direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×C23.20D4, C2.D8⋊9C10, C4.Q8⋊12C10, Q8⋊C4⋊8C10, (C2×C20).462D4, C22⋊C8.5C10, C23.19(C5×D4), C22⋊Q8.7C10, (C22×C10).37D4, C10.129(C4○D8), C20.322(C4○D4), (C2×C20).941C23, (C2×C40).264C22, C42⋊C2.9C10, C22.106(D4×C10), (Q8×C10).171C22, C10.144(C8.C22), (C22×C20).433C22, C10.100(C22.D4), (C5×C4.Q8)⋊27C2, (C5×C2.D8)⋊24C2, C2.16(C5×C4○D8), C4.34(C5×C4○D4), C4⋊C4.62(C2×C10), (C2×C8).11(C2×C10), (C2×C4).108(C5×D4), (C5×Q8⋊C4)⋊36C2, (C2×C10).662(C2×D4), (C5×C22⋊C8).14C2, (C2×Q8).15(C2×C10), C2.19(C5×C8.C22), (C5×C22⋊Q8).17C2, (C5×C4⋊C4).385C22, (C22×C4).51(C2×C10), (C5×C42⋊C2).23C2, (C2×C4).116(C22×C10), C2.16(C5×C22.D4), SmallGroup(320,986)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C23.20D4
G = < a,b,c,d,e,f | a5=b2=c2=d2=1, e4=f2=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ebe-1=bc=cb, bd=db, fbf-1=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce3 >
Subgroups: 162 in 96 conjugacy classes, 50 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, Q8, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C2×Q8, C20, C20, C2×C10, C2×C10, C22⋊C8, Q8⋊C4, C4.Q8, C2.D8, C42⋊C2, C22⋊Q8, C40, C2×C20, C2×C20, C5×Q8, C22×C10, C23.20D4, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C2×C40, C22×C20, Q8×C10, C5×C22⋊C8, C5×Q8⋊C4, C5×C4.Q8, C5×C2.D8, C5×C42⋊C2, C5×C22⋊Q8, C5×C23.20D4
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C4○D4, C2×C10, C22.D4, C4○D8, C8.C22, C5×D4, C22×C10, C23.20D4, D4×C10, C5×C4○D4, C5×C22.D4, C5×C4○D8, C5×C8.C22, C5×C23.20D4
►On 160 points
Generators in S160
(1 127 159 39 151)(2 128 160 40 152)(3 121 153 33 145)(4 122 154 34 146)(5 123 155 35 147)(6 124 156 36 148)(7 125 157 37 149)(8 126 158 38 150)(9 113 137 17 129)(10 114 138 18 130)(11 115 139 19 131)(12 116 140 20 132)(13 117 141 21 133)(14 118 142 22 134)(15 119 143 23 135)(16 120 144 24 136)(25 89 53 81 41)(26 90 54 82 42)(27 91 55 83 43)(28 92 56 84 44)(29 93 49 85 45)(30 94 50 86 46)(31 95 51 87 47)(32 96 52 88 48)(57 73 109 65 101)(58 74 110 66 102)(59 75 111 67 103)(60 76 112 68 104)(61 77 105 69 97)(62 78 106 70 98)(63 79 107 71 99)(64 80 108 72 100)
(2 88)(4 82)(6 84)(8 86)(9 13)(10 101)(11 15)(12 103)(14 97)(16 99)(17 21)(18 109)(19 23)(20 111)(22 105)(24 107)(26 154)(28 156)(30 158)(32 160)(34 90)(36 92)(38 94)(40 96)(42 122)(44 124)(46 126)(48 128)(50 150)(52 152)(54 146)(56 148)(57 114)(58 62)(59 116)(60 64)(61 118)(63 120)(65 130)(66 70)(67 132)(68 72)(69 134)(71 136)(73 138)(74 78)(75 140)(76 80)(77 142)(79 144)(98 102)(100 104)(106 110)(108 112)(113 117)(115 119)(129 133)(131 135)(137 141)(139 143)
(1 87)(2 88)(3 81)(4 82)(5 83)(6 84)(7 85)(8 86)(9 104)(10 97)(11 98)(12 99)(13 100)(14 101)(15 102)(16 103)(17 112)(18 105)(19 106)(20 107)(21 108)(22 109)(23 110)(24 111)(25 153)(26 154)(27 155)(28 156)(29 157)(30 158)(31 159)(32 160)(33 89)(34 90)(35 91)(36 92)(37 93)(38 94)(39 95)(40 96)(41 121)(42 122)(43 123)(44 124)(45 125)(46 126)(47 127)(48 128)(49 149)(50 150)(51 151)(52 152)(53 145)(54 146)(55 147)(56 148)(57 118)(58 119)(59 120)(60 113)(61 114)(62 115)(63 116)(64 117)(65 134)(66 135)(67 136)(68 129)(69 130)(70 131)(71 132)(72 133)(73 142)(74 143)(75 144)(76 137)(77 138)(78 139)(79 140)(80 141)
(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)
(1 118 5 114)(2 60 6 64)(3 116 7 120)(4 58 8 62)(9 56 13 52)(10 151 14 147)(11 54 15 50)(12 149 16 145)(17 28 21 32)(18 159 22 155)(19 26 23 30)(20 157 24 153)(25 107 29 111)(27 105 31 109)(33 132 37 136)(34 66 38 70)(35 130 39 134)(36 72 40 68)(41 79 45 75)(42 143 46 139)(43 77 47 73)(44 141 48 137)(49 103 53 99)(51 101 55 97)(57 83 61 87)(59 81 63 85)(65 91 69 95)(67 89 71 93)(74 126 78 122)(76 124 80 128)(82 119 86 115)(84 117 88 113)(90 135 94 131)(92 133 96 129)(98 146 102 150)(100 152 104 148)(106 154 110 158)(108 160 112 156)(121 140 125 144)(123 138 127 142)
G:=sub<Sym(160)| (1,127,159,39,151)(2,128,160,40,152)(3,121,153,33,145)(4,122,154,34,146)(5,123,155,35,147)(6,124,156,36,148)(7,125,157,37,149)(8,126,158,38,150)(9,113,137,17,129)(10,114,138,18,130)(11,115,139,19,131)(12,116,140,20,132)(13,117,141,21,133)(14,118,142,22,134)(15,119,143,23,135)(16,120,144,24,136)(25,89,53,81,41)(26,90,54,82,42)(27,91,55,83,43)(28,92,56,84,44)(29,93,49,85,45)(30,94,50,86,46)(31,95,51,87,47)(32,96,52,88,48)(57,73,109,65,101)(58,74,110,66,102)(59,75,111,67,103)(60,76,112,68,104)(61,77,105,69,97)(62,78,106,70,98)(63,79,107,71,99)(64,80,108,72,100), (2,88)(4,82)(6,84)(8,86)(9,13)(10,101)(11,15)(12,103)(14,97)(16,99)(17,21)(18,109)(19,23)(20,111)(22,105)(24,107)(26,154)(28,156)(30,158)(32,160)(34,90)(36,92)(38,94)(40,96)(42,122)(44,124)(46,126)(48,128)(50,150)(52,152)(54,146)(56,148)(57,114)(58,62)(59,116)(60,64)(61,118)(63,120)(65,130)(66,70)(67,132)(68,72)(69,134)(71,136)(73,138)(74,78)(75,140)(76,80)(77,142)(79,144)(98,102)(100,104)(106,110)(108,112)(113,117)(115,119)(129,133)(131,135)(137,141)(139,143), (1,87)(2,88)(3,81)(4,82)(5,83)(6,84)(7,85)(8,86)(9,104)(10,97)(11,98)(12,99)(13,100)(14,101)(15,102)(16,103)(17,112)(18,105)(19,106)(20,107)(21,108)(22,109)(23,110)(24,111)(25,153)(26,154)(27,155)(28,156)(29,157)(30,158)(31,159)(32,160)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,121)(42,122)(43,123)(44,124)(45,125)(46,126)(47,127)(48,128)(49,149)(50,150)(51,151)(52,152)(53,145)(54,146)(55,147)(56,148)(57,118)(58,119)(59,120)(60,113)(61,114)(62,115)(63,116)(64,117)(65,134)(66,135)(67,136)(68,129)(69,130)(70,131)(71,132)(72,133)(73,142)(74,143)(75,144)(76,137)(77,138)(78,139)(79,140)(80,141), (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), (1,118,5,114)(2,60,6,64)(3,116,7,120)(4,58,8,62)(9,56,13,52)(10,151,14,147)(11,54,15,50)(12,149,16,145)(17,28,21,32)(18,159,22,155)(19,26,23,30)(20,157,24,153)(25,107,29,111)(27,105,31,109)(33,132,37,136)(34,66,38,70)(35,130,39,134)(36,72,40,68)(41,79,45,75)(42,143,46,139)(43,77,47,73)(44,141,48,137)(49,103,53,99)(51,101,55,97)(57,83,61,87)(59,81,63,85)(65,91,69,95)(67,89,71,93)(74,126,78,122)(76,124,80,128)(82,119,86,115)(84,117,88,113)(90,135,94,131)(92,133,96,129)(98,146,102,150)(100,152,104,148)(106,154,110,158)(108,160,112,156)(121,140,125,144)(123,138,127,142)>;
G:=Group( (1,127,159,39,151)(2,128,160,40,152)(3,121,153,33,145)(4,122,154,34,146)(5,123,155,35,147)(6,124,156,36,148)(7,125,157,37,149)(8,126,158,38,150)(9,113,137,17,129)(10,114,138,18,130)(11,115,139,19,131)(12,116,140,20,132)(13,117,141,21,133)(14,118,142,22,134)(15,119,143,23,135)(16,120,144,24,136)(25,89,53,81,41)(26,90,54,82,42)(27,91,55,83,43)(28,92,56,84,44)(29,93,49,85,45)(30,94,50,86,46)(31,95,51,87,47)(32,96,52,88,48)(57,73,109,65,101)(58,74,110,66,102)(59,75,111,67,103)(60,76,112,68,104)(61,77,105,69,97)(62,78,106,70,98)(63,79,107,71,99)(64,80,108,72,100), (2,88)(4,82)(6,84)(8,86)(9,13)(10,101)(11,15)(12,103)(14,97)(16,99)(17,21)(18,109)(19,23)(20,111)(22,105)(24,107)(26,154)(28,156)(30,158)(32,160)(34,90)(36,92)(38,94)(40,96)(42,122)(44,124)(46,126)(48,128)(50,150)(52,152)(54,146)(56,148)(57,114)(58,62)(59,116)(60,64)(61,118)(63,120)(65,130)(66,70)(67,132)(68,72)(69,134)(71,136)(73,138)(74,78)(75,140)(76,80)(77,142)(79,144)(98,102)(100,104)(106,110)(108,112)(113,117)(115,119)(129,133)(131,135)(137,141)(139,143), (1,87)(2,88)(3,81)(4,82)(5,83)(6,84)(7,85)(8,86)(9,104)(10,97)(11,98)(12,99)(13,100)(14,101)(15,102)(16,103)(17,112)(18,105)(19,106)(20,107)(21,108)(22,109)(23,110)(24,111)(25,153)(26,154)(27,155)(28,156)(29,157)(30,158)(31,159)(32,160)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,121)(42,122)(43,123)(44,124)(45,125)(46,126)(47,127)(48,128)(49,149)(50,150)(51,151)(52,152)(53,145)(54,146)(55,147)(56,148)(57,118)(58,119)(59,120)(60,113)(61,114)(62,115)(63,116)(64,117)(65,134)(66,135)(67,136)(68,129)(69,130)(70,131)(71,132)(72,133)(73,142)(74,143)(75,144)(76,137)(77,138)(78,139)(79,140)(80,141), (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), (1,118,5,114)(2,60,6,64)(3,116,7,120)(4,58,8,62)(9,56,13,52)(10,151,14,147)(11,54,15,50)(12,149,16,145)(17,28,21,32)(18,159,22,155)(19,26,23,30)(20,157,24,153)(25,107,29,111)(27,105,31,109)(33,132,37,136)(34,66,38,70)(35,130,39,134)(36,72,40,68)(41,79,45,75)(42,143,46,139)(43,77,47,73)(44,141,48,137)(49,103,53,99)(51,101,55,97)(57,83,61,87)(59,81,63,85)(65,91,69,95)(67,89,71,93)(74,126,78,122)(76,124,80,128)(82,119,86,115)(84,117,88,113)(90,135,94,131)(92,133,96,129)(98,146,102,150)(100,152,104,148)(106,154,110,158)(108,160,112,156)(121,140,125,144)(123,138,127,142) );
G=PermutationGroup([[(1,127,159,39,151),(2,128,160,40,152),(3,121,153,33,145),(4,122,154,34,146),(5,123,155,35,147),(6,124,156,36,148),(7,125,157,37,149),(8,126,158,38,150),(9,113,137,17,129),(10,114,138,18,130),(11,115,139,19,131),(12,116,140,20,132),(13,117,141,21,133),(14,118,142,22,134),(15,119,143,23,135),(16,120,144,24,136),(25,89,53,81,41),(26,90,54,82,42),(27,91,55,83,43),(28,92,56,84,44),(29,93,49,85,45),(30,94,50,86,46),(31,95,51,87,47),(32,96,52,88,48),(57,73,109,65,101),(58,74,110,66,102),(59,75,111,67,103),(60,76,112,68,104),(61,77,105,69,97),(62,78,106,70,98),(63,79,107,71,99),(64,80,108,72,100)], [(2,88),(4,82),(6,84),(8,86),(9,13),(10,101),(11,15),(12,103),(14,97),(16,99),(17,21),(18,109),(19,23),(20,111),(22,105),(24,107),(26,154),(28,156),(30,158),(32,160),(34,90),(36,92),(38,94),(40,96),(42,122),(44,124),(46,126),(48,128),(50,150),(52,152),(54,146),(56,148),(57,114),(58,62),(59,116),(60,64),(61,118),(63,120),(65,130),(66,70),(67,132),(68,72),(69,134),(71,136),(73,138),(74,78),(75,140),(76,80),(77,142),(79,144),(98,102),(100,104),(106,110),(108,112),(113,117),(115,119),(129,133),(131,135),(137,141),(139,143)], [(1,87),(2,88),(3,81),(4,82),(5,83),(6,84),(7,85),(8,86),(9,104),(10,97),(11,98),(12,99),(13,100),(14,101),(15,102),(16,103),(17,112),(18,105),(19,106),(20,107),(21,108),(22,109),(23,110),(24,111),(25,153),(26,154),(27,155),(28,156),(29,157),(30,158),(31,159),(32,160),(33,89),(34,90),(35,91),(36,92),(37,93),(38,94),(39,95),(40,96),(41,121),(42,122),(43,123),(44,124),(45,125),(46,126),(47,127),(48,128),(49,149),(50,150),(51,151),(52,152),(53,145),(54,146),(55,147),(56,148),(57,118),(58,119),(59,120),(60,113),(61,114),(62,115),(63,116),(64,117),(65,134),(66,135),(67,136),(68,129),(69,130),(70,131),(71,132),(72,133),(73,142),(74,143),(75,144),(76,137),(77,138),(78,139),(79,140),(80,141)], [(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)], [(1,118,5,114),(2,60,6,64),(3,116,7,120),(4,58,8,62),(9,56,13,52),(10,151,14,147),(11,54,15,50),(12,149,16,145),(17,28,21,32),(18,159,22,155),(19,26,23,30),(20,157,24,153),(25,107,29,111),(27,105,31,109),(33,132,37,136),(34,66,38,70),(35,130,39,134),(36,72,40,68),(41,79,45,75),(42,143,46,139),(43,77,47,73),(44,141,48,137),(49,103,53,99),(51,101,55,97),(57,83,61,87),(59,81,63,85),(65,91,69,95),(67,89,71,93),(74,126,78,122),(76,124,80,128),(82,119,86,115),(84,117,88,113),(90,135,94,131),(92,133,96,129),(98,146,102,150),(100,152,104,148),(106,154,110,158),(108,160,112,156),(121,140,125,144),(123,138,127,142)]])
95 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 10A | ··· | 10L | 10M | 10N | 10O | 10P | 20A | ··· | 20P | 20Q | ··· | 20AF | 20AG | ··· | 20AN | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
95 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | C10 | D4 | D4 | C4○D4 | C4○D8 | C5×D4 | C5×D4 | C5×C4○D4 | C5×C4○D8 | C8.C22 | C5×C8.C22 |
| kernel | C5×C23.20D4 | C5×C22⋊C8 | C5×Q8⋊C4 | C5×C4.Q8 | C5×C2.D8 | C5×C42⋊C2 | C5×C22⋊Q8 | C23.20D4 | C22⋊C8 | Q8⋊C4 | C4.Q8 | C2.D8 | C42⋊C2 | C22⋊Q8 | C2×C20 | C22×C10 | C20 | C10 | C2×C4 | C23 | C4 | C2 | C10 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 4 | 4 | 8 | 4 | 4 | 4 | 4 | 1 | 1 | 4 | 4 | 4 | 4 | 16 | 16 | 1 | 4 |
Matrix representation of C5×C23.20D4 ►in GL4(𝔽41) generated by
| 37 | 0 | 0 | 0 |
| 0 | 37 | 0 | 0 |
| 0 | 0 | 37 | 0 |
| 0 | 0 | 0 | 37 |
| 1 | 1 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 4 | 40 |
| 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 |
| 38 | 12 | 0 | 0 |
| 0 | 14 | 0 | 0 |
| 0 | 0 | 5 | 18 |
| 0 | 0 | 35 | 36 |
| 38 | 12 | 0 | 0 |
| 6 | 3 | 0 | 0 |
| 0 | 0 | 4 | 39 |
| 0 | 0 | 28 | 37 |
G:=sub<GL(4,GF(41))| [37,0,0,0,0,37,0,0,0,0,37,0,0,0,0,37],[1,0,0,0,1,40,0,0,0,0,1,4,0,0,0,40],[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],[38,0,0,0,12,14,0,0,0,0,5,35,0,0,18,36],[38,6,0,0,12,3,0,0,0,0,4,28,0,0,39,37] >;
C5×C23.20D4 in GAP, Magma, Sage, TeX
C_5\times C_2^3._{20}D_4 % in TeX
G:=Group("C5xC2^3.20D4"); // GroupNames label
G:=SmallGroup(320,986);
// by ID
G=gap.SmallGroup(320,986);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,1120,589,1128,1766,226,10085,2539,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^5=b^2=c^2=d^2=1,e^4=f^2=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^-1=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^-1=c*e^3>;
// generators/relations