direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×C23.25D4, (C2×C8)⋊6C20, (C2×C40)⋊26C4, C4.3(Q8×C10), C8.16(C2×C20), C4.Q8⋊13C10, C2.D8⋊13C10, C20.92(C2×Q8), (C2×C20).78Q8, C40.125(C2×C4), (C2×C20).537D4, C20.101(C4⋊C4), C23.24(C5×D4), (C22×C8).11C10, (C22×C40).29C2, C4.26(C22×C20), C22.49(D4×C10), C10.115(C4○D8), C42⋊C2.6C10, (C2×C20).900C23, (C2×C40).436C22, C20.243(C22×C4), (C22×C10).128D4, (C22×C20).589C22, C4.21(C5×C4⋊C4), C2.2(C5×C4○D8), C2.13(C10×C4⋊C4), C10.92(C2×C4⋊C4), (C5×C2.D8)⋊28C2, (C5×C4.Q8)⋊28C2, C22.9(C5×C4⋊C4), (C2×C4).20(C5×Q8), C4⋊C4.43(C2×C10), (C2×C4).76(C2×C20), (C2×C8).76(C2×C10), (C2×C4).147(C5×D4), (C2×C10).54(C4⋊C4), (C2×C20).510(C2×C4), (C2×C10).625(C2×D4), (C5×C4⋊C4).364C22, (C2×C4).75(C22×C10), (C22×C4).118(C2×C10), (C5×C42⋊C2).20C2, SmallGroup(320,928)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C23.25D4
G = < a,b,c,d,e,f | a5=b2=c2=d2=1, e4=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=e3 >
Subgroups: 162 in 114 conjugacy classes, 82 normal (26 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, C23, C10, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C20, C20, C20, C2×C10, C2×C10, C2×C10, C4.Q8, C2.D8, C42⋊C2, C22×C8, C40, C2×C20, C2×C20, C2×C20, C22×C10, C23.25D4, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×C40, C2×C40, C22×C20, C5×C4.Q8, C5×C2.D8, C5×C42⋊C2, C22×C40, C5×C23.25D4
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, Q8, C23, C10, C4⋊C4, C22×C4, C2×D4, C2×Q8, C20, C2×C10, C2×C4⋊C4, C4○D8, C2×C20, C5×D4, C5×Q8, C22×C10, C23.25D4, C5×C4⋊C4, C22×C20, D4×C10, Q8×C10, C10×C4⋊C4, C5×C4○D8, C5×C23.25D4
►On 160 points
Generators in S160
(1 23 54 11 63)(2 24 55 12 64)(3 17 56 13 57)(4 18 49 14 58)(5 19 50 15 59)(6 20 51 16 60)(7 21 52 9 61)(8 22 53 10 62)(25 85 73 33 65)(26 86 74 34 66)(27 87 75 35 67)(28 88 76 36 68)(29 81 77 37 69)(30 82 78 38 70)(31 83 79 39 71)(32 84 80 40 72)(41 155 119 147 111)(42 156 120 148 112)(43 157 113 149 105)(44 158 114 150 106)(45 159 115 151 107)(46 160 116 152 108)(47 153 117 145 109)(48 154 118 146 110)(89 122 141 97 133)(90 123 142 98 134)(91 124 143 99 135)(92 125 144 100 136)(93 126 137 101 129)(94 127 138 102 130)(95 128 139 103 131)(96 121 140 104 132)
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 37)(10 38)(11 39)(12 40)(13 33)(14 34)(15 35)(16 36)(17 85)(18 86)(19 87)(20 88)(21 81)(22 82)(23 83)(24 84)(41 126)(42 127)(43 128)(44 121)(45 122)(46 123)(47 124)(48 125)(49 74)(50 75)(51 76)(52 77)(53 78)(54 79)(55 80)(56 73)(57 65)(58 66)(59 67)(60 68)(61 69)(62 70)(63 71)(64 72)(89 107)(90 108)(91 109)(92 110)(93 111)(94 112)(95 105)(96 106)(97 115)(98 116)(99 117)(100 118)(101 119)(102 120)(103 113)(104 114)(129 147)(130 148)(131 149)(132 150)(133 151)(134 152)(135 145)(136 146)(137 155)(138 156)(139 157)(140 158)(141 159)(142 160)(143 153)(144 154)
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 37)(10 38)(11 39)(12 40)(13 33)(14 34)(15 35)(16 36)(17 85)(18 86)(19 87)(20 88)(21 81)(22 82)(23 83)(24 84)(41 122)(42 123)(43 124)(44 125)(45 126)(46 127)(47 128)(48 121)(49 74)(50 75)(51 76)(52 77)(53 78)(54 79)(55 80)(56 73)(57 65)(58 66)(59 67)(60 68)(61 69)(62 70)(63 71)(64 72)(89 111)(90 112)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 119)(98 120)(99 113)(100 114)(101 115)(102 116)(103 117)(104 118)(129 151)(130 152)(131 145)(132 146)(133 147)(134 148)(135 149)(136 150)(137 159)(138 160)(139 153)(140 154)(141 155)(142 156)(143 157)(144 158)
(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)
(1 108 27 90)(2 111 28 93)(3 106 29 96)(4 109 30 91)(5 112 31 94)(6 107 32 89)(7 110 25 92)(8 105 26 95)(9 118 33 100)(10 113 34 103)(11 116 35 98)(12 119 36 101)(13 114 37 104)(14 117 38 99)(15 120 39 102)(16 115 40 97)(17 44 81 121)(18 47 82 124)(19 42 83 127)(20 45 84 122)(21 48 85 125)(22 43 86 128)(23 46 87 123)(24 41 88 126)(49 153 78 143)(50 156 79 138)(51 159 80 141)(52 154 73 144)(53 157 74 139)(54 160 75 142)(55 155 76 137)(56 158 77 140)(57 150 69 132)(58 145 70 135)(59 148 71 130)(60 151 72 133)(61 146 65 136)(62 149 66 131)(63 152 67 134)(64 147 68 129)
G:=sub<Sym(160)| (1,23,54,11,63)(2,24,55,12,64)(3,17,56,13,57)(4,18,49,14,58)(5,19,50,15,59)(6,20,51,16,60)(7,21,52,9,61)(8,22,53,10,62)(25,85,73,33,65)(26,86,74,34,66)(27,87,75,35,67)(28,88,76,36,68)(29,81,77,37,69)(30,82,78,38,70)(31,83,79,39,71)(32,84,80,40,72)(41,155,119,147,111)(42,156,120,148,112)(43,157,113,149,105)(44,158,114,150,106)(45,159,115,151,107)(46,160,116,152,108)(47,153,117,145,109)(48,154,118,146,110)(89,122,141,97,133)(90,123,142,98,134)(91,124,143,99,135)(92,125,144,100,136)(93,126,137,101,129)(94,127,138,102,130)(95,128,139,103,131)(96,121,140,104,132), (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,37)(10,38)(11,39)(12,40)(13,33)(14,34)(15,35)(16,36)(17,85)(18,86)(19,87)(20,88)(21,81)(22,82)(23,83)(24,84)(41,126)(42,127)(43,128)(44,121)(45,122)(46,123)(47,124)(48,125)(49,74)(50,75)(51,76)(52,77)(53,78)(54,79)(55,80)(56,73)(57,65)(58,66)(59,67)(60,68)(61,69)(62,70)(63,71)(64,72)(89,107)(90,108)(91,109)(92,110)(93,111)(94,112)(95,105)(96,106)(97,115)(98,116)(99,117)(100,118)(101,119)(102,120)(103,113)(104,114)(129,147)(130,148)(131,149)(132,150)(133,151)(134,152)(135,145)(136,146)(137,155)(138,156)(139,157)(140,158)(141,159)(142,160)(143,153)(144,154), (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,37)(10,38)(11,39)(12,40)(13,33)(14,34)(15,35)(16,36)(17,85)(18,86)(19,87)(20,88)(21,81)(22,82)(23,83)(24,84)(41,122)(42,123)(43,124)(44,125)(45,126)(46,127)(47,128)(48,121)(49,74)(50,75)(51,76)(52,77)(53,78)(54,79)(55,80)(56,73)(57,65)(58,66)(59,67)(60,68)(61,69)(62,70)(63,71)(64,72)(89,111)(90,112)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,119)(98,120)(99,113)(100,114)(101,115)(102,116)(103,117)(104,118)(129,151)(130,152)(131,145)(132,146)(133,147)(134,148)(135,149)(136,150)(137,159)(138,160)(139,153)(140,154)(141,155)(142,156)(143,157)(144,158), (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), (1,108,27,90)(2,111,28,93)(3,106,29,96)(4,109,30,91)(5,112,31,94)(6,107,32,89)(7,110,25,92)(8,105,26,95)(9,118,33,100)(10,113,34,103)(11,116,35,98)(12,119,36,101)(13,114,37,104)(14,117,38,99)(15,120,39,102)(16,115,40,97)(17,44,81,121)(18,47,82,124)(19,42,83,127)(20,45,84,122)(21,48,85,125)(22,43,86,128)(23,46,87,123)(24,41,88,126)(49,153,78,143)(50,156,79,138)(51,159,80,141)(52,154,73,144)(53,157,74,139)(54,160,75,142)(55,155,76,137)(56,158,77,140)(57,150,69,132)(58,145,70,135)(59,148,71,130)(60,151,72,133)(61,146,65,136)(62,149,66,131)(63,152,67,134)(64,147,68,129)>;
G:=Group( (1,23,54,11,63)(2,24,55,12,64)(3,17,56,13,57)(4,18,49,14,58)(5,19,50,15,59)(6,20,51,16,60)(7,21,52,9,61)(8,22,53,10,62)(25,85,73,33,65)(26,86,74,34,66)(27,87,75,35,67)(28,88,76,36,68)(29,81,77,37,69)(30,82,78,38,70)(31,83,79,39,71)(32,84,80,40,72)(41,155,119,147,111)(42,156,120,148,112)(43,157,113,149,105)(44,158,114,150,106)(45,159,115,151,107)(46,160,116,152,108)(47,153,117,145,109)(48,154,118,146,110)(89,122,141,97,133)(90,123,142,98,134)(91,124,143,99,135)(92,125,144,100,136)(93,126,137,101,129)(94,127,138,102,130)(95,128,139,103,131)(96,121,140,104,132), (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,37)(10,38)(11,39)(12,40)(13,33)(14,34)(15,35)(16,36)(17,85)(18,86)(19,87)(20,88)(21,81)(22,82)(23,83)(24,84)(41,126)(42,127)(43,128)(44,121)(45,122)(46,123)(47,124)(48,125)(49,74)(50,75)(51,76)(52,77)(53,78)(54,79)(55,80)(56,73)(57,65)(58,66)(59,67)(60,68)(61,69)(62,70)(63,71)(64,72)(89,107)(90,108)(91,109)(92,110)(93,111)(94,112)(95,105)(96,106)(97,115)(98,116)(99,117)(100,118)(101,119)(102,120)(103,113)(104,114)(129,147)(130,148)(131,149)(132,150)(133,151)(134,152)(135,145)(136,146)(137,155)(138,156)(139,157)(140,158)(141,159)(142,160)(143,153)(144,154), (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,37)(10,38)(11,39)(12,40)(13,33)(14,34)(15,35)(16,36)(17,85)(18,86)(19,87)(20,88)(21,81)(22,82)(23,83)(24,84)(41,122)(42,123)(43,124)(44,125)(45,126)(46,127)(47,128)(48,121)(49,74)(50,75)(51,76)(52,77)(53,78)(54,79)(55,80)(56,73)(57,65)(58,66)(59,67)(60,68)(61,69)(62,70)(63,71)(64,72)(89,111)(90,112)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,119)(98,120)(99,113)(100,114)(101,115)(102,116)(103,117)(104,118)(129,151)(130,152)(131,145)(132,146)(133,147)(134,148)(135,149)(136,150)(137,159)(138,160)(139,153)(140,154)(141,155)(142,156)(143,157)(144,158), (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), (1,108,27,90)(2,111,28,93)(3,106,29,96)(4,109,30,91)(5,112,31,94)(6,107,32,89)(7,110,25,92)(8,105,26,95)(9,118,33,100)(10,113,34,103)(11,116,35,98)(12,119,36,101)(13,114,37,104)(14,117,38,99)(15,120,39,102)(16,115,40,97)(17,44,81,121)(18,47,82,124)(19,42,83,127)(20,45,84,122)(21,48,85,125)(22,43,86,128)(23,46,87,123)(24,41,88,126)(49,153,78,143)(50,156,79,138)(51,159,80,141)(52,154,73,144)(53,157,74,139)(54,160,75,142)(55,155,76,137)(56,158,77,140)(57,150,69,132)(58,145,70,135)(59,148,71,130)(60,151,72,133)(61,146,65,136)(62,149,66,131)(63,152,67,134)(64,147,68,129) );
G=PermutationGroup([[(1,23,54,11,63),(2,24,55,12,64),(3,17,56,13,57),(4,18,49,14,58),(5,19,50,15,59),(6,20,51,16,60),(7,21,52,9,61),(8,22,53,10,62),(25,85,73,33,65),(26,86,74,34,66),(27,87,75,35,67),(28,88,76,36,68),(29,81,77,37,69),(30,82,78,38,70),(31,83,79,39,71),(32,84,80,40,72),(41,155,119,147,111),(42,156,120,148,112),(43,157,113,149,105),(44,158,114,150,106),(45,159,115,151,107),(46,160,116,152,108),(47,153,117,145,109),(48,154,118,146,110),(89,122,141,97,133),(90,123,142,98,134),(91,124,143,99,135),(92,125,144,100,136),(93,126,137,101,129),(94,127,138,102,130),(95,128,139,103,131),(96,121,140,104,132)], [(1,31),(2,32),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30),(9,37),(10,38),(11,39),(12,40),(13,33),(14,34),(15,35),(16,36),(17,85),(18,86),(19,87),(20,88),(21,81),(22,82),(23,83),(24,84),(41,126),(42,127),(43,128),(44,121),(45,122),(46,123),(47,124),(48,125),(49,74),(50,75),(51,76),(52,77),(53,78),(54,79),(55,80),(56,73),(57,65),(58,66),(59,67),(60,68),(61,69),(62,70),(63,71),(64,72),(89,107),(90,108),(91,109),(92,110),(93,111),(94,112),(95,105),(96,106),(97,115),(98,116),(99,117),(100,118),(101,119),(102,120),(103,113),(104,114),(129,147),(130,148),(131,149),(132,150),(133,151),(134,152),(135,145),(136,146),(137,155),(138,156),(139,157),(140,158),(141,159),(142,160),(143,153),(144,154)], [(1,31),(2,32),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30),(9,37),(10,38),(11,39),(12,40),(13,33),(14,34),(15,35),(16,36),(17,85),(18,86),(19,87),(20,88),(21,81),(22,82),(23,83),(24,84),(41,122),(42,123),(43,124),(44,125),(45,126),(46,127),(47,128),(48,121),(49,74),(50,75),(51,76),(52,77),(53,78),(54,79),(55,80),(56,73),(57,65),(58,66),(59,67),(60,68),(61,69),(62,70),(63,71),(64,72),(89,111),(90,112),(91,105),(92,106),(93,107),(94,108),(95,109),(96,110),(97,119),(98,120),(99,113),(100,114),(101,115),(102,116),(103,117),(104,118),(129,151),(130,152),(131,145),(132,146),(133,147),(134,148),(135,149),(136,150),(137,159),(138,160),(139,153),(140,154),(141,155),(142,156),(143,157),(144,158)], [(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)], [(1,108,27,90),(2,111,28,93),(3,106,29,96),(4,109,30,91),(5,112,31,94),(6,107,32,89),(7,110,25,92),(8,105,26,95),(9,118,33,100),(10,113,34,103),(11,116,35,98),(12,119,36,101),(13,114,37,104),(14,117,38,99),(15,120,39,102),(16,115,40,97),(17,44,81,121),(18,47,82,124),(19,42,83,127),(20,45,84,122),(21,48,85,125),(22,43,86,128),(23,46,87,123),(24,41,88,126),(49,153,78,143),(50,156,79,138),(51,159,80,141),(52,154,73,144),(53,157,74,139),(54,160,75,142),(55,155,76,137),(56,158,77,140),(57,150,69,132),(58,145,70,135),(59,148,71,130),(60,151,72,133),(61,146,65,136),(62,149,66,131),(63,152,67,134),(64,147,68,129)]])
140 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 10A | ··· | 10L | 10M | ··· | 10T | 20A | ··· | 20P | 20Q | ··· | 20X | 20Y | ··· | 20BD | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C10 | C20 | D4 | Q8 | D4 | C4○D8 | C5×D4 | C5×Q8 | C5×D4 | C5×C4○D8 |
| kernel | C5×C23.25D4 | C5×C4.Q8 | C5×C2.D8 | C5×C42⋊C2 | C22×C40 | C2×C40 | C23.25D4 | C4.Q8 | C2.D8 | C42⋊C2 | C22×C8 | C2×C8 | C2×C20 | C2×C20 | C22×C10 | C10 | C2×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 8 | 4 | 8 | 8 | 8 | 4 | 32 | 1 | 2 | 1 | 8 | 4 | 8 | 4 | 32 |
Matrix representation of C5×C23.25D4 ►in GL4(𝔽41) generated by
| 10 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 37 | 0 |
| 0 | 0 | 0 | 37 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 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 | 1 | 0 | 0 |
| 39 | 40 | 0 | 0 |
| 0 | 0 | 38 | 0 |
| 0 | 0 | 0 | 14 |
| 9 | 9 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 14 |
| 0 | 0 | 3 | 0 |
G:=sub<GL(4,GF(41))| [10,0,0,0,0,10,0,0,0,0,37,0,0,0,0,37],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[1,39,0,0,1,40,0,0,0,0,38,0,0,0,0,14],[9,0,0,0,9,32,0,0,0,0,0,3,0,0,14,0] >;
C5×C23.25D4 in GAP, Magma, Sage, TeX
C_5\times C_2^3._{25}D_4 % in TeX
G:=Group("C5xC2^3.25D4"); // GroupNames label
G:=SmallGroup(320,928);
// by ID
G=gap.SmallGroup(320,928);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,288,856,7004,172]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^5=b^2=c^2=d^2=1,e^4=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^3>;
// generators/relations