direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×C23.11D4, (C2×C20).310D4, C24.8(C2×C10), C23.10(C5×D4), C22.73(D4×C10), (C22×C10).29D4, C2.C42⋊5C10, (C23×C10).8C22, C10.140(C4⋊D4), C10.69(C4.4D4), (C22×C20).35C22, C23.80(C22×C10), C10.35(C42⋊2C2), (C22×C10).461C23, C10.91(C22.D4), (C2×C4⋊C4)⋊7C10, (C10×C4⋊C4)⋊34C2, (C2×C4).17(C5×D4), C2.9(C5×C4⋊D4), C2.7(C5×C4.4D4), (C2×C10).613(C2×D4), (C2×C22⋊C4).7C10, (C22×C4).8(C2×C10), C2.5(C5×C42⋊2C2), C22.40(C5×C4○D4), (C10×C22⋊C4).29C2, (C5×C2.C42)⋊7C2, (C2×C10).221(C4○D4), C2.7(C5×C22.D4), SmallGroup(320,898)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C23.11D4
G = < a,b,c,d,e,f | a5=b2=c2=d2=e4=1, f2=c, 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=e-1 >
Subgroups: 314 in 170 conjugacy classes, 70 normal (30 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, C23, C23, C23, C10, C10, C10, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, C20, C2×C10, C2×C10, C2×C10, C2.C42, C2.C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C20, C2×C20, C22×C10, C22×C10, C22×C10, C23.11D4, C5×C22⋊C4, C5×C4⋊C4, C22×C20, C22×C20, C23×C10, C5×C2.C42, C5×C2.C42, C10×C22⋊C4, C10×C22⋊C4, C10×C4⋊C4, C5×C23.11D4
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C4○D4, C2×C10, C4⋊D4, C22.D4, C4.4D4, C42⋊2C2, C5×D4, C22×C10, C23.11D4, D4×C10, C5×C4○D4, C5×C4⋊D4, C5×C22.D4, C5×C4.4D4, C5×C42⋊2C2, C5×C23.11D4
►On 160 points
Generators in S160
(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)
(6 19)(7 20)(8 16)(9 17)(10 18)(21 34)(22 35)(23 31)(24 32)(25 33)(56 94)(57 95)(58 91)(59 92)(60 93)(61 82)(62 83)(63 84)(64 85)(65 81)(71 90)(72 86)(73 87)(74 88)(75 89)(76 110)(77 106)(78 107)(79 108)(80 109)(96 121)(97 122)(98 123)(99 124)(100 125)(101 135)(102 131)(103 132)(104 133)(105 134)(111 149)(112 150)(113 146)(114 147)(115 148)(116 126)(117 127)(118 128)(119 129)(120 130)(136 145)(137 141)(138 142)(139 143)(140 144)(151 160)(152 156)(153 157)(154 158)(155 159)
(1 45)(2 41)(3 42)(4 43)(5 44)(6 140)(7 136)(8 137)(9 138)(10 139)(11 53)(12 54)(13 55)(14 51)(15 52)(16 141)(17 142)(18 143)(19 144)(20 145)(21 155)(22 151)(23 152)(24 153)(25 154)(26 69)(27 70)(28 66)(29 67)(30 68)(31 156)(32 157)(33 158)(34 159)(35 160)(36 46)(37 47)(38 48)(39 49)(40 50)(56 81)(57 82)(58 83)(59 84)(60 85)(61 95)(62 91)(63 92)(64 93)(65 94)(71 109)(72 110)(73 106)(74 107)(75 108)(76 86)(77 87)(78 88)(79 89)(80 90)(96 121)(97 122)(98 123)(99 124)(100 125)(101 135)(102 131)(103 132)(104 133)(105 134)(111 149)(112 150)(113 146)(114 147)(115 148)(116 126)(117 127)(118 128)(119 129)(120 130)
(1 12)(2 13)(3 14)(4 15)(5 11)(6 144)(7 145)(8 141)(9 142)(10 143)(16 137)(17 138)(18 139)(19 140)(20 136)(21 159)(22 160)(23 156)(24 157)(25 158)(26 50)(27 46)(28 47)(29 48)(30 49)(31 152)(32 153)(33 154)(34 155)(35 151)(36 70)(37 66)(38 67)(39 68)(40 69)(41 55)(42 51)(43 52)(44 53)(45 54)(56 94)(57 95)(58 91)(59 92)(60 93)(61 82)(62 83)(63 84)(64 85)(65 81)(71 90)(72 86)(73 87)(74 88)(75 89)(76 110)(77 106)(78 107)(79 108)(80 109)(96 134)(97 135)(98 131)(99 132)(100 133)(101 122)(102 123)(103 124)(104 125)(105 121)(111 130)(112 126)(113 127)(114 128)(115 129)(116 150)(117 146)(118 147)(119 148)(120 149)
(1 146 66 121)(2 147 67 122)(3 148 68 123)(4 149 69 124)(5 150 70 125)(6 85 23 110)(7 81 24 106)(8 82 25 107)(9 83 21 108)(10 84 22 109)(11 116 36 104)(12 117 37 105)(13 118 38 101)(14 119 39 102)(15 120 40 103)(16 95 33 88)(17 91 34 89)(18 92 35 90)(19 93 31 86)(20 94 32 87)(26 99 43 111)(27 100 44 112)(28 96 45 113)(29 97 41 114)(30 98 42 115)(46 133 53 126)(47 134 54 127)(48 135 55 128)(49 131 51 129)(50 132 52 130)(56 153 73 136)(57 154 74 137)(58 155 75 138)(59 151 71 139)(60 152 72 140)(61 158 78 141)(62 159 79 142)(63 160 80 143)(64 156 76 144)(65 157 77 145)
(1 136 45 7)(2 137 41 8)(3 138 42 9)(4 139 43 10)(5 140 44 6)(11 19 53 144)(12 20 54 145)(13 16 55 141)(14 17 51 142)(15 18 52 143)(21 68 155 30)(22 69 151 26)(23 70 152 27)(24 66 153 28)(25 67 154 29)(31 46 156 36)(32 47 157 37)(33 48 158 38)(34 49 159 39)(35 50 160 40)(56 96 81 121)(57 97 82 122)(58 98 83 123)(59 99 84 124)(60 100 85 125)(61 101 95 135)(62 102 91 131)(63 103 92 132)(64 104 93 133)(65 105 94 134)(71 111 109 149)(72 112 110 150)(73 113 106 146)(74 114 107 147)(75 115 108 148)(76 116 86 126)(77 117 87 127)(78 118 88 128)(79 119 89 129)(80 120 90 130)
G:=sub<Sym(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), (6,19)(7,20)(8,16)(9,17)(10,18)(21,34)(22,35)(23,31)(24,32)(25,33)(56,94)(57,95)(58,91)(59,92)(60,93)(61,82)(62,83)(63,84)(64,85)(65,81)(71,90)(72,86)(73,87)(74,88)(75,89)(76,110)(77,106)(78,107)(79,108)(80,109)(96,121)(97,122)(98,123)(99,124)(100,125)(101,135)(102,131)(103,132)(104,133)(105,134)(111,149)(112,150)(113,146)(114,147)(115,148)(116,126)(117,127)(118,128)(119,129)(120,130)(136,145)(137,141)(138,142)(139,143)(140,144)(151,160)(152,156)(153,157)(154,158)(155,159), (1,45)(2,41)(3,42)(4,43)(5,44)(6,140)(7,136)(8,137)(9,138)(10,139)(11,53)(12,54)(13,55)(14,51)(15,52)(16,141)(17,142)(18,143)(19,144)(20,145)(21,155)(22,151)(23,152)(24,153)(25,154)(26,69)(27,70)(28,66)(29,67)(30,68)(31,156)(32,157)(33,158)(34,159)(35,160)(36,46)(37,47)(38,48)(39,49)(40,50)(56,81)(57,82)(58,83)(59,84)(60,85)(61,95)(62,91)(63,92)(64,93)(65,94)(71,109)(72,110)(73,106)(74,107)(75,108)(76,86)(77,87)(78,88)(79,89)(80,90)(96,121)(97,122)(98,123)(99,124)(100,125)(101,135)(102,131)(103,132)(104,133)(105,134)(111,149)(112,150)(113,146)(114,147)(115,148)(116,126)(117,127)(118,128)(119,129)(120,130), (1,12)(2,13)(3,14)(4,15)(5,11)(6,144)(7,145)(8,141)(9,142)(10,143)(16,137)(17,138)(18,139)(19,140)(20,136)(21,159)(22,160)(23,156)(24,157)(25,158)(26,50)(27,46)(28,47)(29,48)(30,49)(31,152)(32,153)(33,154)(34,155)(35,151)(36,70)(37,66)(38,67)(39,68)(40,69)(41,55)(42,51)(43,52)(44,53)(45,54)(56,94)(57,95)(58,91)(59,92)(60,93)(61,82)(62,83)(63,84)(64,85)(65,81)(71,90)(72,86)(73,87)(74,88)(75,89)(76,110)(77,106)(78,107)(79,108)(80,109)(96,134)(97,135)(98,131)(99,132)(100,133)(101,122)(102,123)(103,124)(104,125)(105,121)(111,130)(112,126)(113,127)(114,128)(115,129)(116,150)(117,146)(118,147)(119,148)(120,149), (1,146,66,121)(2,147,67,122)(3,148,68,123)(4,149,69,124)(5,150,70,125)(6,85,23,110)(7,81,24,106)(8,82,25,107)(9,83,21,108)(10,84,22,109)(11,116,36,104)(12,117,37,105)(13,118,38,101)(14,119,39,102)(15,120,40,103)(16,95,33,88)(17,91,34,89)(18,92,35,90)(19,93,31,86)(20,94,32,87)(26,99,43,111)(27,100,44,112)(28,96,45,113)(29,97,41,114)(30,98,42,115)(46,133,53,126)(47,134,54,127)(48,135,55,128)(49,131,51,129)(50,132,52,130)(56,153,73,136)(57,154,74,137)(58,155,75,138)(59,151,71,139)(60,152,72,140)(61,158,78,141)(62,159,79,142)(63,160,80,143)(64,156,76,144)(65,157,77,145), (1,136,45,7)(2,137,41,8)(3,138,42,9)(4,139,43,10)(5,140,44,6)(11,19,53,144)(12,20,54,145)(13,16,55,141)(14,17,51,142)(15,18,52,143)(21,68,155,30)(22,69,151,26)(23,70,152,27)(24,66,153,28)(25,67,154,29)(31,46,156,36)(32,47,157,37)(33,48,158,38)(34,49,159,39)(35,50,160,40)(56,96,81,121)(57,97,82,122)(58,98,83,123)(59,99,84,124)(60,100,85,125)(61,101,95,135)(62,102,91,131)(63,103,92,132)(64,104,93,133)(65,105,94,134)(71,111,109,149)(72,112,110,150)(73,113,106,146)(74,114,107,147)(75,115,108,148)(76,116,86,126)(77,117,87,127)(78,118,88,128)(79,119,89,129)(80,120,90,130)>;
G:=Group( (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), (6,19)(7,20)(8,16)(9,17)(10,18)(21,34)(22,35)(23,31)(24,32)(25,33)(56,94)(57,95)(58,91)(59,92)(60,93)(61,82)(62,83)(63,84)(64,85)(65,81)(71,90)(72,86)(73,87)(74,88)(75,89)(76,110)(77,106)(78,107)(79,108)(80,109)(96,121)(97,122)(98,123)(99,124)(100,125)(101,135)(102,131)(103,132)(104,133)(105,134)(111,149)(112,150)(113,146)(114,147)(115,148)(116,126)(117,127)(118,128)(119,129)(120,130)(136,145)(137,141)(138,142)(139,143)(140,144)(151,160)(152,156)(153,157)(154,158)(155,159), (1,45)(2,41)(3,42)(4,43)(5,44)(6,140)(7,136)(8,137)(9,138)(10,139)(11,53)(12,54)(13,55)(14,51)(15,52)(16,141)(17,142)(18,143)(19,144)(20,145)(21,155)(22,151)(23,152)(24,153)(25,154)(26,69)(27,70)(28,66)(29,67)(30,68)(31,156)(32,157)(33,158)(34,159)(35,160)(36,46)(37,47)(38,48)(39,49)(40,50)(56,81)(57,82)(58,83)(59,84)(60,85)(61,95)(62,91)(63,92)(64,93)(65,94)(71,109)(72,110)(73,106)(74,107)(75,108)(76,86)(77,87)(78,88)(79,89)(80,90)(96,121)(97,122)(98,123)(99,124)(100,125)(101,135)(102,131)(103,132)(104,133)(105,134)(111,149)(112,150)(113,146)(114,147)(115,148)(116,126)(117,127)(118,128)(119,129)(120,130), (1,12)(2,13)(3,14)(4,15)(5,11)(6,144)(7,145)(8,141)(9,142)(10,143)(16,137)(17,138)(18,139)(19,140)(20,136)(21,159)(22,160)(23,156)(24,157)(25,158)(26,50)(27,46)(28,47)(29,48)(30,49)(31,152)(32,153)(33,154)(34,155)(35,151)(36,70)(37,66)(38,67)(39,68)(40,69)(41,55)(42,51)(43,52)(44,53)(45,54)(56,94)(57,95)(58,91)(59,92)(60,93)(61,82)(62,83)(63,84)(64,85)(65,81)(71,90)(72,86)(73,87)(74,88)(75,89)(76,110)(77,106)(78,107)(79,108)(80,109)(96,134)(97,135)(98,131)(99,132)(100,133)(101,122)(102,123)(103,124)(104,125)(105,121)(111,130)(112,126)(113,127)(114,128)(115,129)(116,150)(117,146)(118,147)(119,148)(120,149), (1,146,66,121)(2,147,67,122)(3,148,68,123)(4,149,69,124)(5,150,70,125)(6,85,23,110)(7,81,24,106)(8,82,25,107)(9,83,21,108)(10,84,22,109)(11,116,36,104)(12,117,37,105)(13,118,38,101)(14,119,39,102)(15,120,40,103)(16,95,33,88)(17,91,34,89)(18,92,35,90)(19,93,31,86)(20,94,32,87)(26,99,43,111)(27,100,44,112)(28,96,45,113)(29,97,41,114)(30,98,42,115)(46,133,53,126)(47,134,54,127)(48,135,55,128)(49,131,51,129)(50,132,52,130)(56,153,73,136)(57,154,74,137)(58,155,75,138)(59,151,71,139)(60,152,72,140)(61,158,78,141)(62,159,79,142)(63,160,80,143)(64,156,76,144)(65,157,77,145), (1,136,45,7)(2,137,41,8)(3,138,42,9)(4,139,43,10)(5,140,44,6)(11,19,53,144)(12,20,54,145)(13,16,55,141)(14,17,51,142)(15,18,52,143)(21,68,155,30)(22,69,151,26)(23,70,152,27)(24,66,153,28)(25,67,154,29)(31,46,156,36)(32,47,157,37)(33,48,158,38)(34,49,159,39)(35,50,160,40)(56,96,81,121)(57,97,82,122)(58,98,83,123)(59,99,84,124)(60,100,85,125)(61,101,95,135)(62,102,91,131)(63,103,92,132)(64,104,93,133)(65,105,94,134)(71,111,109,149)(72,112,110,150)(73,113,106,146)(74,114,107,147)(75,115,108,148)(76,116,86,126)(77,117,87,127)(78,118,88,128)(79,119,89,129)(80,120,90,130) );
G=PermutationGroup([[(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)], [(6,19),(7,20),(8,16),(9,17),(10,18),(21,34),(22,35),(23,31),(24,32),(25,33),(56,94),(57,95),(58,91),(59,92),(60,93),(61,82),(62,83),(63,84),(64,85),(65,81),(71,90),(72,86),(73,87),(74,88),(75,89),(76,110),(77,106),(78,107),(79,108),(80,109),(96,121),(97,122),(98,123),(99,124),(100,125),(101,135),(102,131),(103,132),(104,133),(105,134),(111,149),(112,150),(113,146),(114,147),(115,148),(116,126),(117,127),(118,128),(119,129),(120,130),(136,145),(137,141),(138,142),(139,143),(140,144),(151,160),(152,156),(153,157),(154,158),(155,159)], [(1,45),(2,41),(3,42),(4,43),(5,44),(6,140),(7,136),(8,137),(9,138),(10,139),(11,53),(12,54),(13,55),(14,51),(15,52),(16,141),(17,142),(18,143),(19,144),(20,145),(21,155),(22,151),(23,152),(24,153),(25,154),(26,69),(27,70),(28,66),(29,67),(30,68),(31,156),(32,157),(33,158),(34,159),(35,160),(36,46),(37,47),(38,48),(39,49),(40,50),(56,81),(57,82),(58,83),(59,84),(60,85),(61,95),(62,91),(63,92),(64,93),(65,94),(71,109),(72,110),(73,106),(74,107),(75,108),(76,86),(77,87),(78,88),(79,89),(80,90),(96,121),(97,122),(98,123),(99,124),(100,125),(101,135),(102,131),(103,132),(104,133),(105,134),(111,149),(112,150),(113,146),(114,147),(115,148),(116,126),(117,127),(118,128),(119,129),(120,130)], [(1,12),(2,13),(3,14),(4,15),(5,11),(6,144),(7,145),(8,141),(9,142),(10,143),(16,137),(17,138),(18,139),(19,140),(20,136),(21,159),(22,160),(23,156),(24,157),(25,158),(26,50),(27,46),(28,47),(29,48),(30,49),(31,152),(32,153),(33,154),(34,155),(35,151),(36,70),(37,66),(38,67),(39,68),(40,69),(41,55),(42,51),(43,52),(44,53),(45,54),(56,94),(57,95),(58,91),(59,92),(60,93),(61,82),(62,83),(63,84),(64,85),(65,81),(71,90),(72,86),(73,87),(74,88),(75,89),(76,110),(77,106),(78,107),(79,108),(80,109),(96,134),(97,135),(98,131),(99,132),(100,133),(101,122),(102,123),(103,124),(104,125),(105,121),(111,130),(112,126),(113,127),(114,128),(115,129),(116,150),(117,146),(118,147),(119,148),(120,149)], [(1,146,66,121),(2,147,67,122),(3,148,68,123),(4,149,69,124),(5,150,70,125),(6,85,23,110),(7,81,24,106),(8,82,25,107),(9,83,21,108),(10,84,22,109),(11,116,36,104),(12,117,37,105),(13,118,38,101),(14,119,39,102),(15,120,40,103),(16,95,33,88),(17,91,34,89),(18,92,35,90),(19,93,31,86),(20,94,32,87),(26,99,43,111),(27,100,44,112),(28,96,45,113),(29,97,41,114),(30,98,42,115),(46,133,53,126),(47,134,54,127),(48,135,55,128),(49,131,51,129),(50,132,52,130),(56,153,73,136),(57,154,74,137),(58,155,75,138),(59,151,71,139),(60,152,72,140),(61,158,78,141),(62,159,79,142),(63,160,80,143),(64,156,76,144),(65,157,77,145)], [(1,136,45,7),(2,137,41,8),(3,138,42,9),(4,139,43,10),(5,140,44,6),(11,19,53,144),(12,20,54,145),(13,16,55,141),(14,17,51,142),(15,18,52,143),(21,68,155,30),(22,69,151,26),(23,70,152,27),(24,66,153,28),(25,67,154,29),(31,46,156,36),(32,47,157,37),(33,48,158,38),(34,49,159,39),(35,50,160,40),(56,96,81,121),(57,97,82,122),(58,98,83,123),(59,99,84,124),(60,100,85,125),(61,101,95,135),(62,102,91,131),(63,103,92,132),(64,104,93,133),(65,105,94,134),(71,111,109,149),(72,112,110,150),(73,113,106,146),(74,114,107,147),(75,115,108,148),(76,116,86,126),(77,117,87,127),(78,118,88,128),(79,119,89,129),(80,120,90,130)]])
110 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | ··· | 4L | 5A | 5B | 5C | 5D | 10A | ··· | 10AB | 10AC | ··· | 10AJ | 20A | ··· | 20AV |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 4 | ··· | 4 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | D4 | D4 | C4○D4 | C5×D4 | C5×D4 | C5×C4○D4 |
| kernel | C5×C23.11D4 | C5×C2.C42 | C10×C22⋊C4 | C10×C4⋊C4 | C23.11D4 | C2.C42 | C2×C22⋊C4 | C2×C4⋊C4 | C2×C20 | C22×C10 | C2×C10 | C2×C4 | C23 | C22 |
| # reps | 1 | 3 | 3 | 1 | 4 | 12 | 12 | 4 | 2 | 2 | 10 | 8 | 8 | 40 |
Matrix representation of C5×C23.11D4 ►in GL6(𝔽41)
| 10 | 0 | 0 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 0 | 0 | 0 |
| 0 | 0 | 0 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 0 | 0 | 10 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 9 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 40 | 23 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 2 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 18 |
| 0 | 0 | 0 | 0 | 0 | 9 |
G:=sub<GL(6,GF(41))| [10,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10],[1,9,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,1,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,23,1,0,0,0,0,0,0,0,9,0,0,0,0,9,0,0,0,0,0,0,0,40,0,0,0,0,0,2,1],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,9,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,18,9] >;
C5×C23.11D4 in GAP, Magma, Sage, TeX
C_5\times C_2^3._{11}D_4 % in TeX
G:=Group("C5xC2^3.11D4"); // GroupNames label
G:=SmallGroup(320,898);
// by ID
G=gap.SmallGroup(320,898);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,1766,1731,226]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^5=b^2=c^2=d^2=e^4=1,f^2=c,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=e^-1>;
// generators/relations