Copied to
clipboard

G = C5×C23.19D4order 320 = 26·5

Direct product of C5 and C23.19D4

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

Aliases: C5×C23.19D4, C2.D87C10, C22⋊C87C10, C4.Q810C10, C4⋊D4.7C10, (C2×C20).461D4, D4⋊C413C10, C23.18(C5×D4), C42⋊C25C10, (C22×C10).36D4, C10.128(C4○D8), C20.319(C4○D4), (C2×C20).938C23, (C2×C40).306C22, C22.103(D4×C10), C10.143(C8⋊C22), (D4×C10).197C22, (C22×C20).430C22, C10.97(C22.D4), (C5×C4.Q8)⋊25C2, (C5×C2.D8)⋊22C2, C2.15(C5×C4○D8), C4.31(C5×C4○D4), C4⋊C4.59(C2×C10), (C5×C22⋊C8)⋊24C2, (C2×C8).43(C2×C10), (C2×C4).107(C5×D4), C2.18(C5×C8⋊C22), (C5×D4⋊C4)⋊31C2, (C2×D4).20(C2×C10), (C2×C10).659(C2×D4), (C5×C4⋊D4).17C2, (C5×C42⋊C2)⋊26C2, (C5×C4⋊C4).382C22, (C22×C4).48(C2×C10), (C2×C4).113(C22×C10), C2.13(C5×C22.D4), SmallGroup(320,983)

Series: Derived Chief Lower central Upper central

C1C2×C4 — C5×C23.19D4
C1C2C4C2×C4C2×C20D4×C10C5×C4⋊D4 — C5×C23.19D4
C1C2C2×C4 — C5×C23.19D4
C1C2×C10C22×C20 — C5×C23.19D4

Generators and relations for C5×C23.19D4
 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=bc=cb, bd=db, fbf=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=cde3 >

Subgroups: 210 in 106 conjugacy classes, 50 normal (all characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C5, C8 [×2], C2×C4 [×2], C2×C4 [×5], D4 [×4], C23, C23, C10 [×3], C10 [×2], C42, C22⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×2], C22×C4, C2×D4, C2×D4, C20 [×2], C20 [×4], C2×C10, C2×C10 [×6], C22⋊C8, D4⋊C4 [×2], C4.Q8, C2.D8, C42⋊C2, C4⋊D4, C40 [×2], C2×C20 [×2], C2×C20 [×5], C5×D4 [×4], C22×C10, C22×C10, C23.19D4, C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4 [×3], C2×C40 [×2], C22×C20, D4×C10, D4×C10, C5×C22⋊C8, C5×D4⋊C4 [×2], C5×C4.Q8, C5×C2.D8, C5×C42⋊C2, C5×C4⋊D4, C5×C23.19D4
Quotients: C1, C2 [×7], C22 [×7], C5, D4 [×2], C23, C10 [×7], C2×D4, C4○D4 [×2], C2×C10 [×7], C22.D4, C4○D8, C8⋊C22, C5×D4 [×2], C22×C10, C23.19D4, D4×C10, C5×C4○D4 [×2], C5×C22.D4, C5×C4○D8, C5×C8⋊C22, C5×C23.19D4

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

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

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

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

95 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I5A5B5C5D8A8B8C8D10A···10L10M10N10O10P10Q10R10S10T20A···20P20Q···20AF20AG20AH20AI20AJ40A···40P
order1222224444444445555888810···10101010101010101020···2020···202020202040···40
size111148222244448111144441···1444488882···24···488884···4

95 irreducible representations

dim111111111111112222222244
type++++++++++
imageC1C2C2C2C2C2C2C5C10C10C10C10C10C10D4D4C4○D4C4○D8C5×D4C5×D4C5×C4○D4C5×C4○D8C8⋊C22C5×C8⋊C22
kernelC5×C23.19D4C5×C22⋊C8C5×D4⋊C4C5×C4.Q8C5×C2.D8C5×C42⋊C2C5×C4⋊D4C23.19D4C22⋊C8D4⋊C4C4.Q8C2.D8C42⋊C2C4⋊D4C2×C20C22×C10C20C10C2×C4C23C4C2C10C2
# reps11211114484444114444161614

Matrix representation of C5×C23.19D4 in GL4(𝔽41) generated by

10000
01000
00180
00018
,
0900
32000
00018
00160
,
1000
0100
00400
00040
,
40000
04000
0010
0001
,
122900
121200
0002
00210
,
1000
04000
0010
00040
G:=sub<GL(4,GF(41))| [10,0,0,0,0,10,0,0,0,0,18,0,0,0,0,18],[0,32,0,0,9,0,0,0,0,0,0,16,0,0,18,0],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[12,12,0,0,29,12,0,0,0,0,0,21,0,0,2,0],[1,0,0,0,0,40,0,0,0,0,1,0,0,0,0,40] >;

C5×C23.19D4 in GAP, Magma, Sage, TeX

C_5\times C_2^3._{19}D_4
% in TeX

G:=Group("C5xC2^3.19D4");
// GroupNames label

G:=SmallGroup(320,983);
// by ID

G=gap.SmallGroup(320,983);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,589,1128,1766,226,10085,2539,124]);
// 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=b*c=c*b,b*d=d*b,f*b*f=b*c*d,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=c*d*e^3>;
// generators/relations

׿
×
𝔽