direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×C22.D8, C2.D8⋊6C10, C2.8(C10×D8), C22⋊C8⋊4C10, C10.80(C2×D8), (C2×C10).26D8, D4⋊C4⋊7C10, C22.4(C5×D8), (C2×C20).336D4, C4⋊D4.5C10, C23.45(C5×D4), C20.317(C4○D4), (C2×C40).262C22, (C2×C20).936C23, (C22×C10).167D4, C22.101(D4×C10), (D4×C10).195C22, C10.142(C8.C22), (C22×C20).428C22, C10.95(C22.D4), (C10×C4⋊C4)⋊38C2, (C2×C4⋊C4)⋊11C10, (C2×C8).9(C2×C10), (C5×C2.D8)⋊21C2, C4.29(C5×C4○D4), (C2×C4).37(C5×D4), C4⋊C4.57(C2×C10), (C5×C22⋊C8)⋊21C2, (C5×D4⋊C4)⋊30C2, (C2×D4).18(C2×C10), (C5×C4⋊D4).15C2, (C2×C10).657(C2×D4), C2.17(C5×C8.C22), (C5×C4⋊C4).380C22, (C22×C4).46(C2×C10), (C2×C4).111(C22×C10), C2.11(C5×C22.D4), SmallGroup(320,981)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C22.D8
G = < a,b,c,d,e | a5=b2=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe=bc=cb, cd=dc, ce=ec, ede=cd-1 >
Subgroups: 226 in 114 conjugacy classes, 54 normal (30 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, C23, C10, C10, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C20, C20, C2×C10, C2×C10, C2×C10, C22⋊C8, D4⋊C4, C2.D8, C2×C4⋊C4, C4⋊D4, C40, C2×C20, C2×C20, C5×D4, C22×C10, C22×C10, C22.D8, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C5×C4⋊C4, C2×C40, C22×C20, C22×C20, D4×C10, D4×C10, C5×C22⋊C8, C5×D4⋊C4, C5×C2.D8, C10×C4⋊C4, C5×C4⋊D4, C5×C22.D8
Quotients: C1, C2, C22, C5, D4, C23, C10, D8, C2×D4, C4○D4, C2×C10, C22.D4, C2×D8, C8.C22, C5×D4, C22×C10, C22.D8, C5×D8, D4×C10, C5×C4○D4, C5×C22.D4, C10×D8, C5×C8.C22, C5×C22.D8
►On 160 points
Generators in S160
(1 58 24 50 16)(2 59 17 51 9)(3 60 18 52 10)(4 61 19 53 11)(5 62 20 54 12)(6 63 21 55 13)(7 64 22 56 14)(8 57 23 49 15)(25 85 75 33 67)(26 86 76 34 68)(27 87 77 35 69)(28 88 78 36 70)(29 81 79 37 71)(30 82 80 38 72)(31 83 73 39 65)(32 84 74 40 66)(41 156 118 148 110)(42 157 119 149 111)(43 158 120 150 112)(44 159 113 151 105)(45 160 114 152 106)(46 153 115 145 107)(47 154 116 146 108)(48 155 117 147 109)(89 124 143 97 135)(90 125 144 98 136)(91 126 137 99 129)(92 127 138 100 130)(93 128 139 101 131)(94 121 140 102 132)(95 122 141 103 133)(96 123 142 104 134)
(1 107)(2 96)(3 109)(4 90)(5 111)(6 92)(7 105)(8 94)(9 134)(10 147)(11 136)(12 149)(13 130)(14 151)(15 132)(16 145)(17 142)(18 155)(19 144)(20 157)(21 138)(22 159)(23 140)(24 153)(25 93)(26 106)(27 95)(28 108)(29 89)(30 110)(31 91)(32 112)(33 101)(34 114)(35 103)(36 116)(37 97)(38 118)(39 99)(40 120)(41 82)(42 62)(43 84)(44 64)(45 86)(46 58)(47 88)(48 60)(49 102)(50 115)(51 104)(52 117)(53 98)(54 119)(55 100)(56 113)(57 121)(59 123)(61 125)(63 127)(65 129)(66 150)(67 131)(68 152)(69 133)(70 146)(71 135)(72 148)(73 137)(74 158)(75 139)(76 160)(77 141)(78 154)(79 143)(80 156)(81 124)(83 126)(85 128)(87 122)
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 25)(8 26)(9 70)(10 71)(11 72)(12 65)(13 66)(14 67)(15 68)(16 69)(17 78)(18 79)(19 80)(20 73)(21 74)(22 75)(23 76)(24 77)(33 56)(34 49)(35 50)(36 51)(37 52)(38 53)(39 54)(40 55)(41 125)(42 126)(43 127)(44 128)(45 121)(46 122)(47 123)(48 124)(57 86)(58 87)(59 88)(60 81)(61 82)(62 83)(63 84)(64 85)(89 109)(90 110)(91 111)(92 112)(93 105)(94 106)(95 107)(96 108)(97 117)(98 118)(99 119)(100 120)(101 113)(102 114)(103 115)(104 116)(129 149)(130 150)(131 151)(132 152)(133 145)(134 146)(135 147)(136 148)(137 157)(138 158)(139 159)(140 160)(141 153)(142 154)(143 155)(144 156)
(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)
(2 26)(3 7)(4 32)(6 30)(8 28)(9 68)(10 14)(11 66)(13 72)(15 70)(17 76)(18 22)(19 74)(21 80)(23 78)(25 29)(33 37)(34 51)(36 49)(38 55)(40 53)(41 43)(42 126)(44 124)(45 47)(46 122)(48 128)(52 56)(57 88)(59 86)(60 64)(61 84)(63 82)(67 71)(75 79)(81 85)(89 105)(90 92)(91 111)(93 109)(94 96)(95 107)(97 113)(98 100)(99 119)(101 117)(102 104)(103 115)(106 108)(110 112)(114 116)(118 120)(121 123)(125 127)(129 149)(130 136)(131 147)(132 134)(133 145)(135 151)(137 157)(138 144)(139 155)(140 142)(141 153)(143 159)(146 152)(148 150)(154 160)(156 158)
G:=sub<Sym(160)| (1,58,24,50,16)(2,59,17,51,9)(3,60,18,52,10)(4,61,19,53,11)(5,62,20,54,12)(6,63,21,55,13)(7,64,22,56,14)(8,57,23,49,15)(25,85,75,33,67)(26,86,76,34,68)(27,87,77,35,69)(28,88,78,36,70)(29,81,79,37,71)(30,82,80,38,72)(31,83,73,39,65)(32,84,74,40,66)(41,156,118,148,110)(42,157,119,149,111)(43,158,120,150,112)(44,159,113,151,105)(45,160,114,152,106)(46,153,115,145,107)(47,154,116,146,108)(48,155,117,147,109)(89,124,143,97,135)(90,125,144,98,136)(91,126,137,99,129)(92,127,138,100,130)(93,128,139,101,131)(94,121,140,102,132)(95,122,141,103,133)(96,123,142,104,134), (1,107)(2,96)(3,109)(4,90)(5,111)(6,92)(7,105)(8,94)(9,134)(10,147)(11,136)(12,149)(13,130)(14,151)(15,132)(16,145)(17,142)(18,155)(19,144)(20,157)(21,138)(22,159)(23,140)(24,153)(25,93)(26,106)(27,95)(28,108)(29,89)(30,110)(31,91)(32,112)(33,101)(34,114)(35,103)(36,116)(37,97)(38,118)(39,99)(40,120)(41,82)(42,62)(43,84)(44,64)(45,86)(46,58)(47,88)(48,60)(49,102)(50,115)(51,104)(52,117)(53,98)(54,119)(55,100)(56,113)(57,121)(59,123)(61,125)(63,127)(65,129)(66,150)(67,131)(68,152)(69,133)(70,146)(71,135)(72,148)(73,137)(74,158)(75,139)(76,160)(77,141)(78,154)(79,143)(80,156)(81,124)(83,126)(85,128)(87,122), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,25)(8,26)(9,70)(10,71)(11,72)(12,65)(13,66)(14,67)(15,68)(16,69)(17,78)(18,79)(19,80)(20,73)(21,74)(22,75)(23,76)(24,77)(33,56)(34,49)(35,50)(36,51)(37,52)(38,53)(39,54)(40,55)(41,125)(42,126)(43,127)(44,128)(45,121)(46,122)(47,123)(48,124)(57,86)(58,87)(59,88)(60,81)(61,82)(62,83)(63,84)(64,85)(89,109)(90,110)(91,111)(92,112)(93,105)(94,106)(95,107)(96,108)(97,117)(98,118)(99,119)(100,120)(101,113)(102,114)(103,115)(104,116)(129,149)(130,150)(131,151)(132,152)(133,145)(134,146)(135,147)(136,148)(137,157)(138,158)(139,159)(140,160)(141,153)(142,154)(143,155)(144,156), (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), (2,26)(3,7)(4,32)(6,30)(8,28)(9,68)(10,14)(11,66)(13,72)(15,70)(17,76)(18,22)(19,74)(21,80)(23,78)(25,29)(33,37)(34,51)(36,49)(38,55)(40,53)(41,43)(42,126)(44,124)(45,47)(46,122)(48,128)(52,56)(57,88)(59,86)(60,64)(61,84)(63,82)(67,71)(75,79)(81,85)(89,105)(90,92)(91,111)(93,109)(94,96)(95,107)(97,113)(98,100)(99,119)(101,117)(102,104)(103,115)(106,108)(110,112)(114,116)(118,120)(121,123)(125,127)(129,149)(130,136)(131,147)(132,134)(133,145)(135,151)(137,157)(138,144)(139,155)(140,142)(141,153)(143,159)(146,152)(148,150)(154,160)(156,158)>;
G:=Group( (1,58,24,50,16)(2,59,17,51,9)(3,60,18,52,10)(4,61,19,53,11)(5,62,20,54,12)(6,63,21,55,13)(7,64,22,56,14)(8,57,23,49,15)(25,85,75,33,67)(26,86,76,34,68)(27,87,77,35,69)(28,88,78,36,70)(29,81,79,37,71)(30,82,80,38,72)(31,83,73,39,65)(32,84,74,40,66)(41,156,118,148,110)(42,157,119,149,111)(43,158,120,150,112)(44,159,113,151,105)(45,160,114,152,106)(46,153,115,145,107)(47,154,116,146,108)(48,155,117,147,109)(89,124,143,97,135)(90,125,144,98,136)(91,126,137,99,129)(92,127,138,100,130)(93,128,139,101,131)(94,121,140,102,132)(95,122,141,103,133)(96,123,142,104,134), (1,107)(2,96)(3,109)(4,90)(5,111)(6,92)(7,105)(8,94)(9,134)(10,147)(11,136)(12,149)(13,130)(14,151)(15,132)(16,145)(17,142)(18,155)(19,144)(20,157)(21,138)(22,159)(23,140)(24,153)(25,93)(26,106)(27,95)(28,108)(29,89)(30,110)(31,91)(32,112)(33,101)(34,114)(35,103)(36,116)(37,97)(38,118)(39,99)(40,120)(41,82)(42,62)(43,84)(44,64)(45,86)(46,58)(47,88)(48,60)(49,102)(50,115)(51,104)(52,117)(53,98)(54,119)(55,100)(56,113)(57,121)(59,123)(61,125)(63,127)(65,129)(66,150)(67,131)(68,152)(69,133)(70,146)(71,135)(72,148)(73,137)(74,158)(75,139)(76,160)(77,141)(78,154)(79,143)(80,156)(81,124)(83,126)(85,128)(87,122), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,25)(8,26)(9,70)(10,71)(11,72)(12,65)(13,66)(14,67)(15,68)(16,69)(17,78)(18,79)(19,80)(20,73)(21,74)(22,75)(23,76)(24,77)(33,56)(34,49)(35,50)(36,51)(37,52)(38,53)(39,54)(40,55)(41,125)(42,126)(43,127)(44,128)(45,121)(46,122)(47,123)(48,124)(57,86)(58,87)(59,88)(60,81)(61,82)(62,83)(63,84)(64,85)(89,109)(90,110)(91,111)(92,112)(93,105)(94,106)(95,107)(96,108)(97,117)(98,118)(99,119)(100,120)(101,113)(102,114)(103,115)(104,116)(129,149)(130,150)(131,151)(132,152)(133,145)(134,146)(135,147)(136,148)(137,157)(138,158)(139,159)(140,160)(141,153)(142,154)(143,155)(144,156), (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), (2,26)(3,7)(4,32)(6,30)(8,28)(9,68)(10,14)(11,66)(13,72)(15,70)(17,76)(18,22)(19,74)(21,80)(23,78)(25,29)(33,37)(34,51)(36,49)(38,55)(40,53)(41,43)(42,126)(44,124)(45,47)(46,122)(48,128)(52,56)(57,88)(59,86)(60,64)(61,84)(63,82)(67,71)(75,79)(81,85)(89,105)(90,92)(91,111)(93,109)(94,96)(95,107)(97,113)(98,100)(99,119)(101,117)(102,104)(103,115)(106,108)(110,112)(114,116)(118,120)(121,123)(125,127)(129,149)(130,136)(131,147)(132,134)(133,145)(135,151)(137,157)(138,144)(139,155)(140,142)(141,153)(143,159)(146,152)(148,150)(154,160)(156,158) );
G=PermutationGroup([[(1,58,24,50,16),(2,59,17,51,9),(3,60,18,52,10),(4,61,19,53,11),(5,62,20,54,12),(6,63,21,55,13),(7,64,22,56,14),(8,57,23,49,15),(25,85,75,33,67),(26,86,76,34,68),(27,87,77,35,69),(28,88,78,36,70),(29,81,79,37,71),(30,82,80,38,72),(31,83,73,39,65),(32,84,74,40,66),(41,156,118,148,110),(42,157,119,149,111),(43,158,120,150,112),(44,159,113,151,105),(45,160,114,152,106),(46,153,115,145,107),(47,154,116,146,108),(48,155,117,147,109),(89,124,143,97,135),(90,125,144,98,136),(91,126,137,99,129),(92,127,138,100,130),(93,128,139,101,131),(94,121,140,102,132),(95,122,141,103,133),(96,123,142,104,134)], [(1,107),(2,96),(3,109),(4,90),(5,111),(6,92),(7,105),(8,94),(9,134),(10,147),(11,136),(12,149),(13,130),(14,151),(15,132),(16,145),(17,142),(18,155),(19,144),(20,157),(21,138),(22,159),(23,140),(24,153),(25,93),(26,106),(27,95),(28,108),(29,89),(30,110),(31,91),(32,112),(33,101),(34,114),(35,103),(36,116),(37,97),(38,118),(39,99),(40,120),(41,82),(42,62),(43,84),(44,64),(45,86),(46,58),(47,88),(48,60),(49,102),(50,115),(51,104),(52,117),(53,98),(54,119),(55,100),(56,113),(57,121),(59,123),(61,125),(63,127),(65,129),(66,150),(67,131),(68,152),(69,133),(70,146),(71,135),(72,148),(73,137),(74,158),(75,139),(76,160),(77,141),(78,154),(79,143),(80,156),(81,124),(83,126),(85,128),(87,122)], [(1,27),(2,28),(3,29),(4,30),(5,31),(6,32),(7,25),(8,26),(9,70),(10,71),(11,72),(12,65),(13,66),(14,67),(15,68),(16,69),(17,78),(18,79),(19,80),(20,73),(21,74),(22,75),(23,76),(24,77),(33,56),(34,49),(35,50),(36,51),(37,52),(38,53),(39,54),(40,55),(41,125),(42,126),(43,127),(44,128),(45,121),(46,122),(47,123),(48,124),(57,86),(58,87),(59,88),(60,81),(61,82),(62,83),(63,84),(64,85),(89,109),(90,110),(91,111),(92,112),(93,105),(94,106),(95,107),(96,108),(97,117),(98,118),(99,119),(100,120),(101,113),(102,114),(103,115),(104,116),(129,149),(130,150),(131,151),(132,152),(133,145),(134,146),(135,147),(136,148),(137,157),(138,158),(139,159),(140,160),(141,153),(142,154),(143,155),(144,156)], [(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)], [(2,26),(3,7),(4,32),(6,30),(8,28),(9,68),(10,14),(11,66),(13,72),(15,70),(17,76),(18,22),(19,74),(21,80),(23,78),(25,29),(33,37),(34,51),(36,49),(38,55),(40,53),(41,43),(42,126),(44,124),(45,47),(46,122),(48,128),(52,56),(57,88),(59,86),(60,64),(61,84),(63,82),(67,71),(75,79),(81,85),(89,105),(90,92),(91,111),(93,109),(94,96),(95,107),(97,113),(98,100),(99,119),(101,117),(102,104),(103,115),(106,108),(110,112),(114,116),(118,120),(121,123),(125,127),(129,149),(130,136),(131,147),(132,134),(133,145),(135,151),(137,157),(138,144),(139,155),(140,142),(141,153),(143,159),(146,152),(148,150),(154,160),(156,158)]])
95 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | ··· | 4G | 4H | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 10A | ··· | 10L | 10M | ··· | 10T | 10U | 10V | 10W | 10X | 20A | ··· | 20H | 20I | ··· | 20AB | 20AC | 20AD | 20AE | 20AF | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 2 | 2 | 8 | 2 | 2 | 4 | ··· | 4 | 8 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 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 | 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 | C4○D4 | D8 | C5×D4 | C5×D4 | C5×C4○D4 | C5×D8 | C8.C22 | C5×C8.C22 |
| kernel | C5×C22.D8 | C5×C22⋊C8 | C5×D4⋊C4 | C5×C2.D8 | C10×C4⋊C4 | C5×C4⋊D4 | C22.D8 | C22⋊C8 | D4⋊C4 | C2.D8 | C2×C4⋊C4 | C4⋊D4 | C2×C20 | C22×C10 | C20 | C2×C10 | C2×C4 | C23 | C4 | C22 | C10 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 4 | 4 | 8 | 8 | 4 | 4 | 1 | 1 | 4 | 4 | 4 | 4 | 16 | 16 | 1 | 4 |
Matrix representation of C5×C22.D8 ►in GL4(𝔽41) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 10 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 24 | 12 | 0 | 0 |
| 17 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 32 |
| 1 | 1 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [16,0,0,0,0,16,0,0,0,0,10,0,0,0,0,10],[40,0,0,0,0,40,0,0,0,0,0,1,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[24,17,0,0,12,0,0,0,0,0,9,0,0,0,0,32],[1,0,0,0,1,40,0,0,0,0,1,0,0,0,0,40] >;
C5×C22.D8 in GAP, Magma, Sage, TeX
C_5\times C_2^2.D_8
% in TeX
G:=Group("C5xC2^2.D8"); // GroupNames label
G:=SmallGroup(320,981);
// by ID
G=gap.SmallGroup(320,981);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,589,1766,226,10085,2539,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^2=c^2=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=e*b*e=b*c=c*b,c*d=d*c,c*e=e*c,e*d*e=c*d^-1>;
// generators/relations