direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×D4×Q8, C10.1192- 1+4, C4⋊2(Q8×C10), C4⋊Q8⋊14C10, C20⋊10(C2×Q8), (C4×Q8)⋊12C10, (Q8×C20)⋊32C2, C4.43(D4×C10), C22⋊2(Q8×C10), (C4×D4).10C10, (D4×C20).25C2, C20.404(C2×D4), C22⋊Q8⋊14C10, (C22×Q8)⋊7C10, C42.44(C2×C10), C10.63(C22×Q8), (C4×C20).285C22, (C2×C20).676C23, (C2×C10).369C24, C10.197(C22×D4), (D4×C10).334C22, C22.43(C23×C10), C23.43(C22×C10), (Q8×C10).275C22, C2.11(C5×2- 1+4), (C22×C20).455C22, (C22×C10).265C23, C2.9(Q8×C2×C10), (C5×C4⋊Q8)⋊35C2, (Q8×C2×C10)⋊19C2, (C2×C10)⋊8(C2×Q8), C2.21(D4×C2×C10), C4⋊C4.32(C2×C10), (C5×C22⋊Q8)⋊41C2, (C2×D4).80(C2×C10), (C2×Q8).62(C2×C10), C22⋊C4.20(C2×C10), (C5×C4⋊C4).397C22, (C2×C4).34(C22×C10), (C22×C4).67(C2×C10), (C5×C22⋊C4).153C22, SmallGroup(320,1551)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×D4×Q8
G = < a,b,c,d,e | a5=b4=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 378 in 280 conjugacy classes, 182 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C4 [×8], C4 [×9], C22, C22 [×4], C22 [×4], C5, C2×C4, C2×C4 [×12], C2×C4 [×12], D4 [×4], Q8 [×4], Q8 [×12], C23 [×2], C10 [×3], C10 [×4], C42 [×3], C22⋊C4 [×6], C4⋊C4 [×12], C22×C4 [×6], C2×D4, C2×Q8, C2×Q8 [×6], C2×Q8 [×8], C20 [×8], C20 [×9], C2×C10, C2×C10 [×4], C2×C10 [×4], C4×D4 [×3], C4×Q8, C22⋊Q8 [×6], C4⋊Q8 [×3], C22×Q8 [×2], C2×C20, C2×C20 [×12], C2×C20 [×12], C5×D4 [×4], C5×Q8 [×4], C5×Q8 [×12], C22×C10 [×2], D4×Q8, C4×C20 [×3], C5×C22⋊C4 [×6], C5×C4⋊C4 [×12], C22×C20 [×6], D4×C10, Q8×C10, Q8×C10 [×6], Q8×C10 [×8], D4×C20 [×3], Q8×C20, C5×C22⋊Q8 [×6], C5×C4⋊Q8 [×3], Q8×C2×C10 [×2], C5×D4×Q8
Quotients: C1, C2 [×15], C22 [×35], C5, D4 [×4], Q8 [×4], C23 [×15], C10 [×15], C2×D4 [×6], C2×Q8 [×6], C24, C2×C10 [×35], C22×D4, C22×Q8, 2- 1+4, C5×D4 [×4], C5×Q8 [×4], C22×C10 [×15], D4×Q8, D4×C10 [×6], Q8×C10 [×6], C23×C10, D4×C2×C10, Q8×C2×C10, C5×2- 1+4, C5×D4×Q8
►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)
(1 123 30 135)(2 124 26 131)(3 125 27 132)(4 121 28 133)(5 122 29 134)(6 70 156 56)(7 66 157 57)(8 67 158 58)(9 68 159 59)(10 69 160 60)(11 75 16 63)(12 71 17 64)(13 72 18 65)(14 73 19 61)(15 74 20 62)(21 119 33 128)(22 120 34 129)(23 116 35 130)(24 117 31 126)(25 118 32 127)(36 110 50 96)(37 106 46 97)(38 107 47 98)(39 108 48 99)(40 109 49 100)(41 113 53 101)(42 114 54 102)(43 115 55 103)(44 111 51 104)(45 112 52 105)(76 143 90 155)(77 144 86 151)(78 145 87 152)(79 141 88 153)(80 142 89 154)(81 139 93 148)(82 140 94 149)(83 136 95 150)(84 137 91 146)(85 138 92 147)
(1 135)(2 131)(3 132)(4 133)(5 134)(6 70)(7 66)(8 67)(9 68)(10 69)(11 75)(12 71)(13 72)(14 73)(15 74)(16 63)(17 64)(18 65)(19 61)(20 62)(21 128)(22 129)(23 130)(24 126)(25 127)(26 124)(27 125)(28 121)(29 122)(30 123)(31 117)(32 118)(33 119)(34 120)(35 116)(36 96)(37 97)(38 98)(39 99)(40 100)(41 101)(42 102)(43 103)(44 104)(45 105)(46 106)(47 107)(48 108)(49 109)(50 110)(51 111)(52 112)(53 113)(54 114)(55 115)(56 156)(57 157)(58 158)(59 159)(60 160)(76 155)(77 151)(78 152)(79 153)(80 154)(81 148)(82 149)(83 150)(84 146)(85 147)(86 144)(87 145)(88 141)(89 142)(90 143)(91 137)(92 138)(93 139)(94 140)(95 136)
(1 55 35 36)(2 51 31 37)(3 52 32 38)(4 53 33 39)(5 54 34 40)(6 143 16 150)(7 144 17 146)(8 145 18 147)(9 141 19 148)(10 142 20 149)(11 136 156 155)(12 137 157 151)(13 138 158 152)(14 139 159 153)(15 140 160 154)(21 48 28 41)(22 49 29 42)(23 50 30 43)(24 46 26 44)(25 47 27 45)(56 76 75 95)(57 77 71 91)(58 78 72 92)(59 79 73 93)(60 80 74 94)(61 81 68 88)(62 82 69 89)(63 83 70 90)(64 84 66 86)(65 85 67 87)(96 135 115 116)(97 131 111 117)(98 132 112 118)(99 133 113 119)(100 134 114 120)(101 128 108 121)(102 129 109 122)(103 130 110 123)(104 126 106 124)(105 127 107 125)
(1 75 35 56)(2 71 31 57)(3 72 32 58)(4 73 33 59)(5 74 34 60)(6 123 16 130)(7 124 17 126)(8 125 18 127)(9 121 19 128)(10 122 20 129)(11 116 156 135)(12 117 157 131)(13 118 158 132)(14 119 159 133)(15 120 160 134)(21 68 28 61)(22 69 29 62)(23 70 30 63)(24 66 26 64)(25 67 27 65)(36 95 55 76)(37 91 51 77)(38 92 52 78)(39 93 53 79)(40 94 54 80)(41 88 48 81)(42 89 49 82)(43 90 50 83)(44 86 46 84)(45 87 47 85)(96 136 115 155)(97 137 111 151)(98 138 112 152)(99 139 113 153)(100 140 114 154)(101 141 108 148)(102 142 109 149)(103 143 110 150)(104 144 106 146)(105 145 107 147)
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), (1,123,30,135)(2,124,26,131)(3,125,27,132)(4,121,28,133)(5,122,29,134)(6,70,156,56)(7,66,157,57)(8,67,158,58)(9,68,159,59)(10,69,160,60)(11,75,16,63)(12,71,17,64)(13,72,18,65)(14,73,19,61)(15,74,20,62)(21,119,33,128)(22,120,34,129)(23,116,35,130)(24,117,31,126)(25,118,32,127)(36,110,50,96)(37,106,46,97)(38,107,47,98)(39,108,48,99)(40,109,49,100)(41,113,53,101)(42,114,54,102)(43,115,55,103)(44,111,51,104)(45,112,52,105)(76,143,90,155)(77,144,86,151)(78,145,87,152)(79,141,88,153)(80,142,89,154)(81,139,93,148)(82,140,94,149)(83,136,95,150)(84,137,91,146)(85,138,92,147), (1,135)(2,131)(3,132)(4,133)(5,134)(6,70)(7,66)(8,67)(9,68)(10,69)(11,75)(12,71)(13,72)(14,73)(15,74)(16,63)(17,64)(18,65)(19,61)(20,62)(21,128)(22,129)(23,130)(24,126)(25,127)(26,124)(27,125)(28,121)(29,122)(30,123)(31,117)(32,118)(33,119)(34,120)(35,116)(36,96)(37,97)(38,98)(39,99)(40,100)(41,101)(42,102)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,109)(50,110)(51,111)(52,112)(53,113)(54,114)(55,115)(56,156)(57,157)(58,158)(59,159)(60,160)(76,155)(77,151)(78,152)(79,153)(80,154)(81,148)(82,149)(83,150)(84,146)(85,147)(86,144)(87,145)(88,141)(89,142)(90,143)(91,137)(92,138)(93,139)(94,140)(95,136), (1,55,35,36)(2,51,31,37)(3,52,32,38)(4,53,33,39)(5,54,34,40)(6,143,16,150)(7,144,17,146)(8,145,18,147)(9,141,19,148)(10,142,20,149)(11,136,156,155)(12,137,157,151)(13,138,158,152)(14,139,159,153)(15,140,160,154)(21,48,28,41)(22,49,29,42)(23,50,30,43)(24,46,26,44)(25,47,27,45)(56,76,75,95)(57,77,71,91)(58,78,72,92)(59,79,73,93)(60,80,74,94)(61,81,68,88)(62,82,69,89)(63,83,70,90)(64,84,66,86)(65,85,67,87)(96,135,115,116)(97,131,111,117)(98,132,112,118)(99,133,113,119)(100,134,114,120)(101,128,108,121)(102,129,109,122)(103,130,110,123)(104,126,106,124)(105,127,107,125), (1,75,35,56)(2,71,31,57)(3,72,32,58)(4,73,33,59)(5,74,34,60)(6,123,16,130)(7,124,17,126)(8,125,18,127)(9,121,19,128)(10,122,20,129)(11,116,156,135)(12,117,157,131)(13,118,158,132)(14,119,159,133)(15,120,160,134)(21,68,28,61)(22,69,29,62)(23,70,30,63)(24,66,26,64)(25,67,27,65)(36,95,55,76)(37,91,51,77)(38,92,52,78)(39,93,53,79)(40,94,54,80)(41,88,48,81)(42,89,49,82)(43,90,50,83)(44,86,46,84)(45,87,47,85)(96,136,115,155)(97,137,111,151)(98,138,112,152)(99,139,113,153)(100,140,114,154)(101,141,108,148)(102,142,109,149)(103,143,110,150)(104,144,106,146)(105,145,107,147)>;
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), (1,123,30,135)(2,124,26,131)(3,125,27,132)(4,121,28,133)(5,122,29,134)(6,70,156,56)(7,66,157,57)(8,67,158,58)(9,68,159,59)(10,69,160,60)(11,75,16,63)(12,71,17,64)(13,72,18,65)(14,73,19,61)(15,74,20,62)(21,119,33,128)(22,120,34,129)(23,116,35,130)(24,117,31,126)(25,118,32,127)(36,110,50,96)(37,106,46,97)(38,107,47,98)(39,108,48,99)(40,109,49,100)(41,113,53,101)(42,114,54,102)(43,115,55,103)(44,111,51,104)(45,112,52,105)(76,143,90,155)(77,144,86,151)(78,145,87,152)(79,141,88,153)(80,142,89,154)(81,139,93,148)(82,140,94,149)(83,136,95,150)(84,137,91,146)(85,138,92,147), (1,135)(2,131)(3,132)(4,133)(5,134)(6,70)(7,66)(8,67)(9,68)(10,69)(11,75)(12,71)(13,72)(14,73)(15,74)(16,63)(17,64)(18,65)(19,61)(20,62)(21,128)(22,129)(23,130)(24,126)(25,127)(26,124)(27,125)(28,121)(29,122)(30,123)(31,117)(32,118)(33,119)(34,120)(35,116)(36,96)(37,97)(38,98)(39,99)(40,100)(41,101)(42,102)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,109)(50,110)(51,111)(52,112)(53,113)(54,114)(55,115)(56,156)(57,157)(58,158)(59,159)(60,160)(76,155)(77,151)(78,152)(79,153)(80,154)(81,148)(82,149)(83,150)(84,146)(85,147)(86,144)(87,145)(88,141)(89,142)(90,143)(91,137)(92,138)(93,139)(94,140)(95,136), (1,55,35,36)(2,51,31,37)(3,52,32,38)(4,53,33,39)(5,54,34,40)(6,143,16,150)(7,144,17,146)(8,145,18,147)(9,141,19,148)(10,142,20,149)(11,136,156,155)(12,137,157,151)(13,138,158,152)(14,139,159,153)(15,140,160,154)(21,48,28,41)(22,49,29,42)(23,50,30,43)(24,46,26,44)(25,47,27,45)(56,76,75,95)(57,77,71,91)(58,78,72,92)(59,79,73,93)(60,80,74,94)(61,81,68,88)(62,82,69,89)(63,83,70,90)(64,84,66,86)(65,85,67,87)(96,135,115,116)(97,131,111,117)(98,132,112,118)(99,133,113,119)(100,134,114,120)(101,128,108,121)(102,129,109,122)(103,130,110,123)(104,126,106,124)(105,127,107,125), (1,75,35,56)(2,71,31,57)(3,72,32,58)(4,73,33,59)(5,74,34,60)(6,123,16,130)(7,124,17,126)(8,125,18,127)(9,121,19,128)(10,122,20,129)(11,116,156,135)(12,117,157,131)(13,118,158,132)(14,119,159,133)(15,120,160,134)(21,68,28,61)(22,69,29,62)(23,70,30,63)(24,66,26,64)(25,67,27,65)(36,95,55,76)(37,91,51,77)(38,92,52,78)(39,93,53,79)(40,94,54,80)(41,88,48,81)(42,89,49,82)(43,90,50,83)(44,86,46,84)(45,87,47,85)(96,136,115,155)(97,137,111,151)(98,138,112,152)(99,139,113,153)(100,140,114,154)(101,141,108,148)(102,142,109,149)(103,143,110,150)(104,144,106,146)(105,145,107,147) );
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)], [(1,123,30,135),(2,124,26,131),(3,125,27,132),(4,121,28,133),(5,122,29,134),(6,70,156,56),(7,66,157,57),(8,67,158,58),(9,68,159,59),(10,69,160,60),(11,75,16,63),(12,71,17,64),(13,72,18,65),(14,73,19,61),(15,74,20,62),(21,119,33,128),(22,120,34,129),(23,116,35,130),(24,117,31,126),(25,118,32,127),(36,110,50,96),(37,106,46,97),(38,107,47,98),(39,108,48,99),(40,109,49,100),(41,113,53,101),(42,114,54,102),(43,115,55,103),(44,111,51,104),(45,112,52,105),(76,143,90,155),(77,144,86,151),(78,145,87,152),(79,141,88,153),(80,142,89,154),(81,139,93,148),(82,140,94,149),(83,136,95,150),(84,137,91,146),(85,138,92,147)], [(1,135),(2,131),(3,132),(4,133),(5,134),(6,70),(7,66),(8,67),(9,68),(10,69),(11,75),(12,71),(13,72),(14,73),(15,74),(16,63),(17,64),(18,65),(19,61),(20,62),(21,128),(22,129),(23,130),(24,126),(25,127),(26,124),(27,125),(28,121),(29,122),(30,123),(31,117),(32,118),(33,119),(34,120),(35,116),(36,96),(37,97),(38,98),(39,99),(40,100),(41,101),(42,102),(43,103),(44,104),(45,105),(46,106),(47,107),(48,108),(49,109),(50,110),(51,111),(52,112),(53,113),(54,114),(55,115),(56,156),(57,157),(58,158),(59,159),(60,160),(76,155),(77,151),(78,152),(79,153),(80,154),(81,148),(82,149),(83,150),(84,146),(85,147),(86,144),(87,145),(88,141),(89,142),(90,143),(91,137),(92,138),(93,139),(94,140),(95,136)], [(1,55,35,36),(2,51,31,37),(3,52,32,38),(4,53,33,39),(5,54,34,40),(6,143,16,150),(7,144,17,146),(8,145,18,147),(9,141,19,148),(10,142,20,149),(11,136,156,155),(12,137,157,151),(13,138,158,152),(14,139,159,153),(15,140,160,154),(21,48,28,41),(22,49,29,42),(23,50,30,43),(24,46,26,44),(25,47,27,45),(56,76,75,95),(57,77,71,91),(58,78,72,92),(59,79,73,93),(60,80,74,94),(61,81,68,88),(62,82,69,89),(63,83,70,90),(64,84,66,86),(65,85,67,87),(96,135,115,116),(97,131,111,117),(98,132,112,118),(99,133,113,119),(100,134,114,120),(101,128,108,121),(102,129,109,122),(103,130,110,123),(104,126,106,124),(105,127,107,125)], [(1,75,35,56),(2,71,31,57),(3,72,32,58),(4,73,33,59),(5,74,34,60),(6,123,16,130),(7,124,17,126),(8,125,18,127),(9,121,19,128),(10,122,20,129),(11,116,156,135),(12,117,157,131),(13,118,158,132),(14,119,159,133),(15,120,160,134),(21,68,28,61),(22,69,29,62),(23,70,30,63),(24,66,26,64),(25,67,27,65),(36,95,55,76),(37,91,51,77),(38,92,52,78),(39,93,53,79),(40,94,54,80),(41,88,48,81),(42,89,49,82),(43,90,50,83),(44,86,46,84),(45,87,47,85),(96,136,115,155),(97,137,111,151),(98,138,112,152),(99,139,113,153),(100,140,114,154),(101,141,108,148),(102,142,109,149),(103,143,110,150),(104,144,106,146),(105,145,107,147)])
125 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4H | 4I | ··· | 4Q | 5A | 5B | 5C | 5D | 10A | ··· | 10L | 10M | ··· | 10AB | 20A | ··· | 20AF | 20AG | ··· | 20BP |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
125 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | Q8 | D4 | C5×Q8 | C5×D4 | 2- 1+4 | C5×2- 1+4 |
| kernel | C5×D4×Q8 | D4×C20 | Q8×C20 | C5×C22⋊Q8 | C5×C4⋊Q8 | Q8×C2×C10 | D4×Q8 | C4×D4 | C4×Q8 | C22⋊Q8 | C4⋊Q8 | C22×Q8 | C5×D4 | C5×Q8 | D4 | Q8 | C10 | C2 |
| # reps | 1 | 3 | 1 | 6 | 3 | 2 | 4 | 12 | 4 | 24 | 12 | 8 | 4 | 4 | 16 | 16 | 1 | 4 |
Matrix representation of C5×D4×Q8 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 37 | 0 |
| 0 | 0 | 0 | 37 |
| 40 | 2 | 0 | 0 |
| 40 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 40 | 2 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 9 | 2 |
| 0 | 0 | 0 | 32 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 7 | 6 |
| 0 | 0 | 19 | 34 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,37,0,0,0,0,37],[40,40,0,0,2,1,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,2,1,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,9,0,0,0,2,32],[40,0,0,0,0,40,0,0,0,0,7,19,0,0,6,34] >;
C5×D4×Q8 in GAP, Magma, Sage, TeX
C_5\times D_4\times Q_8
% in TeX
G:=Group("C5xD4xQ8"); // GroupNames label
G:=SmallGroup(320,1551);
// by ID
G=gap.SmallGroup(320,1551);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,560,1149,568,3446,1242,304]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^4=c^2=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations