direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×D4.D4, C20⋊12SD16, C4⋊C8⋊9C10, C4⋊Q8⋊2C10, D4.1(C5×D4), C4⋊2(C5×SD16), (C4×D4).6C10, (C5×D4).26D4, C4.33(D4×C10), (D4×C20).21C2, C20.394(C2×D4), (C2×C20).323D4, C2.8(C10×SD16), Q8⋊C4⋊11C10, C42.16(C2×C10), (C2×SD16).3C10, (C10×SD16).8C2, C10.88(C2×SD16), C22.85(D4×C10), C20.343(C4○D4), (C2×C20).920C23, (C2×C40).300C22, (C4×C20).258C22, C10.144(C4⋊D4), (D4×C10).297C22, (Q8×C10).159C22, C10.134(C8.C22), (C5×C4⋊C8)⋊28C2, (C5×C4⋊Q8)⋊23C2, C4.42(C5×C4○D4), C4⋊C4.53(C2×C10), (C2×C8).37(C2×C10), (C2×C4).129(C5×D4), C2.13(C5×C4⋊D4), (C2×Q8).5(C2×C10), C2.9(C5×C8.C22), (C5×Q8⋊C4)⋊33C2, (C2×D4).56(C2×C10), (C2×C10).641(C2×D4), (C5×C4⋊C4).374C22, (C2×C4).95(C22×C10), SmallGroup(320,962)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×D4.D4
G = < a,b,c,d,e | a5=b4=c2=d4=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d-1 >
Subgroups: 218 in 120 conjugacy classes, 58 normal (34 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, C20, C20, C20, C2×C10, C2×C10, Q8⋊C4, C4⋊C8, C4×D4, C4⋊Q8, C2×SD16, C40, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C22×C10, D4.D4, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C2×C40, C5×SD16, C22×C20, D4×C10, Q8×C10, C5×Q8⋊C4, C5×C4⋊C8, D4×C20, C5×C4⋊Q8, C10×SD16, C5×D4.D4
Quotients: C1, C2, C22, C5, D4, C23, C10, SD16, C2×D4, C4○D4, C2×C10, C4⋊D4, C2×SD16, C8.C22, C5×D4, C22×C10, D4.D4, C5×SD16, D4×C10, C5×C4○D4, C5×C4⋊D4, C10×SD16, C5×C8.C22, C5×D4.D4
►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)
(1 37 12 66)(2 38 13 67)(3 39 14 68)(4 40 15 69)(5 36 11 70)(6 23 144 156)(7 24 145 157)(8 25 141 158)(9 21 142 159)(10 22 143 160)(16 33 137 154)(17 34 138 155)(18 35 139 151)(19 31 140 152)(20 32 136 153)(26 43 50 52)(27 44 46 53)(28 45 47 54)(29 41 48 55)(30 42 49 51)(56 87 94 73)(57 88 95 74)(58 89 91 75)(59 90 92 71)(60 86 93 72)(61 107 82 78)(62 108 83 79)(63 109 84 80)(64 110 85 76)(65 106 81 77)(96 113 134 127)(97 114 135 128)(98 115 131 129)(99 111 132 130)(100 112 133 126)(101 118 122 147)(102 119 123 148)(103 120 124 149)(104 116 125 150)(105 117 121 146)
(1 66)(2 67)(3 68)(4 69)(5 70)(6 144)(7 145)(8 141)(9 142)(10 143)(11 36)(12 37)(13 38)(14 39)(15 40)(16 137)(17 138)(18 139)(19 140)(20 136)(26 43)(27 44)(28 45)(29 41)(30 42)(46 53)(47 54)(48 55)(49 51)(50 52)(56 73)(57 74)(58 75)(59 71)(60 72)(61 78)(62 79)(63 80)(64 76)(65 77)(81 106)(82 107)(83 108)(84 109)(85 110)(86 93)(87 94)(88 95)(89 91)(90 92)(96 134)(97 135)(98 131)(99 132)(100 133)(101 122)(102 123)(103 124)(104 125)(105 121)
(1 56 45 81)(2 57 41 82)(3 58 42 83)(4 59 43 84)(5 60 44 85)(6 100 140 125)(7 96 136 121)(8 97 137 122)(9 98 138 123)(10 99 139 124)(11 93 53 64)(12 94 54 65)(13 95 55 61)(14 91 51 62)(15 92 52 63)(16 101 141 135)(17 102 142 131)(18 103 143 132)(19 104 144 133)(20 105 145 134)(21 115 155 148)(22 111 151 149)(23 112 152 150)(24 113 153 146)(25 114 154 147)(26 109 69 71)(27 110 70 72)(28 106 66 73)(29 107 67 74)(30 108 68 75)(31 116 156 126)(32 117 157 127)(33 118 158 128)(34 119 159 129)(35 120 160 130)(36 86 46 76)(37 87 47 77)(38 88 48 78)(39 89 49 79)(40 90 50 80)
(1 20 12 136)(2 16 13 137)(3 17 14 138)(4 18 15 139)(5 19 11 140)(6 44 144 53)(7 45 145 54)(8 41 141 55)(9 42 142 51)(10 43 143 52)(21 30 159 49)(22 26 160 50)(23 27 156 46)(24 28 157 47)(25 29 158 48)(31 36 152 70)(32 37 153 66)(33 38 154 67)(34 39 155 68)(35 40 151 69)(56 134 94 96)(57 135 95 97)(58 131 91 98)(59 132 92 99)(60 133 93 100)(61 122 82 101)(62 123 83 102)(63 124 84 103)(64 125 85 104)(65 121 81 105)(71 130 90 111)(72 126 86 112)(73 127 87 113)(74 128 88 114)(75 129 89 115)(76 150 110 116)(77 146 106 117)(78 147 107 118)(79 148 108 119)(80 149 109 120)
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), (1,37,12,66)(2,38,13,67)(3,39,14,68)(4,40,15,69)(5,36,11,70)(6,23,144,156)(7,24,145,157)(8,25,141,158)(9,21,142,159)(10,22,143,160)(16,33,137,154)(17,34,138,155)(18,35,139,151)(19,31,140,152)(20,32,136,153)(26,43,50,52)(27,44,46,53)(28,45,47,54)(29,41,48,55)(30,42,49,51)(56,87,94,73)(57,88,95,74)(58,89,91,75)(59,90,92,71)(60,86,93,72)(61,107,82,78)(62,108,83,79)(63,109,84,80)(64,110,85,76)(65,106,81,77)(96,113,134,127)(97,114,135,128)(98,115,131,129)(99,111,132,130)(100,112,133,126)(101,118,122,147)(102,119,123,148)(103,120,124,149)(104,116,125,150)(105,117,121,146), (1,66)(2,67)(3,68)(4,69)(5,70)(6,144)(7,145)(8,141)(9,142)(10,143)(11,36)(12,37)(13,38)(14,39)(15,40)(16,137)(17,138)(18,139)(19,140)(20,136)(26,43)(27,44)(28,45)(29,41)(30,42)(46,53)(47,54)(48,55)(49,51)(50,52)(56,73)(57,74)(58,75)(59,71)(60,72)(61,78)(62,79)(63,80)(64,76)(65,77)(81,106)(82,107)(83,108)(84,109)(85,110)(86,93)(87,94)(88,95)(89,91)(90,92)(96,134)(97,135)(98,131)(99,132)(100,133)(101,122)(102,123)(103,124)(104,125)(105,121), (1,56,45,81)(2,57,41,82)(3,58,42,83)(4,59,43,84)(5,60,44,85)(6,100,140,125)(7,96,136,121)(8,97,137,122)(9,98,138,123)(10,99,139,124)(11,93,53,64)(12,94,54,65)(13,95,55,61)(14,91,51,62)(15,92,52,63)(16,101,141,135)(17,102,142,131)(18,103,143,132)(19,104,144,133)(20,105,145,134)(21,115,155,148)(22,111,151,149)(23,112,152,150)(24,113,153,146)(25,114,154,147)(26,109,69,71)(27,110,70,72)(28,106,66,73)(29,107,67,74)(30,108,68,75)(31,116,156,126)(32,117,157,127)(33,118,158,128)(34,119,159,129)(35,120,160,130)(36,86,46,76)(37,87,47,77)(38,88,48,78)(39,89,49,79)(40,90,50,80), (1,20,12,136)(2,16,13,137)(3,17,14,138)(4,18,15,139)(5,19,11,140)(6,44,144,53)(7,45,145,54)(8,41,141,55)(9,42,142,51)(10,43,143,52)(21,30,159,49)(22,26,160,50)(23,27,156,46)(24,28,157,47)(25,29,158,48)(31,36,152,70)(32,37,153,66)(33,38,154,67)(34,39,155,68)(35,40,151,69)(56,134,94,96)(57,135,95,97)(58,131,91,98)(59,132,92,99)(60,133,93,100)(61,122,82,101)(62,123,83,102)(63,124,84,103)(64,125,85,104)(65,121,81,105)(71,130,90,111)(72,126,86,112)(73,127,87,113)(74,128,88,114)(75,129,89,115)(76,150,110,116)(77,146,106,117)(78,147,107,118)(79,148,108,119)(80,149,109,120)>;
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), (1,37,12,66)(2,38,13,67)(3,39,14,68)(4,40,15,69)(5,36,11,70)(6,23,144,156)(7,24,145,157)(8,25,141,158)(9,21,142,159)(10,22,143,160)(16,33,137,154)(17,34,138,155)(18,35,139,151)(19,31,140,152)(20,32,136,153)(26,43,50,52)(27,44,46,53)(28,45,47,54)(29,41,48,55)(30,42,49,51)(56,87,94,73)(57,88,95,74)(58,89,91,75)(59,90,92,71)(60,86,93,72)(61,107,82,78)(62,108,83,79)(63,109,84,80)(64,110,85,76)(65,106,81,77)(96,113,134,127)(97,114,135,128)(98,115,131,129)(99,111,132,130)(100,112,133,126)(101,118,122,147)(102,119,123,148)(103,120,124,149)(104,116,125,150)(105,117,121,146), (1,66)(2,67)(3,68)(4,69)(5,70)(6,144)(7,145)(8,141)(9,142)(10,143)(11,36)(12,37)(13,38)(14,39)(15,40)(16,137)(17,138)(18,139)(19,140)(20,136)(26,43)(27,44)(28,45)(29,41)(30,42)(46,53)(47,54)(48,55)(49,51)(50,52)(56,73)(57,74)(58,75)(59,71)(60,72)(61,78)(62,79)(63,80)(64,76)(65,77)(81,106)(82,107)(83,108)(84,109)(85,110)(86,93)(87,94)(88,95)(89,91)(90,92)(96,134)(97,135)(98,131)(99,132)(100,133)(101,122)(102,123)(103,124)(104,125)(105,121), (1,56,45,81)(2,57,41,82)(3,58,42,83)(4,59,43,84)(5,60,44,85)(6,100,140,125)(7,96,136,121)(8,97,137,122)(9,98,138,123)(10,99,139,124)(11,93,53,64)(12,94,54,65)(13,95,55,61)(14,91,51,62)(15,92,52,63)(16,101,141,135)(17,102,142,131)(18,103,143,132)(19,104,144,133)(20,105,145,134)(21,115,155,148)(22,111,151,149)(23,112,152,150)(24,113,153,146)(25,114,154,147)(26,109,69,71)(27,110,70,72)(28,106,66,73)(29,107,67,74)(30,108,68,75)(31,116,156,126)(32,117,157,127)(33,118,158,128)(34,119,159,129)(35,120,160,130)(36,86,46,76)(37,87,47,77)(38,88,48,78)(39,89,49,79)(40,90,50,80), (1,20,12,136)(2,16,13,137)(3,17,14,138)(4,18,15,139)(5,19,11,140)(6,44,144,53)(7,45,145,54)(8,41,141,55)(9,42,142,51)(10,43,143,52)(21,30,159,49)(22,26,160,50)(23,27,156,46)(24,28,157,47)(25,29,158,48)(31,36,152,70)(32,37,153,66)(33,38,154,67)(34,39,155,68)(35,40,151,69)(56,134,94,96)(57,135,95,97)(58,131,91,98)(59,132,92,99)(60,133,93,100)(61,122,82,101)(62,123,83,102)(63,124,84,103)(64,125,85,104)(65,121,81,105)(71,130,90,111)(72,126,86,112)(73,127,87,113)(74,128,88,114)(75,129,89,115)(76,150,110,116)(77,146,106,117)(78,147,107,118)(79,148,108,119)(80,149,109,120) );
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)], [(1,37,12,66),(2,38,13,67),(3,39,14,68),(4,40,15,69),(5,36,11,70),(6,23,144,156),(7,24,145,157),(8,25,141,158),(9,21,142,159),(10,22,143,160),(16,33,137,154),(17,34,138,155),(18,35,139,151),(19,31,140,152),(20,32,136,153),(26,43,50,52),(27,44,46,53),(28,45,47,54),(29,41,48,55),(30,42,49,51),(56,87,94,73),(57,88,95,74),(58,89,91,75),(59,90,92,71),(60,86,93,72),(61,107,82,78),(62,108,83,79),(63,109,84,80),(64,110,85,76),(65,106,81,77),(96,113,134,127),(97,114,135,128),(98,115,131,129),(99,111,132,130),(100,112,133,126),(101,118,122,147),(102,119,123,148),(103,120,124,149),(104,116,125,150),(105,117,121,146)], [(1,66),(2,67),(3,68),(4,69),(5,70),(6,144),(7,145),(8,141),(9,142),(10,143),(11,36),(12,37),(13,38),(14,39),(15,40),(16,137),(17,138),(18,139),(19,140),(20,136),(26,43),(27,44),(28,45),(29,41),(30,42),(46,53),(47,54),(48,55),(49,51),(50,52),(56,73),(57,74),(58,75),(59,71),(60,72),(61,78),(62,79),(63,80),(64,76),(65,77),(81,106),(82,107),(83,108),(84,109),(85,110),(86,93),(87,94),(88,95),(89,91),(90,92),(96,134),(97,135),(98,131),(99,132),(100,133),(101,122),(102,123),(103,124),(104,125),(105,121)], [(1,56,45,81),(2,57,41,82),(3,58,42,83),(4,59,43,84),(5,60,44,85),(6,100,140,125),(7,96,136,121),(8,97,137,122),(9,98,138,123),(10,99,139,124),(11,93,53,64),(12,94,54,65),(13,95,55,61),(14,91,51,62),(15,92,52,63),(16,101,141,135),(17,102,142,131),(18,103,143,132),(19,104,144,133),(20,105,145,134),(21,115,155,148),(22,111,151,149),(23,112,152,150),(24,113,153,146),(25,114,154,147),(26,109,69,71),(27,110,70,72),(28,106,66,73),(29,107,67,74),(30,108,68,75),(31,116,156,126),(32,117,157,127),(33,118,158,128),(34,119,159,129),(35,120,160,130),(36,86,46,76),(37,87,47,77),(38,88,48,78),(39,89,49,79),(40,90,50,80)], [(1,20,12,136),(2,16,13,137),(3,17,14,138),(4,18,15,139),(5,19,11,140),(6,44,144,53),(7,45,145,54),(8,41,141,55),(9,42,142,51),(10,43,143,52),(21,30,159,49),(22,26,160,50),(23,27,156,46),(24,28,157,47),(25,29,158,48),(31,36,152,70),(32,37,153,66),(33,38,154,67),(34,39,155,68),(35,40,151,69),(56,134,94,96),(57,135,95,97),(58,131,91,98),(59,132,92,99),(60,133,93,100),(61,122,82,101),(62,123,83,102),(63,124,84,103),(64,125,85,104),(65,121,81,105),(71,130,90,111),(72,126,86,112),(73,127,87,113),(74,128,88,114),(75,129,89,115),(76,150,110,116),(77,146,106,117),(78,147,107,118),(79,148,108,119),(80,149,109,120)]])
95 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 10A | ··· | 10L | 10M | ··· | 10T | 20A | ··· | 20P | 20Q | ··· | 20AB | 20AC | ··· | 20AJ | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | D4 | D4 | SD16 | C4○D4 | C5×D4 | C5×D4 | C5×SD16 | C5×C4○D4 | C8.C22 | C5×C8.C22 |
| kernel | C5×D4.D4 | C5×Q8⋊C4 | C5×C4⋊C8 | D4×C20 | C5×C4⋊Q8 | C10×SD16 | D4.D4 | Q8⋊C4 | C4⋊C8 | C4×D4 | C4⋊Q8 | C2×SD16 | C2×C20 | C5×D4 | C20 | C20 | C2×C4 | D4 | C4 | C4 | C10 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 4 | 8 | 4 | 4 | 4 | 8 | 2 | 2 | 4 | 2 | 8 | 8 | 16 | 8 | 1 | 4 |
Matrix representation of C5×D4.D4 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 1 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 39 |
| 0 | 0 | 1 | 1 |
| 15 | 26 | 0 | 0 |
| 26 | 26 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 40 | 40 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[0,40,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,40,1,0,0,39,1],[15,26,0,0,26,26,0,0,0,0,1,40,0,0,0,40] >;
C5×D4.D4 in GAP, Magma, Sage, TeX
C_5\times D_4.D_4
% in TeX
G:=Group("C5xD4.D4"); // GroupNames label
G:=SmallGroup(320,962);
// by ID
G=gap.SmallGroup(320,962);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,1120,589,288,1766,10085,2539,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^4=c^2=d^4=1,e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b*c,e*d*e^-1=d^-1>;
// generators/relations