direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C10×C22.D4, (C23×C20)⋊7C2, (C23×C4)⋊4C10, C23.49(C5×D4), C24.32(C2×C10), C22.61(D4×C10), (C2×C20).657C23, (C2×C10).344C24, (C22×C20)⋊59C22, (C22×D4).10C10, (C22×C10).171D4, C10.183(C22×D4), C23.5(C22×C10), (D4×C10).316C22, C22.18(C23×C10), (C23×C10).92C22, (C22×C10).259C23, C2.7(D4×C2×C10), (C2×C4⋊C4)⋊16C10, (C10×C4⋊C4)⋊43C2, C4⋊C4⋊11(C2×C10), (D4×C2×C10).23C2, C2.7(C10×C4○D4), (C5×C4⋊C4)⋊67C22, (C10×C22⋊C4)⋊30C2, (C2×C22⋊C4)⋊10C10, C22⋊C4⋊12(C2×C10), (C22×C4)⋊17(C2×C10), (C2×D4).61(C2×C10), C10.226(C2×C4○D4), (C2×C10).415(C2×D4), C22.31(C5×C4○D4), (C5×C22⋊C4)⋊66C22, (C2×C4).13(C22×C10), (C2×C10).231(C4○D4), SmallGroup(320,1526)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10×C22.D4
G = < a,b,c,d,e | a10=b2=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe=bc=cb, cd=dc, ce=ec, ede=cd-1 >
Subgroups: 530 in 342 conjugacy classes, 178 normal (22 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, C23, C10, C10, C10, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C20, C2×C10, C2×C10, C2×C10, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C22.D4, C23×C4, C22×D4, C2×C20, C2×C20, C5×D4, C22×C10, C22×C10, C22×C10, C2×C22.D4, C5×C22⋊C4, C5×C4⋊C4, C22×C20, C22×C20, C22×C20, D4×C10, D4×C10, C23×C10, C10×C22⋊C4, C10×C22⋊C4, C10×C4⋊C4, C5×C22.D4, C23×C20, D4×C2×C10, C10×C22.D4
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C4○D4, C24, C2×C10, C22.D4, C22×D4, C2×C4○D4, C5×D4, C22×C10, C2×C22.D4, D4×C10, C5×C4○D4, C23×C10, C5×C22.D4, D4×C2×C10, C10×C4○D4, C10×C22.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)
(1 131)(2 132)(3 133)(4 134)(5 135)(6 136)(7 137)(8 138)(9 139)(10 140)(11 82)(12 83)(13 84)(14 85)(15 86)(16 87)(17 88)(18 89)(19 90)(20 81)(21 78)(22 79)(23 80)(24 71)(25 72)(26 73)(27 74)(28 75)(29 76)(30 77)(31 97)(32 98)(33 99)(34 100)(35 91)(36 92)(37 93)(38 94)(39 95)(40 96)(41 150)(42 141)(43 142)(44 143)(45 144)(46 145)(47 146)(48 147)(49 148)(50 149)(51 128)(52 129)(53 130)(54 121)(55 122)(56 123)(57 124)(58 125)(59 126)(60 127)(61 118)(62 119)(63 120)(64 111)(65 112)(66 113)(67 114)(68 115)(69 116)(70 117)(101 158)(102 159)(103 160)(104 151)(105 152)(106 153)(107 154)(108 155)(109 156)(110 157)
(1 56)(2 57)(3 58)(4 59)(5 60)(6 51)(7 52)(8 53)(9 54)(10 55)(11 34)(12 35)(13 36)(14 37)(15 38)(16 39)(17 40)(18 31)(19 32)(20 33)(21 153)(22 154)(23 155)(24 156)(25 157)(26 158)(27 159)(28 160)(29 151)(30 152)(41 65)(42 66)(43 67)(44 68)(45 69)(46 70)(47 61)(48 62)(49 63)(50 64)(71 109)(72 110)(73 101)(74 102)(75 103)(76 104)(77 105)(78 106)(79 107)(80 108)(81 99)(82 100)(83 91)(84 92)(85 93)(86 94)(87 95)(88 96)(89 97)(90 98)(111 149)(112 150)(113 141)(114 142)(115 143)(116 144)(117 145)(118 146)(119 147)(120 148)(121 139)(122 140)(123 131)(124 132)(125 133)(126 134)(127 135)(128 136)(129 137)(130 138)
(1 96 62 74)(2 97 63 75)(3 98 64 76)(4 99 65 77)(5 100 66 78)(6 91 67 79)(7 92 68 80)(8 93 69 71)(9 94 70 72)(10 95 61 73)(11 113 153 135)(12 114 154 136)(13 115 155 137)(14 116 156 138)(15 117 157 139)(16 118 158 140)(17 119 159 131)(18 120 160 132)(19 111 151 133)(20 112 152 134)(21 127 34 141)(22 128 35 142)(23 129 36 143)(24 130 37 144)(25 121 38 145)(26 122 39 146)(27 123 40 147)(28 124 31 148)(29 125 32 149)(30 126 33 150)(41 105 59 81)(42 106 60 82)(43 107 51 83)(44 108 52 84)(45 109 53 85)(46 110 54 86)(47 101 55 87)(48 102 56 88)(49 103 57 89)(50 104 58 90)
(1 67)(2 68)(3 69)(4 70)(5 61)(6 62)(7 63)(8 64)(9 65)(10 66)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)(41 54)(42 55)(43 56)(44 57)(45 58)(46 59)(47 60)(48 51)(49 52)(50 53)(71 104)(72 105)(73 106)(74 107)(75 108)(76 109)(77 110)(78 101)(79 102)(80 103)(81 94)(82 95)(83 96)(84 97)(85 98)(86 99)(87 100)(88 91)(89 92)(90 93)(111 130)(112 121)(113 122)(114 123)(115 124)(116 125)(117 126)(118 127)(119 128)(120 129)(131 142)(132 143)(133 144)(134 145)(135 146)(136 147)(137 148)(138 149)(139 150)(140 141)(151 156)(152 157)(153 158)(154 159)(155 160)
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), (1,131)(2,132)(3,133)(4,134)(5,135)(6,136)(7,137)(8,138)(9,139)(10,140)(11,82)(12,83)(13,84)(14,85)(15,86)(16,87)(17,88)(18,89)(19,90)(20,81)(21,78)(22,79)(23,80)(24,71)(25,72)(26,73)(27,74)(28,75)(29,76)(30,77)(31,97)(32,98)(33,99)(34,100)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,150)(42,141)(43,142)(44,143)(45,144)(46,145)(47,146)(48,147)(49,148)(50,149)(51,128)(52,129)(53,130)(54,121)(55,122)(56,123)(57,124)(58,125)(59,126)(60,127)(61,118)(62,119)(63,120)(64,111)(65,112)(66,113)(67,114)(68,115)(69,116)(70,117)(101,158)(102,159)(103,160)(104,151)(105,152)(106,153)(107,154)(108,155)(109,156)(110,157), (1,56)(2,57)(3,58)(4,59)(5,60)(6,51)(7,52)(8,53)(9,54)(10,55)(11,34)(12,35)(13,36)(14,37)(15,38)(16,39)(17,40)(18,31)(19,32)(20,33)(21,153)(22,154)(23,155)(24,156)(25,157)(26,158)(27,159)(28,160)(29,151)(30,152)(41,65)(42,66)(43,67)(44,68)(45,69)(46,70)(47,61)(48,62)(49,63)(50,64)(71,109)(72,110)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,99)(82,100)(83,91)(84,92)(85,93)(86,94)(87,95)(88,96)(89,97)(90,98)(111,149)(112,150)(113,141)(114,142)(115,143)(116,144)(117,145)(118,146)(119,147)(120,148)(121,139)(122,140)(123,131)(124,132)(125,133)(126,134)(127,135)(128,136)(129,137)(130,138), (1,96,62,74)(2,97,63,75)(3,98,64,76)(4,99,65,77)(5,100,66,78)(6,91,67,79)(7,92,68,80)(8,93,69,71)(9,94,70,72)(10,95,61,73)(11,113,153,135)(12,114,154,136)(13,115,155,137)(14,116,156,138)(15,117,157,139)(16,118,158,140)(17,119,159,131)(18,120,160,132)(19,111,151,133)(20,112,152,134)(21,127,34,141)(22,128,35,142)(23,129,36,143)(24,130,37,144)(25,121,38,145)(26,122,39,146)(27,123,40,147)(28,124,31,148)(29,125,32,149)(30,126,33,150)(41,105,59,81)(42,106,60,82)(43,107,51,83)(44,108,52,84)(45,109,53,85)(46,110,54,86)(47,101,55,87)(48,102,56,88)(49,103,57,89)(50,104,58,90), (1,67)(2,68)(3,69)(4,70)(5,61)(6,62)(7,63)(8,64)(9,65)(10,66)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)(41,54)(42,55)(43,56)(44,57)(45,58)(46,59)(47,60)(48,51)(49,52)(50,53)(71,104)(72,105)(73,106)(74,107)(75,108)(76,109)(77,110)(78,101)(79,102)(80,103)(81,94)(82,95)(83,96)(84,97)(85,98)(86,99)(87,100)(88,91)(89,92)(90,93)(111,130)(112,121)(113,122)(114,123)(115,124)(116,125)(117,126)(118,127)(119,128)(120,129)(131,142)(132,143)(133,144)(134,145)(135,146)(136,147)(137,148)(138,149)(139,150)(140,141)(151,156)(152,157)(153,158)(154,159)(155,160)>;
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), (1,131)(2,132)(3,133)(4,134)(5,135)(6,136)(7,137)(8,138)(9,139)(10,140)(11,82)(12,83)(13,84)(14,85)(15,86)(16,87)(17,88)(18,89)(19,90)(20,81)(21,78)(22,79)(23,80)(24,71)(25,72)(26,73)(27,74)(28,75)(29,76)(30,77)(31,97)(32,98)(33,99)(34,100)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,150)(42,141)(43,142)(44,143)(45,144)(46,145)(47,146)(48,147)(49,148)(50,149)(51,128)(52,129)(53,130)(54,121)(55,122)(56,123)(57,124)(58,125)(59,126)(60,127)(61,118)(62,119)(63,120)(64,111)(65,112)(66,113)(67,114)(68,115)(69,116)(70,117)(101,158)(102,159)(103,160)(104,151)(105,152)(106,153)(107,154)(108,155)(109,156)(110,157), (1,56)(2,57)(3,58)(4,59)(5,60)(6,51)(7,52)(8,53)(9,54)(10,55)(11,34)(12,35)(13,36)(14,37)(15,38)(16,39)(17,40)(18,31)(19,32)(20,33)(21,153)(22,154)(23,155)(24,156)(25,157)(26,158)(27,159)(28,160)(29,151)(30,152)(41,65)(42,66)(43,67)(44,68)(45,69)(46,70)(47,61)(48,62)(49,63)(50,64)(71,109)(72,110)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,99)(82,100)(83,91)(84,92)(85,93)(86,94)(87,95)(88,96)(89,97)(90,98)(111,149)(112,150)(113,141)(114,142)(115,143)(116,144)(117,145)(118,146)(119,147)(120,148)(121,139)(122,140)(123,131)(124,132)(125,133)(126,134)(127,135)(128,136)(129,137)(130,138), (1,96,62,74)(2,97,63,75)(3,98,64,76)(4,99,65,77)(5,100,66,78)(6,91,67,79)(7,92,68,80)(8,93,69,71)(9,94,70,72)(10,95,61,73)(11,113,153,135)(12,114,154,136)(13,115,155,137)(14,116,156,138)(15,117,157,139)(16,118,158,140)(17,119,159,131)(18,120,160,132)(19,111,151,133)(20,112,152,134)(21,127,34,141)(22,128,35,142)(23,129,36,143)(24,130,37,144)(25,121,38,145)(26,122,39,146)(27,123,40,147)(28,124,31,148)(29,125,32,149)(30,126,33,150)(41,105,59,81)(42,106,60,82)(43,107,51,83)(44,108,52,84)(45,109,53,85)(46,110,54,86)(47,101,55,87)(48,102,56,88)(49,103,57,89)(50,104,58,90), (1,67)(2,68)(3,69)(4,70)(5,61)(6,62)(7,63)(8,64)(9,65)(10,66)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)(41,54)(42,55)(43,56)(44,57)(45,58)(46,59)(47,60)(48,51)(49,52)(50,53)(71,104)(72,105)(73,106)(74,107)(75,108)(76,109)(77,110)(78,101)(79,102)(80,103)(81,94)(82,95)(83,96)(84,97)(85,98)(86,99)(87,100)(88,91)(89,92)(90,93)(111,130)(112,121)(113,122)(114,123)(115,124)(116,125)(117,126)(118,127)(119,128)(120,129)(131,142)(132,143)(133,144)(134,145)(135,146)(136,147)(137,148)(138,149)(139,150)(140,141)(151,156)(152,157)(153,158)(154,159)(155,160) );
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)], [(1,131),(2,132),(3,133),(4,134),(5,135),(6,136),(7,137),(8,138),(9,139),(10,140),(11,82),(12,83),(13,84),(14,85),(15,86),(16,87),(17,88),(18,89),(19,90),(20,81),(21,78),(22,79),(23,80),(24,71),(25,72),(26,73),(27,74),(28,75),(29,76),(30,77),(31,97),(32,98),(33,99),(34,100),(35,91),(36,92),(37,93),(38,94),(39,95),(40,96),(41,150),(42,141),(43,142),(44,143),(45,144),(46,145),(47,146),(48,147),(49,148),(50,149),(51,128),(52,129),(53,130),(54,121),(55,122),(56,123),(57,124),(58,125),(59,126),(60,127),(61,118),(62,119),(63,120),(64,111),(65,112),(66,113),(67,114),(68,115),(69,116),(70,117),(101,158),(102,159),(103,160),(104,151),(105,152),(106,153),(107,154),(108,155),(109,156),(110,157)], [(1,56),(2,57),(3,58),(4,59),(5,60),(6,51),(7,52),(8,53),(9,54),(10,55),(11,34),(12,35),(13,36),(14,37),(15,38),(16,39),(17,40),(18,31),(19,32),(20,33),(21,153),(22,154),(23,155),(24,156),(25,157),(26,158),(27,159),(28,160),(29,151),(30,152),(41,65),(42,66),(43,67),(44,68),(45,69),(46,70),(47,61),(48,62),(49,63),(50,64),(71,109),(72,110),(73,101),(74,102),(75,103),(76,104),(77,105),(78,106),(79,107),(80,108),(81,99),(82,100),(83,91),(84,92),(85,93),(86,94),(87,95),(88,96),(89,97),(90,98),(111,149),(112,150),(113,141),(114,142),(115,143),(116,144),(117,145),(118,146),(119,147),(120,148),(121,139),(122,140),(123,131),(124,132),(125,133),(126,134),(127,135),(128,136),(129,137),(130,138)], [(1,96,62,74),(2,97,63,75),(3,98,64,76),(4,99,65,77),(5,100,66,78),(6,91,67,79),(7,92,68,80),(8,93,69,71),(9,94,70,72),(10,95,61,73),(11,113,153,135),(12,114,154,136),(13,115,155,137),(14,116,156,138),(15,117,157,139),(16,118,158,140),(17,119,159,131),(18,120,160,132),(19,111,151,133),(20,112,152,134),(21,127,34,141),(22,128,35,142),(23,129,36,143),(24,130,37,144),(25,121,38,145),(26,122,39,146),(27,123,40,147),(28,124,31,148),(29,125,32,149),(30,126,33,150),(41,105,59,81),(42,106,60,82),(43,107,51,83),(44,108,52,84),(45,109,53,85),(46,110,54,86),(47,101,55,87),(48,102,56,88),(49,103,57,89),(50,104,58,90)], [(1,67),(2,68),(3,69),(4,70),(5,61),(6,62),(7,63),(8,64),(9,65),(10,66),(11,16),(12,17),(13,18),(14,19),(15,20),(21,26),(22,27),(23,28),(24,29),(25,30),(31,36),(32,37),(33,38),(34,39),(35,40),(41,54),(42,55),(43,56),(44,57),(45,58),(46,59),(47,60),(48,51),(49,52),(50,53),(71,104),(72,105),(73,106),(74,107),(75,108),(76,109),(77,110),(78,101),(79,102),(80,103),(81,94),(82,95),(83,96),(84,97),(85,98),(86,99),(87,100),(88,91),(89,92),(90,93),(111,130),(112,121),(113,122),(114,123),(115,124),(116,125),(117,126),(118,127),(119,128),(120,129),(131,142),(132,143),(133,144),(134,145),(135,146),(136,147),(137,148),(138,149),(139,150),(140,141),(151,156),(152,157),(153,158),(154,159),(155,160)]])
140 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 4A | ··· | 4H | 4I | ··· | 4N | 5A | 5B | 5C | 5D | 10A | ··· | 10AB | 10AC | ··· | 10AR | 10AS | ··· | 10AZ | 20A | ··· | 20AF | 20AG | ··· | 20BD |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | D4 | C4○D4 | C5×D4 | C5×C4○D4 |
| kernel | C10×C22.D4 | C10×C22⋊C4 | C10×C4⋊C4 | C5×C22.D4 | C23×C20 | D4×C2×C10 | C2×C22.D4 | C2×C22⋊C4 | C2×C4⋊C4 | C22.D4 | C23×C4 | C22×D4 | C22×C10 | C2×C10 | C23 | C22 |
| # reps | 1 | 3 | 2 | 8 | 1 | 1 | 4 | 12 | 8 | 32 | 4 | 4 | 4 | 8 | 16 | 32 |
Matrix representation of C10×C22.D4 ►in GL5(𝔽41)
| 40 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4],[40,0,0,0,0,0,0,32,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,40,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,40,0],[1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,40] >;
C10×C22.D4 in GAP, Magma, Sage, TeX
C_{10}\times C_2^2.D_4 % in TeX
G:=Group("C10xC2^2.D4"); // GroupNames label
G:=SmallGroup(320,1526);
// by ID
G=gap.SmallGroup(320,1526);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1149,3446,436]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^2=c^2=d^4=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