metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊2D5⋊2C4, (C4×D5).94D4, D4.10(C4×D5), C4.157(D4×D5), D4⋊C4⋊22D5, C4⋊C4.134D10, C10.Q16⋊5C2, D4⋊Dic5⋊6C2, (C2×C8).201D10, C20.106(C2×D4), C22.71(D4×D5), (C2×D4).133D10, C10.39(C4○D8), C2.2(D8⋊3D5), C20.42(C22×C4), (C22×D5).77D4, C20.44D4⋊19C2, (C2×C20).212C23, (C2×C40).183C22, Dic10.16(C2×C4), (C2×Dic5).269D4, (D4×C10).33C22, C5⋊2(C23.24D4), C4⋊Dic5.68C22, D10.24(C22⋊C4), C2.2(SD16⋊3D5), Dic5.56(C22⋊C4), (C2×Dic10).60C22, C4.7(C2×C4×D5), (D5×C2×C8)⋊17C2, C4⋊C4⋊7D5⋊2C2, (C4×D5).49(C2×C4), (C5×D4).18(C2×C4), C2.20(D5×C22⋊C4), (C5×D4⋊C4)⋊21C2, (C2×D4⋊2D5).4C2, (C2×C10).225(C2×D4), (C5×C4⋊C4).15C22, C10.60(C2×C22⋊C4), (C2×C4×D5).293C22, (C2×C4).319(C22×D5), (C2×C5⋊2C8).219C22, SmallGroup(320,399)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊2D5⋊C4
G = < a,b,c,d,e | a4=b2=c5=d2=e4=1, bab=eae-1=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, ebe-1=ab, dcd=c-1, ce=ec, ede-1=a2d >
Subgroups: 590 in 158 conjugacy classes, 55 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, D4⋊C4, D4⋊C4, Q8⋊C4, C42⋊C2, C22×C8, C2×C4○D4, C5⋊2C8, C40, Dic10, Dic10, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C22×D5, C22×C10, C23.24D4, C8×D5, C2×C5⋊2C8, C4×Dic5, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C2×C40, C2×Dic10, C2×C4×D5, D4⋊2D5, D4⋊2D5, C22×Dic5, C2×C5⋊D4, D4×C10, C10.Q16, C20.44D4, D4⋊Dic5, C5×D4⋊C4, C4⋊C4⋊7D5, D5×C2×C8, C2×D4⋊2D5, D4⋊2D5⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, D10, C2×C22⋊C4, C4○D8, C4×D5, C22×D5, C23.24D4, C2×C4×D5, D4×D5, D5×C22⋊C4, D8⋊3D5, SD16⋊3D5, D4⋊2D5⋊C4
►On 160 points
Generators in S160
(1 16 6 11)(2 17 7 12)(3 18 8 13)(4 19 9 14)(5 20 10 15)(21 36 26 31)(22 37 27 32)(23 38 28 33)(24 39 29 34)(25 40 30 35)(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 91 86 96)(82 92 87 97)(83 93 88 98)(84 94 89 99)(85 95 90 100)(101 111 106 116)(102 112 107 117)(103 113 108 118)(104 114 109 119)(105 115 110 120)(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 81)(2 82)(3 83)(4 84)(5 85)(6 86)(7 87)(8 88)(9 89)(10 90)(11 91)(12 92)(13 93)(14 94)(15 95)(16 96)(17 97)(18 98)(19 99)(20 100)(21 101)(22 102)(23 103)(24 104)(25 105)(26 106)(27 107)(28 108)(29 109)(30 110)(31 111)(32 112)(33 113)(34 114)(35 115)(36 116)(37 117)(38 118)(39 119)(40 120)(41 121)(42 122)(43 123)(44 124)(45 125)(46 126)(47 127)(48 128)(49 129)(50 130)(51 131)(52 132)(53 133)(54 134)(55 135)(56 136)(57 137)(58 138)(59 139)(60 140)(61 141)(62 142)(63 143)(64 144)(65 145)(66 146)(67 147)(68 148)(69 149)(70 150)(71 151)(72 152)(73 153)(74 154)(75 155)(76 156)(77 157)(78 158)(79 159)(80 160)
(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 35)(2 34)(3 33)(4 32)(5 31)(6 40)(7 39)(8 38)(9 37)(10 36)(11 30)(12 29)(13 28)(14 27)(15 26)(16 25)(17 24)(18 23)(19 22)(20 21)(41 75)(42 74)(43 73)(44 72)(45 71)(46 80)(47 79)(48 78)(49 77)(50 76)(51 70)(52 69)(53 68)(54 67)(55 66)(56 65)(57 64)(58 63)(59 62)(60 61)(81 120)(82 119)(83 118)(84 117)(85 116)(86 115)(87 114)(88 113)(89 112)(90 111)(91 105)(92 104)(93 103)(94 102)(95 101)(96 110)(97 109)(98 108)(99 107)(100 106)(121 160)(122 159)(123 158)(124 157)(125 156)(126 155)(127 154)(128 153)(129 152)(130 151)(131 145)(132 144)(133 143)(134 142)(135 141)(136 150)(137 149)(138 148)(139 147)(140 146)
(1 151 31 121)(2 152 32 122)(3 153 33 123)(4 154 34 124)(5 155 35 125)(6 156 36 126)(7 157 37 127)(8 158 38 128)(9 159 39 129)(10 160 40 130)(11 146 26 131)(12 147 27 132)(13 148 28 133)(14 149 29 134)(15 150 30 135)(16 141 21 136)(17 142 22 137)(18 143 23 138)(19 144 24 139)(20 145 25 140)(41 96 71 101)(42 97 72 102)(43 98 73 103)(44 99 74 104)(45 100 75 105)(46 91 76 106)(47 92 77 107)(48 93 78 108)(49 94 79 109)(50 95 80 110)(51 81 66 111)(52 82 67 112)(53 83 68 113)(54 84 69 114)(55 85 70 115)(56 86 61 116)(57 87 62 117)(58 88 63 118)(59 89 64 119)(60 90 65 120)
G:=sub<Sym(160)| (1,16,6,11)(2,17,7,12)(3,18,8,13)(4,19,9,14)(5,20,10,15)(21,36,26,31)(22,37,27,32)(23,38,28,33)(24,39,29,34)(25,40,30,35)(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)(81,91,86,96)(82,92,87,97)(83,93,88,98)(84,94,89,99)(85,95,90,100)(101,111,106,116)(102,112,107,117)(103,113,108,118)(104,114,109,119)(105,115,110,120)(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), (1,81)(2,82)(3,83)(4,84)(5,85)(6,86)(7,87)(8,88)(9,89)(10,90)(11,91)(12,92)(13,93)(14,94)(15,95)(16,96)(17,97)(18,98)(19,99)(20,100)(21,101)(22,102)(23,103)(24,104)(25,105)(26,106)(27,107)(28,108)(29,109)(30,110)(31,111)(32,112)(33,113)(34,114)(35,115)(36,116)(37,117)(38,118)(39,119)(40,120)(41,121)(42,122)(43,123)(44,124)(45,125)(46,126)(47,127)(48,128)(49,129)(50,130)(51,131)(52,132)(53,133)(54,134)(55,135)(56,136)(57,137)(58,138)(59,139)(60,140)(61,141)(62,142)(63,143)(64,144)(65,145)(66,146)(67,147)(68,148)(69,149)(70,150)(71,151)(72,152)(73,153)(74,154)(75,155)(76,156)(77,157)(78,158)(79,159)(80,160), (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), (1,35)(2,34)(3,33)(4,32)(5,31)(6,40)(7,39)(8,38)(9,37)(10,36)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(41,75)(42,74)(43,73)(44,72)(45,71)(46,80)(47,79)(48,78)(49,77)(50,76)(51,70)(52,69)(53,68)(54,67)(55,66)(56,65)(57,64)(58,63)(59,62)(60,61)(81,120)(82,119)(83,118)(84,117)(85,116)(86,115)(87,114)(88,113)(89,112)(90,111)(91,105)(92,104)(93,103)(94,102)(95,101)(96,110)(97,109)(98,108)(99,107)(100,106)(121,160)(122,159)(123,158)(124,157)(125,156)(126,155)(127,154)(128,153)(129,152)(130,151)(131,145)(132,144)(133,143)(134,142)(135,141)(136,150)(137,149)(138,148)(139,147)(140,146), (1,151,31,121)(2,152,32,122)(3,153,33,123)(4,154,34,124)(5,155,35,125)(6,156,36,126)(7,157,37,127)(8,158,38,128)(9,159,39,129)(10,160,40,130)(11,146,26,131)(12,147,27,132)(13,148,28,133)(14,149,29,134)(15,150,30,135)(16,141,21,136)(17,142,22,137)(18,143,23,138)(19,144,24,139)(20,145,25,140)(41,96,71,101)(42,97,72,102)(43,98,73,103)(44,99,74,104)(45,100,75,105)(46,91,76,106)(47,92,77,107)(48,93,78,108)(49,94,79,109)(50,95,80,110)(51,81,66,111)(52,82,67,112)(53,83,68,113)(54,84,69,114)(55,85,70,115)(56,86,61,116)(57,87,62,117)(58,88,63,118)(59,89,64,119)(60,90,65,120)>;
G:=Group( (1,16,6,11)(2,17,7,12)(3,18,8,13)(4,19,9,14)(5,20,10,15)(21,36,26,31)(22,37,27,32)(23,38,28,33)(24,39,29,34)(25,40,30,35)(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)(81,91,86,96)(82,92,87,97)(83,93,88,98)(84,94,89,99)(85,95,90,100)(101,111,106,116)(102,112,107,117)(103,113,108,118)(104,114,109,119)(105,115,110,120)(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), (1,81)(2,82)(3,83)(4,84)(5,85)(6,86)(7,87)(8,88)(9,89)(10,90)(11,91)(12,92)(13,93)(14,94)(15,95)(16,96)(17,97)(18,98)(19,99)(20,100)(21,101)(22,102)(23,103)(24,104)(25,105)(26,106)(27,107)(28,108)(29,109)(30,110)(31,111)(32,112)(33,113)(34,114)(35,115)(36,116)(37,117)(38,118)(39,119)(40,120)(41,121)(42,122)(43,123)(44,124)(45,125)(46,126)(47,127)(48,128)(49,129)(50,130)(51,131)(52,132)(53,133)(54,134)(55,135)(56,136)(57,137)(58,138)(59,139)(60,140)(61,141)(62,142)(63,143)(64,144)(65,145)(66,146)(67,147)(68,148)(69,149)(70,150)(71,151)(72,152)(73,153)(74,154)(75,155)(76,156)(77,157)(78,158)(79,159)(80,160), (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), (1,35)(2,34)(3,33)(4,32)(5,31)(6,40)(7,39)(8,38)(9,37)(10,36)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(41,75)(42,74)(43,73)(44,72)(45,71)(46,80)(47,79)(48,78)(49,77)(50,76)(51,70)(52,69)(53,68)(54,67)(55,66)(56,65)(57,64)(58,63)(59,62)(60,61)(81,120)(82,119)(83,118)(84,117)(85,116)(86,115)(87,114)(88,113)(89,112)(90,111)(91,105)(92,104)(93,103)(94,102)(95,101)(96,110)(97,109)(98,108)(99,107)(100,106)(121,160)(122,159)(123,158)(124,157)(125,156)(126,155)(127,154)(128,153)(129,152)(130,151)(131,145)(132,144)(133,143)(134,142)(135,141)(136,150)(137,149)(138,148)(139,147)(140,146), (1,151,31,121)(2,152,32,122)(3,153,33,123)(4,154,34,124)(5,155,35,125)(6,156,36,126)(7,157,37,127)(8,158,38,128)(9,159,39,129)(10,160,40,130)(11,146,26,131)(12,147,27,132)(13,148,28,133)(14,149,29,134)(15,150,30,135)(16,141,21,136)(17,142,22,137)(18,143,23,138)(19,144,24,139)(20,145,25,140)(41,96,71,101)(42,97,72,102)(43,98,73,103)(44,99,74,104)(45,100,75,105)(46,91,76,106)(47,92,77,107)(48,93,78,108)(49,94,79,109)(50,95,80,110)(51,81,66,111)(52,82,67,112)(53,83,68,113)(54,84,69,114)(55,85,70,115)(56,86,61,116)(57,87,62,117)(58,88,63,118)(59,89,64,119)(60,90,65,120) );
G=PermutationGroup([[(1,16,6,11),(2,17,7,12),(3,18,8,13),(4,19,9,14),(5,20,10,15),(21,36,26,31),(22,37,27,32),(23,38,28,33),(24,39,29,34),(25,40,30,35),(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),(81,91,86,96),(82,92,87,97),(83,93,88,98),(84,94,89,99),(85,95,90,100),(101,111,106,116),(102,112,107,117),(103,113,108,118),(104,114,109,119),(105,115,110,120),(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)], [(1,81),(2,82),(3,83),(4,84),(5,85),(6,86),(7,87),(8,88),(9,89),(10,90),(11,91),(12,92),(13,93),(14,94),(15,95),(16,96),(17,97),(18,98),(19,99),(20,100),(21,101),(22,102),(23,103),(24,104),(25,105),(26,106),(27,107),(28,108),(29,109),(30,110),(31,111),(32,112),(33,113),(34,114),(35,115),(36,116),(37,117),(38,118),(39,119),(40,120),(41,121),(42,122),(43,123),(44,124),(45,125),(46,126),(47,127),(48,128),(49,129),(50,130),(51,131),(52,132),(53,133),(54,134),(55,135),(56,136),(57,137),(58,138),(59,139),(60,140),(61,141),(62,142),(63,143),(64,144),(65,145),(66,146),(67,147),(68,148),(69,149),(70,150),(71,151),(72,152),(73,153),(74,154),(75,155),(76,156),(77,157),(78,158),(79,159),(80,160)], [(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)], [(1,35),(2,34),(3,33),(4,32),(5,31),(6,40),(7,39),(8,38),(9,37),(10,36),(11,30),(12,29),(13,28),(14,27),(15,26),(16,25),(17,24),(18,23),(19,22),(20,21),(41,75),(42,74),(43,73),(44,72),(45,71),(46,80),(47,79),(48,78),(49,77),(50,76),(51,70),(52,69),(53,68),(54,67),(55,66),(56,65),(57,64),(58,63),(59,62),(60,61),(81,120),(82,119),(83,118),(84,117),(85,116),(86,115),(87,114),(88,113),(89,112),(90,111),(91,105),(92,104),(93,103),(94,102),(95,101),(96,110),(97,109),(98,108),(99,107),(100,106),(121,160),(122,159),(123,158),(124,157),(125,156),(126,155),(127,154),(128,153),(129,152),(130,151),(131,145),(132,144),(133,143),(134,142),(135,141),(136,150),(137,149),(138,148),(139,147),(140,146)], [(1,151,31,121),(2,152,32,122),(3,153,33,123),(4,154,34,124),(5,155,35,125),(6,156,36,126),(7,157,37,127),(8,158,38,128),(9,159,39,129),(10,160,40,130),(11,146,26,131),(12,147,27,132),(13,148,28,133),(14,149,29,134),(15,150,30,135),(16,141,21,136),(17,142,22,137),(18,143,23,138),(19,144,24,139),(20,145,25,140),(41,96,71,101),(42,97,72,102),(43,98,73,103),(44,99,74,104),(45,100,75,105),(46,91,76,106),(47,92,77,107),(48,93,78,108),(49,94,79,109),(50,95,80,110),(51,81,66,111),(52,82,67,112),(53,83,68,113),(54,84,69,114),(55,85,70,115),(56,86,61,116),(57,87,62,117),(58,88,63,118),(59,89,64,119),(60,90,65,120)]])
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 | 10G | 10H | 10I | 10J | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 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 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 10 | 10 | 2 | 2 | 4 | 4 | 5 | 5 | 5 | 5 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 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 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D4 | D5 | D10 | D10 | D10 | C4○D8 | C4×D5 | D4×D5 | D4×D5 | D8⋊3D5 | SD16⋊3D5 |
| kernel | D4⋊2D5⋊C4 | C10.Q16 | C20.44D4 | D4⋊Dic5 | C5×D4⋊C4 | C4⋊C4⋊7D5 | D5×C2×C8 | C2×D4⋊2D5 | D4⋊2D5 | C4×D5 | C2×Dic5 | C22×D5 | D4⋊C4 | C4⋊C4 | C2×C8 | C2×D4 | C10 | D4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 8 | 8 | 2 | 2 | 4 | 4 |
Matrix representation of D4⋊2D5⋊C4 ►in GL4(𝔽41) generated by
| 32 | 2 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 9 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 0 | 1 | 6 |
| 40 | 23 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 6 | 1 |
| 0 | 0 | 6 | 35 |
| 38 | 11 | 0 | 0 |
| 14 | 3 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
G:=sub<GL(4,GF(41))| [32,0,0,0,2,9,0,0,0,0,1,0,0,0,0,1],[1,9,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,40,6],[40,0,0,0,23,1,0,0,0,0,6,6,0,0,1,35],[38,14,0,0,11,3,0,0,0,0,9,0,0,0,0,9] >;
D4⋊2D5⋊C4 in GAP, Magma, Sage, TeX
D_4\rtimes_2D_5\rtimes C_4
% in TeX
G:=Group("D4:2D5:C4"); // GroupNames label
G:=SmallGroup(320,399);
// by ID
G=gap.SmallGroup(320,399);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,219,58,570,136,851,102,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^2=c^5=d^2=e^4=1,b*a*b=e*a*e^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^2*b,e*b*e^-1=a*b,d*c*d=c^-1,c*e=e*c,e*d*e^-1=a^2*d>;
// generators/relations