direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D5×Q8⋊C4, D10.11Q16, D10.21SD16, (Q8×D5)⋊1C4, Q8⋊5(C4×D5), C2.2(D5×Q16), (C4×D5).95D4, C4.162(D4×D5), C2.4(D5×SD16), C4⋊C4.147D10, Q8⋊Dic5⋊3C2, C10.Q16⋊9C2, (C2×C8).208D10, C20.116(C2×D4), C10.16(C2×Q16), C22.76(D4×D5), Dic10⋊13(C2×C4), C20.47(C22×C4), (C2×Q8).103D10, C10.29(C2×SD16), C20.44D4⋊22C2, (C2×C40).196C22, (C2×C20).241C23, (C2×Dic5).138D4, (C22×D5).153D4, C4⋊Dic5.89C22, (Q8×C10).24C22, D10.56(C22⋊C4), Dic5.25(C22⋊C4), (C2×Dic10).72C22, C4.12(C2×C4×D5), (C2×Q8×D5).2C2, (D5×C4⋊C4).1C2, C5⋊2(C2×Q8⋊C4), (D5×C2×C8).15C2, (C5×Q8)⋊11(C2×C4), (C4×D5).50(C2×C4), C2.21(D5×C22⋊C4), (C5×Q8⋊C4)⋊18C2, (C2×C10).254(C2×D4), (C5×C4⋊C4).42C22, C10.61(C2×C22⋊C4), (C2×C4×D5).296C22, (C2×C4).348(C22×D5), (C2×C5⋊2C8).226C22, SmallGroup(320,428)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D5×Q8⋊C4
G = < a,b,c,d,e | a5=b2=c4=e4=1, d2=c2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece-1=c-1, ede-1=c-1d >
Subgroups: 606 in 162 conjugacy classes, 63 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, Q8, Q8, C23, D5, C10, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×Q8, C2×Q8, Dic5, Dic5, C20, C20, D10, C2×C10, Q8⋊C4, Q8⋊C4, C2×C4⋊C4, C22×C8, C22×Q8, C5⋊2C8, C40, Dic10, Dic10, C4×D5, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C5×Q8, C22×D5, C2×Q8⋊C4, C8×D5, C2×C5⋊2C8, C10.D4, C4⋊Dic5, C5×C4⋊C4, C2×C40, C2×Dic10, C2×Dic10, C2×C4×D5, C2×C4×D5, Q8×D5, Q8×D5, Q8×C10, C10.Q16, C20.44D4, Q8⋊Dic5, C5×Q8⋊C4, D5×C4⋊C4, D5×C2×C8, C2×Q8×D5, D5×Q8⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, SD16, Q16, C22×C4, C2×D4, D10, Q8⋊C4, C2×C22⋊C4, C2×SD16, C2×Q16, C4×D5, C22×D5, C2×Q8⋊C4, C2×C4×D5, D4×D5, D5×C22⋊C4, D5×SD16, D5×Q16, D5×Q8⋊C4
►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 28)(2 27)(3 26)(4 30)(5 29)(6 21)(7 25)(8 24)(9 23)(10 22)(11 36)(12 40)(13 39)(14 38)(15 37)(16 31)(17 35)(18 34)(19 33)(20 32)(41 66)(42 70)(43 69)(44 68)(45 67)(46 61)(47 65)(48 64)(49 63)(50 62)(51 76)(52 80)(53 79)(54 78)(55 77)(56 71)(57 75)(58 74)(59 73)(60 72)(81 106)(82 110)(83 109)(84 108)(85 107)(86 101)(87 105)(88 104)(89 103)(90 102)(91 116)(92 120)(93 119)(94 118)(95 117)(96 111)(97 115)(98 114)(99 113)(100 112)(121 146)(122 150)(123 149)(124 148)(125 147)(126 141)(127 145)(128 144)(129 143)(130 142)(131 156)(132 160)(133 159)(134 158)(135 157)(136 151)(137 155)(138 154)(139 153)(140 152)
(1 14 9 19)(2 15 10 20)(3 11 6 16)(4 12 7 17)(5 13 8 18)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)(41 56 46 51)(42 57 47 52)(43 58 48 53)(44 59 49 54)(45 60 50 55)(61 76 66 71)(62 77 67 72)(63 78 68 73)(64 79 69 74)(65 80 70 75)(81 96 86 91)(82 97 87 92)(83 98 88 93)(84 99 89 94)(85 100 90 95)(101 116 106 111)(102 117 107 112)(103 118 108 113)(104 119 109 114)(105 120 110 115)(121 131 126 136)(122 132 127 137)(123 133 128 138)(124 134 129 139)(125 135 130 140)(141 151 146 156)(142 152 147 157)(143 153 148 158)(144 154 149 159)(145 155 150 160)
(1 109 9 104)(2 110 10 105)(3 106 6 101)(4 107 7 102)(5 108 8 103)(11 116 16 111)(12 117 17 112)(13 118 18 113)(14 119 19 114)(15 120 20 115)(21 86 26 81)(22 87 27 82)(23 88 28 83)(24 89 29 84)(25 90 30 85)(31 96 36 91)(32 97 37 92)(33 98 38 93)(34 99 39 94)(35 100 40 95)(41 146 46 141)(42 147 47 142)(43 148 48 143)(44 149 49 144)(45 150 50 145)(51 156 56 151)(52 157 57 152)(53 158 58 153)(54 159 59 154)(55 160 60 155)(61 126 66 121)(62 127 67 122)(63 128 68 123)(64 129 69 124)(65 130 70 125)(71 136 76 131)(72 137 77 132)(73 138 78 133)(74 139 79 134)(75 140 80 135)
(1 64 24 44)(2 65 25 45)(3 61 21 41)(4 62 22 42)(5 63 23 43)(6 66 26 46)(7 67 27 47)(8 68 28 48)(9 69 29 49)(10 70 30 50)(11 71 31 51)(12 72 32 52)(13 73 33 53)(14 74 34 54)(15 75 35 55)(16 76 36 56)(17 77 37 57)(18 78 38 58)(19 79 39 59)(20 80 40 60)(81 151 101 131)(82 152 102 132)(83 153 103 133)(84 154 104 134)(85 155 105 135)(86 156 106 136)(87 157 107 137)(88 158 108 138)(89 159 109 139)(90 160 110 140)(91 146 111 126)(92 147 112 127)(93 148 113 128)(94 149 114 129)(95 150 115 130)(96 141 116 121)(97 142 117 122)(98 143 118 123)(99 144 119 124)(100 145 120 125)
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), (1,28)(2,27)(3,26)(4,30)(5,29)(6,21)(7,25)(8,24)(9,23)(10,22)(11,36)(12,40)(13,39)(14,38)(15,37)(16,31)(17,35)(18,34)(19,33)(20,32)(41,66)(42,70)(43,69)(44,68)(45,67)(46,61)(47,65)(48,64)(49,63)(50,62)(51,76)(52,80)(53,79)(54,78)(55,77)(56,71)(57,75)(58,74)(59,73)(60,72)(81,106)(82,110)(83,109)(84,108)(85,107)(86,101)(87,105)(88,104)(89,103)(90,102)(91,116)(92,120)(93,119)(94,118)(95,117)(96,111)(97,115)(98,114)(99,113)(100,112)(121,146)(122,150)(123,149)(124,148)(125,147)(126,141)(127,145)(128,144)(129,143)(130,142)(131,156)(132,160)(133,159)(134,158)(135,157)(136,151)(137,155)(138,154)(139,153)(140,152), (1,14,9,19)(2,15,10,20)(3,11,6,16)(4,12,7,17)(5,13,8,18)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40)(41,56,46,51)(42,57,47,52)(43,58,48,53)(44,59,49,54)(45,60,50,55)(61,76,66,71)(62,77,67,72)(63,78,68,73)(64,79,69,74)(65,80,70,75)(81,96,86,91)(82,97,87,92)(83,98,88,93)(84,99,89,94)(85,100,90,95)(101,116,106,111)(102,117,107,112)(103,118,108,113)(104,119,109,114)(105,120,110,115)(121,131,126,136)(122,132,127,137)(123,133,128,138)(124,134,129,139)(125,135,130,140)(141,151,146,156)(142,152,147,157)(143,153,148,158)(144,154,149,159)(145,155,150,160), (1,109,9,104)(2,110,10,105)(3,106,6,101)(4,107,7,102)(5,108,8,103)(11,116,16,111)(12,117,17,112)(13,118,18,113)(14,119,19,114)(15,120,20,115)(21,86,26,81)(22,87,27,82)(23,88,28,83)(24,89,29,84)(25,90,30,85)(31,96,36,91)(32,97,37,92)(33,98,38,93)(34,99,39,94)(35,100,40,95)(41,146,46,141)(42,147,47,142)(43,148,48,143)(44,149,49,144)(45,150,50,145)(51,156,56,151)(52,157,57,152)(53,158,58,153)(54,159,59,154)(55,160,60,155)(61,126,66,121)(62,127,67,122)(63,128,68,123)(64,129,69,124)(65,130,70,125)(71,136,76,131)(72,137,77,132)(73,138,78,133)(74,139,79,134)(75,140,80,135), (1,64,24,44)(2,65,25,45)(3,61,21,41)(4,62,22,42)(5,63,23,43)(6,66,26,46)(7,67,27,47)(8,68,28,48)(9,69,29,49)(10,70,30,50)(11,71,31,51)(12,72,32,52)(13,73,33,53)(14,74,34,54)(15,75,35,55)(16,76,36,56)(17,77,37,57)(18,78,38,58)(19,79,39,59)(20,80,40,60)(81,151,101,131)(82,152,102,132)(83,153,103,133)(84,154,104,134)(85,155,105,135)(86,156,106,136)(87,157,107,137)(88,158,108,138)(89,159,109,139)(90,160,110,140)(91,146,111,126)(92,147,112,127)(93,148,113,128)(94,149,114,129)(95,150,115,130)(96,141,116,121)(97,142,117,122)(98,143,118,123)(99,144,119,124)(100,145,120,125)>;
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), (1,28)(2,27)(3,26)(4,30)(5,29)(6,21)(7,25)(8,24)(9,23)(10,22)(11,36)(12,40)(13,39)(14,38)(15,37)(16,31)(17,35)(18,34)(19,33)(20,32)(41,66)(42,70)(43,69)(44,68)(45,67)(46,61)(47,65)(48,64)(49,63)(50,62)(51,76)(52,80)(53,79)(54,78)(55,77)(56,71)(57,75)(58,74)(59,73)(60,72)(81,106)(82,110)(83,109)(84,108)(85,107)(86,101)(87,105)(88,104)(89,103)(90,102)(91,116)(92,120)(93,119)(94,118)(95,117)(96,111)(97,115)(98,114)(99,113)(100,112)(121,146)(122,150)(123,149)(124,148)(125,147)(126,141)(127,145)(128,144)(129,143)(130,142)(131,156)(132,160)(133,159)(134,158)(135,157)(136,151)(137,155)(138,154)(139,153)(140,152), (1,14,9,19)(2,15,10,20)(3,11,6,16)(4,12,7,17)(5,13,8,18)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40)(41,56,46,51)(42,57,47,52)(43,58,48,53)(44,59,49,54)(45,60,50,55)(61,76,66,71)(62,77,67,72)(63,78,68,73)(64,79,69,74)(65,80,70,75)(81,96,86,91)(82,97,87,92)(83,98,88,93)(84,99,89,94)(85,100,90,95)(101,116,106,111)(102,117,107,112)(103,118,108,113)(104,119,109,114)(105,120,110,115)(121,131,126,136)(122,132,127,137)(123,133,128,138)(124,134,129,139)(125,135,130,140)(141,151,146,156)(142,152,147,157)(143,153,148,158)(144,154,149,159)(145,155,150,160), (1,109,9,104)(2,110,10,105)(3,106,6,101)(4,107,7,102)(5,108,8,103)(11,116,16,111)(12,117,17,112)(13,118,18,113)(14,119,19,114)(15,120,20,115)(21,86,26,81)(22,87,27,82)(23,88,28,83)(24,89,29,84)(25,90,30,85)(31,96,36,91)(32,97,37,92)(33,98,38,93)(34,99,39,94)(35,100,40,95)(41,146,46,141)(42,147,47,142)(43,148,48,143)(44,149,49,144)(45,150,50,145)(51,156,56,151)(52,157,57,152)(53,158,58,153)(54,159,59,154)(55,160,60,155)(61,126,66,121)(62,127,67,122)(63,128,68,123)(64,129,69,124)(65,130,70,125)(71,136,76,131)(72,137,77,132)(73,138,78,133)(74,139,79,134)(75,140,80,135), (1,64,24,44)(2,65,25,45)(3,61,21,41)(4,62,22,42)(5,63,23,43)(6,66,26,46)(7,67,27,47)(8,68,28,48)(9,69,29,49)(10,70,30,50)(11,71,31,51)(12,72,32,52)(13,73,33,53)(14,74,34,54)(15,75,35,55)(16,76,36,56)(17,77,37,57)(18,78,38,58)(19,79,39,59)(20,80,40,60)(81,151,101,131)(82,152,102,132)(83,153,103,133)(84,154,104,134)(85,155,105,135)(86,156,106,136)(87,157,107,137)(88,158,108,138)(89,159,109,139)(90,160,110,140)(91,146,111,126)(92,147,112,127)(93,148,113,128)(94,149,114,129)(95,150,115,130)(96,141,116,121)(97,142,117,122)(98,143,118,123)(99,144,119,124)(100,145,120,125) );
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)], [(1,28),(2,27),(3,26),(4,30),(5,29),(6,21),(7,25),(8,24),(9,23),(10,22),(11,36),(12,40),(13,39),(14,38),(15,37),(16,31),(17,35),(18,34),(19,33),(20,32),(41,66),(42,70),(43,69),(44,68),(45,67),(46,61),(47,65),(48,64),(49,63),(50,62),(51,76),(52,80),(53,79),(54,78),(55,77),(56,71),(57,75),(58,74),(59,73),(60,72),(81,106),(82,110),(83,109),(84,108),(85,107),(86,101),(87,105),(88,104),(89,103),(90,102),(91,116),(92,120),(93,119),(94,118),(95,117),(96,111),(97,115),(98,114),(99,113),(100,112),(121,146),(122,150),(123,149),(124,148),(125,147),(126,141),(127,145),(128,144),(129,143),(130,142),(131,156),(132,160),(133,159),(134,158),(135,157),(136,151),(137,155),(138,154),(139,153),(140,152)], [(1,14,9,19),(2,15,10,20),(3,11,6,16),(4,12,7,17),(5,13,8,18),(21,31,26,36),(22,32,27,37),(23,33,28,38),(24,34,29,39),(25,35,30,40),(41,56,46,51),(42,57,47,52),(43,58,48,53),(44,59,49,54),(45,60,50,55),(61,76,66,71),(62,77,67,72),(63,78,68,73),(64,79,69,74),(65,80,70,75),(81,96,86,91),(82,97,87,92),(83,98,88,93),(84,99,89,94),(85,100,90,95),(101,116,106,111),(102,117,107,112),(103,118,108,113),(104,119,109,114),(105,120,110,115),(121,131,126,136),(122,132,127,137),(123,133,128,138),(124,134,129,139),(125,135,130,140),(141,151,146,156),(142,152,147,157),(143,153,148,158),(144,154,149,159),(145,155,150,160)], [(1,109,9,104),(2,110,10,105),(3,106,6,101),(4,107,7,102),(5,108,8,103),(11,116,16,111),(12,117,17,112),(13,118,18,113),(14,119,19,114),(15,120,20,115),(21,86,26,81),(22,87,27,82),(23,88,28,83),(24,89,29,84),(25,90,30,85),(31,96,36,91),(32,97,37,92),(33,98,38,93),(34,99,39,94),(35,100,40,95),(41,146,46,141),(42,147,47,142),(43,148,48,143),(44,149,49,144),(45,150,50,145),(51,156,56,151),(52,157,57,152),(53,158,58,153),(54,159,59,154),(55,160,60,155),(61,126,66,121),(62,127,67,122),(63,128,68,123),(64,129,69,124),(65,130,70,125),(71,136,76,131),(72,137,77,132),(73,138,78,133),(74,139,79,134),(75,140,80,135)], [(1,64,24,44),(2,65,25,45),(3,61,21,41),(4,62,22,42),(5,63,23,43),(6,66,26,46),(7,67,27,47),(8,68,28,48),(9,69,29,49),(10,70,30,50),(11,71,31,51),(12,72,32,52),(13,73,33,53),(14,74,34,54),(15,75,35,55),(16,76,36,56),(17,77,37,57),(18,78,38,58),(19,79,39,59),(20,80,40,60),(81,151,101,131),(82,152,102,132),(83,153,103,133),(84,154,104,134),(85,155,105,135),(86,156,106,136),(87,157,107,137),(88,158,108,138),(89,159,109,139),(90,160,110,140),(91,146,111,126),(92,147,112,127),(93,148,113,128),(94,149,114,129),(95,150,115,130),(96,141,116,121),(97,142,117,122),(98,143,118,123),(99,144,119,124),(100,145,120,125)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10F | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D4 | D5 | SD16 | Q16 | D10 | D10 | D10 | C4×D5 | D4×D5 | D4×D5 | D5×SD16 | D5×Q16 |
| kernel | D5×Q8⋊C4 | C10.Q16 | C20.44D4 | Q8⋊Dic5 | C5×Q8⋊C4 | D5×C4⋊C4 | D5×C2×C8 | C2×Q8×D5 | Q8×D5 | C4×D5 | C2×Dic5 | C22×D5 | Q8⋊C4 | D10 | D10 | C4⋊C4 | C2×C8 | C2×Q8 | Q8 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 2 | 1 | 1 | 2 | 4 | 4 | 2 | 2 | 2 | 8 | 2 | 2 | 4 | 4 |
Matrix representation of D5×Q8⋊C4 ►in GL5(𝔽41)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 40 | 6 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 35 | 39 | 0 | 0 |
| 0 | 39 | 6 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 9 | 0 | 0 | 0 | 0 |
| 0 | 19 | 38 | 0 | 0 |
| 0 | 38 | 22 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,1,6],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,35,39,0,0,0,39,6,0,0,0,0,0,40,0,0,0,0,0,40],[9,0,0,0,0,0,19,38,0,0,0,38,22,0,0,0,0,0,40,0,0,0,0,0,40] >;
D5×Q8⋊C4 in GAP, Magma, Sage, TeX
D_5\times Q_8\rtimes C_4
% in TeX
G:=Group("D5xQ8:C4"); // GroupNames label
G:=SmallGroup(320,428);
// by ID
G=gap.SmallGroup(320,428);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,219,58,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^2=c^4=e^4=1,d^2=c^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e^-1=c^-1,e*d*e^-1=c^-1*d>;
// generators/relations