,1,0,0,0,0,40] >;

C5×C23.46D4 in GAP, Magma, Sage, TeX

C_5\times C_2^3._{46}D_4
% in TeX

G:=Group("C5xC2^3.46D4");
// GroupNames label

G:=SmallGroup(320,982);
// by ID

G=gap.SmallGroup(320,982);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,589,1766,1066,10085,2539,124]);
// Polycyclic

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

׿
×
𝔽