direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×D4⋊2Q8, C20.45SD16, C4⋊Q8⋊4C10, C4⋊C8⋊10C10, (C5×D4)⋊9Q8, D4⋊2(C5×Q8), C4.Q8⋊7C10, (C4×D4).8C10, C4.14(Q8×C10), (D4×C20).23C2, (C2×C20).332D4, C20.120(C2×Q8), C4.10(C5×SD16), D4⋊C4.5C10, C42.22(C2×C10), C2.11(C10×SD16), C10.91(C2×SD16), C22.97(D4×C10), C20.313(C4○D4), (C4×C20).264C22, (C2×C20).932C23, (C2×C40).303C22, C10.95(C22⋊Q8), C10.139(C8⋊C22), (D4×C10).299C22, (C5×C4⋊C8)⋊29C2, (C5×C4⋊Q8)⋊25C2, (C5×C4.Q8)⋊22C2, C4.25(C5×C4○D4), C4⋊C4.13(C2×C10), (C2×C8).40(C2×C10), (C2×C4).133(C5×D4), C2.14(C5×C8⋊C22), C2.14(C5×C22⋊Q8), (C2×D4).59(C2×C10), (C2×C10).653(C2×D4), (C5×D4⋊C4).14C2, (C5×C4⋊C4).235C22, (C2×C4).107(C22×C10), SmallGroup(320,977)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×D4⋊2Q8
G = < a,b,c,d,e | a5=b4=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, dcd-1=b2c, ece-1=bc, ede-1=d-1 >
Subgroups: 202 in 108 conjugacy classes, 58 normal (34 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, C2×Q8, C20, C20, C20, C2×C10, C2×C10, D4⋊C4, C4⋊C8, C4.Q8, C4×D4, C4⋊Q8, C40, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C22×C10, D4⋊2Q8, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C5×C4⋊C4, C2×C40, C22×C20, D4×C10, Q8×C10, C5×D4⋊C4, C5×C4⋊C8, C5×C4.Q8, D4×C20, C5×C4⋊Q8, C5×D4⋊2Q8
Quotients: C1, C2, C22, C5, D4, Q8, C23, C10, SD16, C2×D4, C2×Q8, C4○D4, C2×C10, C22⋊Q8, C2×SD16, C8⋊C22, C5×D4, C5×Q8, C22×C10, D4⋊2Q8, C5×SD16, D4×C10, Q8×C10, C5×C4○D4, C5×C22⋊Q8, C10×SD16, C5×C8⋊C22, C5×D4⋊2Q8
►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 37 12 66)(2 38 13 67)(3 39 14 68)(4 40 15 69)(5 36 11 70)(6 23 144 156)(7 24 145 157)(8 25 141 158)(9 21 142 159)(10 22 143 160)(16 33 137 154)(17 34 138 155)(18 35 139 151)(19 31 140 152)(20 32 136 153)(26 43 50 52)(27 44 46 53)(28 45 47 54)(29 41 48 55)(30 42 49 51)(56 87 94 73)(57 88 95 74)(58 89 91 75)(59 90 92 71)(60 86 93 72)(61 107 82 78)(62 108 83 79)(63 109 84 80)(64 110 85 76)(65 106 81 77)(96 113 134 127)(97 114 135 128)(98 115 131 129)(99 111 132 130)(100 112 133 126)(101 118 122 147)(102 119 123 148)(103 120 124 149)(104 116 125 150)(105 117 121 146)
(1 47)(2 48)(3 49)(4 50)(5 46)(6 140)(7 136)(8 137)(9 138)(10 139)(11 27)(12 28)(13 29)(14 30)(15 26)(16 141)(17 142)(18 143)(19 144)(20 145)(21 34)(22 35)(23 31)(24 32)(25 33)(36 44)(37 45)(38 41)(39 42)(40 43)(51 68)(52 69)(53 70)(54 66)(55 67)(56 77)(57 78)(58 79)(59 80)(60 76)(61 74)(62 75)(63 71)(64 72)(65 73)(81 87)(82 88)(83 89)(84 90)(85 86)(91 108)(92 109)(93 110)(94 106)(95 107)(96 121)(97 122)(98 123)(99 124)(100 125)(101 135)(102 131)(103 132)(104 133)(105 134)(111 120)(112 116)(113 117)(114 118)(115 119)(126 150)(127 146)(128 147)(129 148)(130 149)(151 160)(152 156)(153 157)(154 158)(155 159)
(1 87 54 106)(2 88 55 107)(3 89 51 108)(4 90 52 109)(5 86 53 110)(6 112 19 116)(7 113 20 117)(8 114 16 118)(9 115 17 119)(10 111 18 120)(11 72 44 76)(12 73 45 77)(13 74 41 78)(14 75 42 79)(15 71 43 80)(21 131 34 123)(22 132 35 124)(23 133 31 125)(24 134 32 121)(25 135 33 122)(26 84 40 92)(27 85 36 93)(28 81 37 94)(29 82 38 95)(30 83 39 91)(46 64 70 60)(47 65 66 56)(48 61 67 57)(49 62 68 58)(50 63 69 59)(96 153 105 157)(97 154 101 158)(98 155 102 159)(99 151 103 160)(100 152 104 156)(126 140 150 144)(127 136 146 145)(128 137 147 141)(129 138 148 142)(130 139 149 143)
(1 134 54 121)(2 135 55 122)(3 131 51 123)(4 132 52 124)(5 133 53 125)(6 85 19 93)(7 81 20 94)(8 82 16 95)(9 83 17 91)(10 84 18 92)(11 100 44 104)(12 96 45 105)(13 97 41 101)(14 98 42 102)(15 99 43 103)(21 108 34 89)(22 109 35 90)(23 110 31 86)(24 106 32 87)(25 107 33 88)(26 120 40 111)(27 116 36 112)(28 117 37 113)(29 118 38 114)(30 119 39 115)(46 150 70 126)(47 146 66 127)(48 147 67 128)(49 148 68 129)(50 149 69 130)(56 145 65 136)(57 141 61 137)(58 142 62 138)(59 143 63 139)(60 144 64 140)(71 160 80 151)(72 156 76 152)(73 157 77 153)(74 158 78 154)(75 159 79 155)
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), (1,37,12,66)(2,38,13,67)(3,39,14,68)(4,40,15,69)(5,36,11,70)(6,23,144,156)(7,24,145,157)(8,25,141,158)(9,21,142,159)(10,22,143,160)(16,33,137,154)(17,34,138,155)(18,35,139,151)(19,31,140,152)(20,32,136,153)(26,43,50,52)(27,44,46,53)(28,45,47,54)(29,41,48,55)(30,42,49,51)(56,87,94,73)(57,88,95,74)(58,89,91,75)(59,90,92,71)(60,86,93,72)(61,107,82,78)(62,108,83,79)(63,109,84,80)(64,110,85,76)(65,106,81,77)(96,113,134,127)(97,114,135,128)(98,115,131,129)(99,111,132,130)(100,112,133,126)(101,118,122,147)(102,119,123,148)(103,120,124,149)(104,116,125,150)(105,117,121,146), (1,47)(2,48)(3,49)(4,50)(5,46)(6,140)(7,136)(8,137)(9,138)(10,139)(11,27)(12,28)(13,29)(14,30)(15,26)(16,141)(17,142)(18,143)(19,144)(20,145)(21,34)(22,35)(23,31)(24,32)(25,33)(36,44)(37,45)(38,41)(39,42)(40,43)(51,68)(52,69)(53,70)(54,66)(55,67)(56,77)(57,78)(58,79)(59,80)(60,76)(61,74)(62,75)(63,71)(64,72)(65,73)(81,87)(82,88)(83,89)(84,90)(85,86)(91,108)(92,109)(93,110)(94,106)(95,107)(96,121)(97,122)(98,123)(99,124)(100,125)(101,135)(102,131)(103,132)(104,133)(105,134)(111,120)(112,116)(113,117)(114,118)(115,119)(126,150)(127,146)(128,147)(129,148)(130,149)(151,160)(152,156)(153,157)(154,158)(155,159), (1,87,54,106)(2,88,55,107)(3,89,51,108)(4,90,52,109)(5,86,53,110)(6,112,19,116)(7,113,20,117)(8,114,16,118)(9,115,17,119)(10,111,18,120)(11,72,44,76)(12,73,45,77)(13,74,41,78)(14,75,42,79)(15,71,43,80)(21,131,34,123)(22,132,35,124)(23,133,31,125)(24,134,32,121)(25,135,33,122)(26,84,40,92)(27,85,36,93)(28,81,37,94)(29,82,38,95)(30,83,39,91)(46,64,70,60)(47,65,66,56)(48,61,67,57)(49,62,68,58)(50,63,69,59)(96,153,105,157)(97,154,101,158)(98,155,102,159)(99,151,103,160)(100,152,104,156)(126,140,150,144)(127,136,146,145)(128,137,147,141)(129,138,148,142)(130,139,149,143), (1,134,54,121)(2,135,55,122)(3,131,51,123)(4,132,52,124)(5,133,53,125)(6,85,19,93)(7,81,20,94)(8,82,16,95)(9,83,17,91)(10,84,18,92)(11,100,44,104)(12,96,45,105)(13,97,41,101)(14,98,42,102)(15,99,43,103)(21,108,34,89)(22,109,35,90)(23,110,31,86)(24,106,32,87)(25,107,33,88)(26,120,40,111)(27,116,36,112)(28,117,37,113)(29,118,38,114)(30,119,39,115)(46,150,70,126)(47,146,66,127)(48,147,67,128)(49,148,68,129)(50,149,69,130)(56,145,65,136)(57,141,61,137)(58,142,62,138)(59,143,63,139)(60,144,64,140)(71,160,80,151)(72,156,76,152)(73,157,77,153)(74,158,78,154)(75,159,79,155)>;
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), (1,37,12,66)(2,38,13,67)(3,39,14,68)(4,40,15,69)(5,36,11,70)(6,23,144,156)(7,24,145,157)(8,25,141,158)(9,21,142,159)(10,22,143,160)(16,33,137,154)(17,34,138,155)(18,35,139,151)(19,31,140,152)(20,32,136,153)(26,43,50,52)(27,44,46,53)(28,45,47,54)(29,41,48,55)(30,42,49,51)(56,87,94,73)(57,88,95,74)(58,89,91,75)(59,90,92,71)(60,86,93,72)(61,107,82,78)(62,108,83,79)(63,109,84,80)(64,110,85,76)(65,106,81,77)(96,113,134,127)(97,114,135,128)(98,115,131,129)(99,111,132,130)(100,112,133,126)(101,118,122,147)(102,119,123,148)(103,120,124,149)(104,116,125,150)(105,117,121,146), (1,47)(2,48)(3,49)(4,50)(5,46)(6,140)(7,136)(8,137)(9,138)(10,139)(11,27)(12,28)(13,29)(14,30)(15,26)(16,141)(17,142)(18,143)(19,144)(20,145)(21,34)(22,35)(23,31)(24,32)(25,33)(36,44)(37,45)(38,41)(39,42)(40,43)(51,68)(52,69)(53,70)(54,66)(55,67)(56,77)(57,78)(58,79)(59,80)(60,76)(61,74)(62,75)(63,71)(64,72)(65,73)(81,87)(82,88)(83,89)(84,90)(85,86)(91,108)(92,109)(93,110)(94,106)(95,107)(96,121)(97,122)(98,123)(99,124)(100,125)(101,135)(102,131)(103,132)(104,133)(105,134)(111,120)(112,116)(113,117)(114,118)(115,119)(126,150)(127,146)(128,147)(129,148)(130,149)(151,160)(152,156)(153,157)(154,158)(155,159), (1,87,54,106)(2,88,55,107)(3,89,51,108)(4,90,52,109)(5,86,53,110)(6,112,19,116)(7,113,20,117)(8,114,16,118)(9,115,17,119)(10,111,18,120)(11,72,44,76)(12,73,45,77)(13,74,41,78)(14,75,42,79)(15,71,43,80)(21,131,34,123)(22,132,35,124)(23,133,31,125)(24,134,32,121)(25,135,33,122)(26,84,40,92)(27,85,36,93)(28,81,37,94)(29,82,38,95)(30,83,39,91)(46,64,70,60)(47,65,66,56)(48,61,67,57)(49,62,68,58)(50,63,69,59)(96,153,105,157)(97,154,101,158)(98,155,102,159)(99,151,103,160)(100,152,104,156)(126,140,150,144)(127,136,146,145)(128,137,147,141)(129,138,148,142)(130,139,149,143), (1,134,54,121)(2,135,55,122)(3,131,51,123)(4,132,52,124)(5,133,53,125)(6,85,19,93)(7,81,20,94)(8,82,16,95)(9,83,17,91)(10,84,18,92)(11,100,44,104)(12,96,45,105)(13,97,41,101)(14,98,42,102)(15,99,43,103)(21,108,34,89)(22,109,35,90)(23,110,31,86)(24,106,32,87)(25,107,33,88)(26,120,40,111)(27,116,36,112)(28,117,37,113)(29,118,38,114)(30,119,39,115)(46,150,70,126)(47,146,66,127)(48,147,67,128)(49,148,68,129)(50,149,69,130)(56,145,65,136)(57,141,61,137)(58,142,62,138)(59,143,63,139)(60,144,64,140)(71,160,80,151)(72,156,76,152)(73,157,77,153)(74,158,78,154)(75,159,79,155) );
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)], [(1,37,12,66),(2,38,13,67),(3,39,14,68),(4,40,15,69),(5,36,11,70),(6,23,144,156),(7,24,145,157),(8,25,141,158),(9,21,142,159),(10,22,143,160),(16,33,137,154),(17,34,138,155),(18,35,139,151),(19,31,140,152),(20,32,136,153),(26,43,50,52),(27,44,46,53),(28,45,47,54),(29,41,48,55),(30,42,49,51),(56,87,94,73),(57,88,95,74),(58,89,91,75),(59,90,92,71),(60,86,93,72),(61,107,82,78),(62,108,83,79),(63,109,84,80),(64,110,85,76),(65,106,81,77),(96,113,134,127),(97,114,135,128),(98,115,131,129),(99,111,132,130),(100,112,133,126),(101,118,122,147),(102,119,123,148),(103,120,124,149),(104,116,125,150),(105,117,121,146)], [(1,47),(2,48),(3,49),(4,50),(5,46),(6,140),(7,136),(8,137),(9,138),(10,139),(11,27),(12,28),(13,29),(14,30),(15,26),(16,141),(17,142),(18,143),(19,144),(20,145),(21,34),(22,35),(23,31),(24,32),(25,33),(36,44),(37,45),(38,41),(39,42),(40,43),(51,68),(52,69),(53,70),(54,66),(55,67),(56,77),(57,78),(58,79),(59,80),(60,76),(61,74),(62,75),(63,71),(64,72),(65,73),(81,87),(82,88),(83,89),(84,90),(85,86),(91,108),(92,109),(93,110),(94,106),(95,107),(96,121),(97,122),(98,123),(99,124),(100,125),(101,135),(102,131),(103,132),(104,133),(105,134),(111,120),(112,116),(113,117),(114,118),(115,119),(126,150),(127,146),(128,147),(129,148),(130,149),(151,160),(152,156),(153,157),(154,158),(155,159)], [(1,87,54,106),(2,88,55,107),(3,89,51,108),(4,90,52,109),(5,86,53,110),(6,112,19,116),(7,113,20,117),(8,114,16,118),(9,115,17,119),(10,111,18,120),(11,72,44,76),(12,73,45,77),(13,74,41,78),(14,75,42,79),(15,71,43,80),(21,131,34,123),(22,132,35,124),(23,133,31,125),(24,134,32,121),(25,135,33,122),(26,84,40,92),(27,85,36,93),(28,81,37,94),(29,82,38,95),(30,83,39,91),(46,64,70,60),(47,65,66,56),(48,61,67,57),(49,62,68,58),(50,63,69,59),(96,153,105,157),(97,154,101,158),(98,155,102,159),(99,151,103,160),(100,152,104,156),(126,140,150,144),(127,136,146,145),(128,137,147,141),(129,138,148,142),(130,139,149,143)], [(1,134,54,121),(2,135,55,122),(3,131,51,123),(4,132,52,124),(5,133,53,125),(6,85,19,93),(7,81,20,94),(8,82,16,95),(9,83,17,91),(10,84,18,92),(11,100,44,104),(12,96,45,105),(13,97,41,101),(14,98,42,102),(15,99,43,103),(21,108,34,89),(22,109,35,90),(23,110,31,86),(24,106,32,87),(25,107,33,88),(26,120,40,111),(27,116,36,112),(28,117,37,113),(29,118,38,114),(30,119,39,115),(46,150,70,126),(47,146,66,127),(48,147,67,128),(49,148,68,129),(50,149,69,130),(56,145,65,136),(57,141,61,137),(58,142,62,138),(59,143,63,139),(60,144,64,140),(71,160,80,151),(72,156,76,152),(73,157,77,153),(74,158,78,154),(75,159,79,155)]])
95 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 10A | ··· | 10L | 10M | ··· | 10T | 20A | ··· | 20P | 20Q | ··· | 20AB | 20AC | ··· | 20AJ | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 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 | Q8 | SD16 | C4○D4 | C5×D4 | C5×Q8 | C5×SD16 | C5×C4○D4 | C8⋊C22 | C5×C8⋊C22 |
| kernel | C5×D4⋊2Q8 | C5×D4⋊C4 | C5×C4⋊C8 | C5×C4.Q8 | D4×C20 | C5×C4⋊Q8 | D4⋊2Q8 | D4⋊C4 | C4⋊C8 | C4.Q8 | C4×D4 | C4⋊Q8 | C2×C20 | C5×D4 | C20 | C20 | C2×C4 | D4 | C4 | C4 | C10 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 4 | 8 | 4 | 8 | 4 | 4 | 2 | 2 | 4 | 2 | 8 | 8 | 16 | 8 | 1 | 4 |
Matrix representation of C5×D4⋊2Q8 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 10 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 0 |
| 1 | 0 | 0 | 0 |
| 21 | 40 | 0 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 0 | 40 | 0 |
| 32 | 0 | 0 | 0 |
| 16 | 9 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 0 |
| 21 | 39 | 0 | 0 |
| 16 | 20 | 0 | 0 |
| 0 | 0 | 26 | 15 |
| 0 | 0 | 15 | 15 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,10,0,0,0,0,10],[40,0,0,0,0,40,0,0,0,0,0,40,0,0,1,0],[1,21,0,0,0,40,0,0,0,0,0,40,0,0,40,0],[32,16,0,0,0,9,0,0,0,0,0,40,0,0,1,0],[21,16,0,0,39,20,0,0,0,0,26,15,0,0,15,15] >;
C5×D4⋊2Q8 in GAP, Magma, Sage, TeX
C_5\times D_4\rtimes_2Q_8
% in TeX
G:=Group("C5xD4:2Q8"); // GroupNames label
G:=SmallGroup(320,977);
// by ID
G=gap.SmallGroup(320,977);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,1400,589,288,1766,10085,2539,124]);
// 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=e*b*e^-1=b^-1,b*d=d*b,d*c*d^-1=b^2*c,e*c*e^-1=b*c,e*d*e^-1=d^-1>;
// generators/relations