direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×D4⋊D4, D4⋊3(C5×D4), Q8⋊3(C5×D4), (C2×D8)⋊2C10, (C5×D4)⋊21D4, (C5×Q8)⋊21D4, (C10×D8)⋊16C2, C4⋊D4⋊2C10, C22⋊C8⋊5C10, C4.23(D4×C10), D4⋊C4⋊8C10, Q8⋊C4⋊4C10, (C2×SD16)⋊8C10, C20.384(C2×D4), (C2×C20).459D4, C23.13(C5×D4), C10.96C22≀C2, (C10×SD16)⋊25C2, C22.79(D4×C10), (C22×C10).31D4, C10.119(C4○D8), (C2×C20).914C23, (C2×C40).298C22, C10.132(C8⋊C22), (D4×C10).294C22, (Q8×C10).259C22, (C22×C20).421C22, C2.6(C5×C4○D8), (C2×C4○D4)⋊1C10, C4⋊C4.2(C2×C10), (C2×C8).1(C2×C10), C2.7(C5×C8⋊C22), (C10×C4○D4)⋊17C2, (C5×C4⋊D4)⋊29C2, (C5×C22⋊C8)⋊22C2, (C2×C4).105(C5×D4), (C5×D4⋊C4)⋊32C2, (C5×Q8⋊C4)⋊27C2, C2.10(C5×C22≀C2), (C2×D4).52(C2×C10), (C2×C10).635(C2×D4), (C2×Q8).44(C2×C10), (C5×C4⋊C4).224C22, (C22×C4).39(C2×C10), (C2×C4).89(C22×C10), SmallGroup(320,950)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×D4⋊D4
G = < a,b,c,d,e | a5=b4=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=ebe=b-1, dcd-1=b-1c, ece=bc, ede=d-1 >
Subgroups: 322 in 162 conjugacy classes, 58 normal (50 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C10, C10, C22⋊C4, C4⋊C4, C2×C8, D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C20, C20, C2×C10, C2×C10, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊D4, C2×D8, C2×SD16, C2×C4○D4, C40, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, C22×C10, C22×C10, D4⋊D4, C5×C22⋊C4, C5×C4⋊C4, C2×C40, C5×D8, C5×SD16, C22×C20, C22×C20, D4×C10, D4×C10, Q8×C10, C5×C4○D4, C5×C22⋊C8, C5×D4⋊C4, C5×Q8⋊C4, C5×C4⋊D4, C10×D8, C10×SD16, C10×C4○D4, C5×D4⋊D4
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C2×C10, C22≀C2, C4○D8, C8⋊C22, C5×D4, C22×C10, D4⋊D4, D4×C10, C5×C22≀C2, C5×C4○D8, 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 54 12 45)(2 55 13 41)(3 51 14 42)(4 52 15 43)(5 53 11 44)(6 140 144 19)(7 136 145 20)(8 137 141 16)(9 138 142 17)(10 139 143 18)(21 155 159 34)(22 151 160 35)(23 152 156 31)(24 153 157 32)(25 154 158 33)(26 69 50 40)(27 70 46 36)(28 66 47 37)(29 67 48 38)(30 68 49 39)(56 81 94 65)(57 82 95 61)(58 83 91 62)(59 84 92 63)(60 85 93 64)(71 109 90 80)(72 110 86 76)(73 106 87 77)(74 107 88 78)(75 108 89 79)(96 105 134 121)(97 101 135 122)(98 102 131 123)(99 103 132 124)(100 104 133 125)(111 120 130 149)(112 116 126 150)(113 117 127 146)(114 118 128 147)(115 119 129 148)
(1 127)(2 128)(3 129)(4 130)(5 126)(6 72)(7 73)(8 74)(9 75)(10 71)(11 112)(12 113)(13 114)(14 115)(15 111)(16 107)(17 108)(18 109)(19 110)(20 106)(21 58)(22 59)(23 60)(24 56)(25 57)(26 124)(27 125)(28 121)(29 122)(30 123)(31 85)(32 81)(33 82)(34 83)(35 84)(36 100)(37 96)(38 97)(39 98)(40 99)(41 147)(42 148)(43 149)(44 150)(45 146)(46 104)(47 105)(48 101)(49 102)(50 103)(51 119)(52 120)(53 116)(54 117)(55 118)(61 154)(62 155)(63 151)(64 152)(65 153)(66 134)(67 135)(68 131)(69 132)(70 133)(76 140)(77 136)(78 137)(79 138)(80 139)(86 144)(87 145)(88 141)(89 142)(90 143)(91 159)(92 160)(93 156)(94 157)(95 158)
(1 56 37 87)(2 57 38 88)(3 58 39 89)(4 59 40 90)(5 60 36 86)(6 116 156 125)(7 117 157 121)(8 118 158 122)(9 119 159 123)(10 120 160 124)(11 93 70 72)(12 94 66 73)(13 95 67 74)(14 91 68 75)(15 92 69 71)(16 128 154 97)(17 129 155 98)(18 130 151 99)(19 126 152 100)(20 127 153 96)(21 102 142 148)(22 103 143 149)(23 104 144 150)(24 105 145 146)(25 101 141 147)(26 109 52 63)(27 110 53 64)(28 106 54 65)(29 107 55 61)(30 108 51 62)(31 133 140 112)(32 134 136 113)(33 135 137 114)(34 131 138 115)(35 132 139 111)(41 82 48 78)(42 83 49 79)(43 84 50 80)(44 85 46 76)(45 81 47 77)
(6 152)(7 153)(8 154)(9 155)(10 151)(16 158)(17 159)(18 160)(19 156)(20 157)(21 138)(22 139)(23 140)(24 136)(25 137)(26 50)(27 46)(28 47)(29 48)(30 49)(31 144)(32 145)(33 141)(34 142)(35 143)(41 55)(42 51)(43 52)(44 53)(45 54)(56 87)(57 88)(58 89)(59 90)(60 86)(61 78)(62 79)(63 80)(64 76)(65 77)(71 92)(72 93)(73 94)(74 95)(75 91)(81 106)(82 107)(83 108)(84 109)(85 110)(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)
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), (1,54,12,45)(2,55,13,41)(3,51,14,42)(4,52,15,43)(5,53,11,44)(6,140,144,19)(7,136,145,20)(8,137,141,16)(9,138,142,17)(10,139,143,18)(21,155,159,34)(22,151,160,35)(23,152,156,31)(24,153,157,32)(25,154,158,33)(26,69,50,40)(27,70,46,36)(28,66,47,37)(29,67,48,38)(30,68,49,39)(56,81,94,65)(57,82,95,61)(58,83,91,62)(59,84,92,63)(60,85,93,64)(71,109,90,80)(72,110,86,76)(73,106,87,77)(74,107,88,78)(75,108,89,79)(96,105,134,121)(97,101,135,122)(98,102,131,123)(99,103,132,124)(100,104,133,125)(111,120,130,149)(112,116,126,150)(113,117,127,146)(114,118,128,147)(115,119,129,148), (1,127)(2,128)(3,129)(4,130)(5,126)(6,72)(7,73)(8,74)(9,75)(10,71)(11,112)(12,113)(13,114)(14,115)(15,111)(16,107)(17,108)(18,109)(19,110)(20,106)(21,58)(22,59)(23,60)(24,56)(25,57)(26,124)(27,125)(28,121)(29,122)(30,123)(31,85)(32,81)(33,82)(34,83)(35,84)(36,100)(37,96)(38,97)(39,98)(40,99)(41,147)(42,148)(43,149)(44,150)(45,146)(46,104)(47,105)(48,101)(49,102)(50,103)(51,119)(52,120)(53,116)(54,117)(55,118)(61,154)(62,155)(63,151)(64,152)(65,153)(66,134)(67,135)(68,131)(69,132)(70,133)(76,140)(77,136)(78,137)(79,138)(80,139)(86,144)(87,145)(88,141)(89,142)(90,143)(91,159)(92,160)(93,156)(94,157)(95,158), (1,56,37,87)(2,57,38,88)(3,58,39,89)(4,59,40,90)(5,60,36,86)(6,116,156,125)(7,117,157,121)(8,118,158,122)(9,119,159,123)(10,120,160,124)(11,93,70,72)(12,94,66,73)(13,95,67,74)(14,91,68,75)(15,92,69,71)(16,128,154,97)(17,129,155,98)(18,130,151,99)(19,126,152,100)(20,127,153,96)(21,102,142,148)(22,103,143,149)(23,104,144,150)(24,105,145,146)(25,101,141,147)(26,109,52,63)(27,110,53,64)(28,106,54,65)(29,107,55,61)(30,108,51,62)(31,133,140,112)(32,134,136,113)(33,135,137,114)(34,131,138,115)(35,132,139,111)(41,82,48,78)(42,83,49,79)(43,84,50,80)(44,85,46,76)(45,81,47,77), (6,152)(7,153)(8,154)(9,155)(10,151)(16,158)(17,159)(18,160)(19,156)(20,157)(21,138)(22,139)(23,140)(24,136)(25,137)(26,50)(27,46)(28,47)(29,48)(30,49)(31,144)(32,145)(33,141)(34,142)(35,143)(41,55)(42,51)(43,52)(44,53)(45,54)(56,87)(57,88)(58,89)(59,90)(60,86)(61,78)(62,79)(63,80)(64,76)(65,77)(71,92)(72,93)(73,94)(74,95)(75,91)(81,106)(82,107)(83,108)(84,109)(85,110)(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)>;
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), (1,54,12,45)(2,55,13,41)(3,51,14,42)(4,52,15,43)(5,53,11,44)(6,140,144,19)(7,136,145,20)(8,137,141,16)(9,138,142,17)(10,139,143,18)(21,155,159,34)(22,151,160,35)(23,152,156,31)(24,153,157,32)(25,154,158,33)(26,69,50,40)(27,70,46,36)(28,66,47,37)(29,67,48,38)(30,68,49,39)(56,81,94,65)(57,82,95,61)(58,83,91,62)(59,84,92,63)(60,85,93,64)(71,109,90,80)(72,110,86,76)(73,106,87,77)(74,107,88,78)(75,108,89,79)(96,105,134,121)(97,101,135,122)(98,102,131,123)(99,103,132,124)(100,104,133,125)(111,120,130,149)(112,116,126,150)(113,117,127,146)(114,118,128,147)(115,119,129,148), (1,127)(2,128)(3,129)(4,130)(5,126)(6,72)(7,73)(8,74)(9,75)(10,71)(11,112)(12,113)(13,114)(14,115)(15,111)(16,107)(17,108)(18,109)(19,110)(20,106)(21,58)(22,59)(23,60)(24,56)(25,57)(26,124)(27,125)(28,121)(29,122)(30,123)(31,85)(32,81)(33,82)(34,83)(35,84)(36,100)(37,96)(38,97)(39,98)(40,99)(41,147)(42,148)(43,149)(44,150)(45,146)(46,104)(47,105)(48,101)(49,102)(50,103)(51,119)(52,120)(53,116)(54,117)(55,118)(61,154)(62,155)(63,151)(64,152)(65,153)(66,134)(67,135)(68,131)(69,132)(70,133)(76,140)(77,136)(78,137)(79,138)(80,139)(86,144)(87,145)(88,141)(89,142)(90,143)(91,159)(92,160)(93,156)(94,157)(95,158), (1,56,37,87)(2,57,38,88)(3,58,39,89)(4,59,40,90)(5,60,36,86)(6,116,156,125)(7,117,157,121)(8,118,158,122)(9,119,159,123)(10,120,160,124)(11,93,70,72)(12,94,66,73)(13,95,67,74)(14,91,68,75)(15,92,69,71)(16,128,154,97)(17,129,155,98)(18,130,151,99)(19,126,152,100)(20,127,153,96)(21,102,142,148)(22,103,143,149)(23,104,144,150)(24,105,145,146)(25,101,141,147)(26,109,52,63)(27,110,53,64)(28,106,54,65)(29,107,55,61)(30,108,51,62)(31,133,140,112)(32,134,136,113)(33,135,137,114)(34,131,138,115)(35,132,139,111)(41,82,48,78)(42,83,49,79)(43,84,50,80)(44,85,46,76)(45,81,47,77), (6,152)(7,153)(8,154)(9,155)(10,151)(16,158)(17,159)(18,160)(19,156)(20,157)(21,138)(22,139)(23,140)(24,136)(25,137)(26,50)(27,46)(28,47)(29,48)(30,49)(31,144)(32,145)(33,141)(34,142)(35,143)(41,55)(42,51)(43,52)(44,53)(45,54)(56,87)(57,88)(58,89)(59,90)(60,86)(61,78)(62,79)(63,80)(64,76)(65,77)(71,92)(72,93)(73,94)(74,95)(75,91)(81,106)(82,107)(83,108)(84,109)(85,110)(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) );
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)], [(1,54,12,45),(2,55,13,41),(3,51,14,42),(4,52,15,43),(5,53,11,44),(6,140,144,19),(7,136,145,20),(8,137,141,16),(9,138,142,17),(10,139,143,18),(21,155,159,34),(22,151,160,35),(23,152,156,31),(24,153,157,32),(25,154,158,33),(26,69,50,40),(27,70,46,36),(28,66,47,37),(29,67,48,38),(30,68,49,39),(56,81,94,65),(57,82,95,61),(58,83,91,62),(59,84,92,63),(60,85,93,64),(71,109,90,80),(72,110,86,76),(73,106,87,77),(74,107,88,78),(75,108,89,79),(96,105,134,121),(97,101,135,122),(98,102,131,123),(99,103,132,124),(100,104,133,125),(111,120,130,149),(112,116,126,150),(113,117,127,146),(114,118,128,147),(115,119,129,148)], [(1,127),(2,128),(3,129),(4,130),(5,126),(6,72),(7,73),(8,74),(9,75),(10,71),(11,112),(12,113),(13,114),(14,115),(15,111),(16,107),(17,108),(18,109),(19,110),(20,106),(21,58),(22,59),(23,60),(24,56),(25,57),(26,124),(27,125),(28,121),(29,122),(30,123),(31,85),(32,81),(33,82),(34,83),(35,84),(36,100),(37,96),(38,97),(39,98),(40,99),(41,147),(42,148),(43,149),(44,150),(45,146),(46,104),(47,105),(48,101),(49,102),(50,103),(51,119),(52,120),(53,116),(54,117),(55,118),(61,154),(62,155),(63,151),(64,152),(65,153),(66,134),(67,135),(68,131),(69,132),(70,133),(76,140),(77,136),(78,137),(79,138),(80,139),(86,144),(87,145),(88,141),(89,142),(90,143),(91,159),(92,160),(93,156),(94,157),(95,158)], [(1,56,37,87),(2,57,38,88),(3,58,39,89),(4,59,40,90),(5,60,36,86),(6,116,156,125),(7,117,157,121),(8,118,158,122),(9,119,159,123),(10,120,160,124),(11,93,70,72),(12,94,66,73),(13,95,67,74),(14,91,68,75),(15,92,69,71),(16,128,154,97),(17,129,155,98),(18,130,151,99),(19,126,152,100),(20,127,153,96),(21,102,142,148),(22,103,143,149),(23,104,144,150),(24,105,145,146),(25,101,141,147),(26,109,52,63),(27,110,53,64),(28,106,54,65),(29,107,55,61),(30,108,51,62),(31,133,140,112),(32,134,136,113),(33,135,137,114),(34,131,138,115),(35,132,139,111),(41,82,48,78),(42,83,49,79),(43,84,50,80),(44,85,46,76),(45,81,47,77)], [(6,152),(7,153),(8,154),(9,155),(10,151),(16,158),(17,159),(18,160),(19,156),(20,157),(21,138),(22,139),(23,140),(24,136),(25,137),(26,50),(27,46),(28,47),(29,48),(30,49),(31,144),(32,145),(33,141),(34,142),(35,143),(41,55),(42,51),(43,52),(44,53),(45,54),(56,87),(57,88),(58,89),(59,90),(60,86),(61,78),(62,79),(63,80),(64,76),(65,77),(71,92),(72,93),(73,94),(74,95),(75,91),(81,106),(82,107),(83,108),(84,109),(85,110),(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)]])
95 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 10A | ··· | 10L | 10M | ··· | 10X | 10Y | 10Z | 10AA | 10AB | 20A | ··· | 20P | 20Q | ··· | 20X | 20Y | 20Z | 20AA | 20AB | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
95 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | C10 | C10 | D4 | D4 | D4 | D4 | C4○D8 | C5×D4 | C5×D4 | C5×D4 | C5×D4 | C5×C4○D8 | C8⋊C22 | C5×C8⋊C22 |
| kernel | C5×D4⋊D4 | C5×C22⋊C8 | C5×D4⋊C4 | C5×Q8⋊C4 | C5×C4⋊D4 | C10×D8 | C10×SD16 | C10×C4○D4 | D4⋊D4 | C22⋊C8 | D4⋊C4 | Q8⋊C4 | C4⋊D4 | C2×D8 | C2×SD16 | C2×C4○D4 | C2×C20 | C5×D4 | C5×Q8 | C22×C10 | C10 | C2×C4 | D4 | Q8 | C23 | C2 | C10 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 1 | 2 | 2 | 1 | 4 | 4 | 8 | 8 | 4 | 16 | 1 | 4 |
Matrix representation of C5×D4⋊D4 ►in GL6(𝔽41)
| 10 | 0 | 0 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 34 | 4 | 0 | 0 | 0 | 0 |
| 29 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 29 | 0 | 0 |
| 0 | 0 | 29 | 29 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 2 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 24 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 39 |
| 0 | 0 | 0 | 0 | 1 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 24 | 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 |
G:=sub<GL(6,GF(41))| [10,0,0,0,0,0,0,10,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[34,29,0,0,0,0,4,7,0,0,0,0,0,0,12,29,0,0,0,0,29,29,0,0,0,0,0,0,40,0,0,0,0,0,2,1],[1,24,0,0,0,0,0,40,0,0,0,0,0,0,0,9,0,0,0,0,9,0,0,0,0,0,0,0,1,1,0,0,0,0,39,40],[1,24,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,1,0,0,0,0,0,40] >;
C5×D4⋊D4 in GAP, Magma, Sage, TeX
C_5\times D_4\rtimes D_4
% in TeX
G:=Group("C5xD4:D4"); // GroupNames label
G:=SmallGroup(320,950);
// by ID
G=gap.SmallGroup(320,950);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,589,1766,856,7004,3511,172]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^4=c^2=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=d*b*d^-1=e*b*e=b^-1,d*c*d^-1=b^-1*c,e*c*e=b*c,e*d*e=d^-1>;
// generators/relations