direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×D4⋊3Q8, C10.1652+ 1+4, D4⋊3(C5×Q8), C4⋊Q8⋊15C10, (C5×D4)⋊10Q8, (C4×Q8)⋊15C10, (Q8×C20)⋊35C2, C4.18(Q8×C10), (C4×D4).12C10, (D4×C20).27C2, C22⋊Q8⋊17C10, C20.124(C2×Q8), C22.6(Q8×C10), C42.48(C2×C10), C42.C2⋊10C10, C20.325(C4○D4), C10.64(C22×Q8), (C2×C20).680C23, (C2×C10).374C24, (C4×C20).289C22, (D4×C10).335C22, C22.48(C23×C10), C23.45(C22×C10), (Q8×C10).184C22, C2.17(C5×2+ 1+4), (C22×C20).459C22, (C22×C10).268C23, (C5×C4⋊Q8)⋊36C2, (C2×C4⋊C4)⋊23C10, (C10×C4⋊C4)⋊50C2, C2.10(Q8×C2×C10), C4.37(C5×C4○D4), C4⋊C4.74(C2×C10), C2.27(C10×C4○D4), (C5×C22⋊Q8)⋊44C2, (C2×C10).55(C2×Q8), (C2×D4).81(C2×C10), C10.246(C2×C4○D4), (C2×Q8).64(C2×C10), (C5×C42.C2)⋊27C2, (C5×C4⋊C4).400C22, C22⋊C4.24(C2×C10), (C2×C4).36(C22×C10), (C22×C4).70(C2×C10), (C5×C22⋊C4).156C22, SmallGroup(320,1556)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×D4⋊3Q8
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, ece-1=b2c, ede-1=d-1 >
Subgroups: 314 in 228 conjugacy classes, 166 normal (34 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C20, C20, C2×C10, C2×C10, C2×C10, C2×C4⋊C4, C4×D4, C4×D4, C4×Q8, C22⋊Q8, C42.C2, C4⋊Q8, C2×C20, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, D4⋊3Q8, C4×C20, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C22×C20, D4×C10, Q8×C10, Q8×C10, C10×C4⋊C4, D4×C20, D4×C20, Q8×C20, C5×C22⋊Q8, C5×C42.C2, C5×C4⋊Q8, C5×D4⋊3Q8
Quotients: C1, C2, C22, C5, Q8, C23, C10, C2×Q8, C4○D4, C24, C2×C10, C22×Q8, C2×C4○D4, 2+ 1+4, C5×Q8, C22×C10, D4⋊3Q8, Q8×C10, C5×C4○D4, C23×C10, Q8×C2×C10, C10×C4○D4, C5×2+ 1+4, C5×D4⋊3Q8
►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)
(1 127 21 135)(2 128 22 131)(3 129 23 132)(4 130 24 133)(5 126 25 134)(6 74 20 66)(7 75 16 67)(8 71 17 68)(9 72 18 69)(10 73 19 70)(11 62 157 57)(12 63 158 58)(13 64 159 59)(14 65 160 60)(15 61 156 56)(26 120 34 125)(27 116 35 121)(28 117 31 122)(29 118 32 123)(30 119 33 124)(36 101 41 96)(37 102 42 97)(38 103 43 98)(39 104 44 99)(40 105 45 100)(46 114 54 106)(47 115 55 107)(48 111 51 108)(49 112 52 109)(50 113 53 110)(76 147 81 155)(77 148 82 151)(78 149 83 152)(79 150 84 153)(80 146 85 154)(86 140 94 145)(87 136 95 141)(88 137 91 142)(89 138 92 143)(90 139 93 144)
(1 135)(2 131)(3 132)(4 133)(5 134)(6 74)(7 75)(8 71)(9 72)(10 73)(11 62)(12 63)(13 64)(14 65)(15 61)(16 67)(17 68)(18 69)(19 70)(20 66)(21 127)(22 128)(23 129)(24 130)(25 126)(26 125)(27 121)(28 122)(29 123)(30 124)(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)(81 147)(82 148)(83 149)(84 150)(85 146)(86 145)(87 141)(88 142)(89 143)(90 144)(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)(6 140 160 154)(7 136 156 155)(8 137 157 151)(9 138 158 152)(10 139 159 153)(11 148 17 142)(12 149 18 143)(13 150 19 144)(14 146 20 145)(15 147 16 141)(21 47 27 41)(22 48 28 42)(23 49 29 43)(24 50 30 44)(25 46 26 45)(56 76 75 95)(57 77 71 91)(58 78 72 92)(59 79 73 93)(60 80 74 94)(61 81 67 87)(62 82 68 88)(63 83 69 89)(64 84 70 90)(65 85 66 86)(96 135 115 116)(97 131 111 117)(98 132 112 118)(99 133 113 119)(100 134 114 120)(101 127 107 121)(102 128 108 122)(103 129 109 123)(104 130 110 124)(105 126 106 125)
(1 7 35 156)(2 8 31 157)(3 9 32 158)(4 10 33 159)(5 6 34 160)(11 22 17 28)(12 23 18 29)(13 24 19 30)(14 25 20 26)(15 21 16 27)(36 136 55 155)(37 137 51 151)(38 138 52 152)(39 139 53 153)(40 140 54 154)(41 141 47 147)(42 142 48 148)(43 143 49 149)(44 144 50 150)(45 145 46 146)(56 127 75 121)(57 128 71 122)(58 129 72 123)(59 130 73 124)(60 126 74 125)(61 135 67 116)(62 131 68 117)(63 132 69 118)(64 133 70 119)(65 134 66 120)(76 101 95 107)(77 102 91 108)(78 103 92 109)(79 104 93 110)(80 105 94 106)(81 96 87 115)(82 97 88 111)(83 98 89 112)(84 99 90 113)(85 100 86 114)
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), (1,127,21,135)(2,128,22,131)(3,129,23,132)(4,130,24,133)(5,126,25,134)(6,74,20,66)(7,75,16,67)(8,71,17,68)(9,72,18,69)(10,73,19,70)(11,62,157,57)(12,63,158,58)(13,64,159,59)(14,65,160,60)(15,61,156,56)(26,120,34,125)(27,116,35,121)(28,117,31,122)(29,118,32,123)(30,119,33,124)(36,101,41,96)(37,102,42,97)(38,103,43,98)(39,104,44,99)(40,105,45,100)(46,114,54,106)(47,115,55,107)(48,111,51,108)(49,112,52,109)(50,113,53,110)(76,147,81,155)(77,148,82,151)(78,149,83,152)(79,150,84,153)(80,146,85,154)(86,140,94,145)(87,136,95,141)(88,137,91,142)(89,138,92,143)(90,139,93,144), (1,135)(2,131)(3,132)(4,133)(5,134)(6,74)(7,75)(8,71)(9,72)(10,73)(11,62)(12,63)(13,64)(14,65)(15,61)(16,67)(17,68)(18,69)(19,70)(20,66)(21,127)(22,128)(23,129)(24,130)(25,126)(26,125)(27,121)(28,122)(29,123)(30,124)(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)(81,147)(82,148)(83,149)(84,150)(85,146)(86,145)(87,141)(88,142)(89,143)(90,144)(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)(6,140,160,154)(7,136,156,155)(8,137,157,151)(9,138,158,152)(10,139,159,153)(11,148,17,142)(12,149,18,143)(13,150,19,144)(14,146,20,145)(15,147,16,141)(21,47,27,41)(22,48,28,42)(23,49,29,43)(24,50,30,44)(25,46,26,45)(56,76,75,95)(57,77,71,91)(58,78,72,92)(59,79,73,93)(60,80,74,94)(61,81,67,87)(62,82,68,88)(63,83,69,89)(64,84,70,90)(65,85,66,86)(96,135,115,116)(97,131,111,117)(98,132,112,118)(99,133,113,119)(100,134,114,120)(101,127,107,121)(102,128,108,122)(103,129,109,123)(104,130,110,124)(105,126,106,125), (1,7,35,156)(2,8,31,157)(3,9,32,158)(4,10,33,159)(5,6,34,160)(11,22,17,28)(12,23,18,29)(13,24,19,30)(14,25,20,26)(15,21,16,27)(36,136,55,155)(37,137,51,151)(38,138,52,152)(39,139,53,153)(40,140,54,154)(41,141,47,147)(42,142,48,148)(43,143,49,149)(44,144,50,150)(45,145,46,146)(56,127,75,121)(57,128,71,122)(58,129,72,123)(59,130,73,124)(60,126,74,125)(61,135,67,116)(62,131,68,117)(63,132,69,118)(64,133,70,119)(65,134,66,120)(76,101,95,107)(77,102,91,108)(78,103,92,109)(79,104,93,110)(80,105,94,106)(81,96,87,115)(82,97,88,111)(83,98,89,112)(84,99,90,113)(85,100,86,114)>;
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), (1,127,21,135)(2,128,22,131)(3,129,23,132)(4,130,24,133)(5,126,25,134)(6,74,20,66)(7,75,16,67)(8,71,17,68)(9,72,18,69)(10,73,19,70)(11,62,157,57)(12,63,158,58)(13,64,159,59)(14,65,160,60)(15,61,156,56)(26,120,34,125)(27,116,35,121)(28,117,31,122)(29,118,32,123)(30,119,33,124)(36,101,41,96)(37,102,42,97)(38,103,43,98)(39,104,44,99)(40,105,45,100)(46,114,54,106)(47,115,55,107)(48,111,51,108)(49,112,52,109)(50,113,53,110)(76,147,81,155)(77,148,82,151)(78,149,83,152)(79,150,84,153)(80,146,85,154)(86,140,94,145)(87,136,95,141)(88,137,91,142)(89,138,92,143)(90,139,93,144), (1,135)(2,131)(3,132)(4,133)(5,134)(6,74)(7,75)(8,71)(9,72)(10,73)(11,62)(12,63)(13,64)(14,65)(15,61)(16,67)(17,68)(18,69)(19,70)(20,66)(21,127)(22,128)(23,129)(24,130)(25,126)(26,125)(27,121)(28,122)(29,123)(30,124)(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)(81,147)(82,148)(83,149)(84,150)(85,146)(86,145)(87,141)(88,142)(89,143)(90,144)(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)(6,140,160,154)(7,136,156,155)(8,137,157,151)(9,138,158,152)(10,139,159,153)(11,148,17,142)(12,149,18,143)(13,150,19,144)(14,146,20,145)(15,147,16,141)(21,47,27,41)(22,48,28,42)(23,49,29,43)(24,50,30,44)(25,46,26,45)(56,76,75,95)(57,77,71,91)(58,78,72,92)(59,79,73,93)(60,80,74,94)(61,81,67,87)(62,82,68,88)(63,83,69,89)(64,84,70,90)(65,85,66,86)(96,135,115,116)(97,131,111,117)(98,132,112,118)(99,133,113,119)(100,134,114,120)(101,127,107,121)(102,128,108,122)(103,129,109,123)(104,130,110,124)(105,126,106,125), (1,7,35,156)(2,8,31,157)(3,9,32,158)(4,10,33,159)(5,6,34,160)(11,22,17,28)(12,23,18,29)(13,24,19,30)(14,25,20,26)(15,21,16,27)(36,136,55,155)(37,137,51,151)(38,138,52,152)(39,139,53,153)(40,140,54,154)(41,141,47,147)(42,142,48,148)(43,143,49,149)(44,144,50,150)(45,145,46,146)(56,127,75,121)(57,128,71,122)(58,129,72,123)(59,130,73,124)(60,126,74,125)(61,135,67,116)(62,131,68,117)(63,132,69,118)(64,133,70,119)(65,134,66,120)(76,101,95,107)(77,102,91,108)(78,103,92,109)(79,104,93,110)(80,105,94,106)(81,96,87,115)(82,97,88,111)(83,98,89,112)(84,99,90,113)(85,100,86,114) );
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)], [(1,127,21,135),(2,128,22,131),(3,129,23,132),(4,130,24,133),(5,126,25,134),(6,74,20,66),(7,75,16,67),(8,71,17,68),(9,72,18,69),(10,73,19,70),(11,62,157,57),(12,63,158,58),(13,64,159,59),(14,65,160,60),(15,61,156,56),(26,120,34,125),(27,116,35,121),(28,117,31,122),(29,118,32,123),(30,119,33,124),(36,101,41,96),(37,102,42,97),(38,103,43,98),(39,104,44,99),(40,105,45,100),(46,114,54,106),(47,115,55,107),(48,111,51,108),(49,112,52,109),(50,113,53,110),(76,147,81,155),(77,148,82,151),(78,149,83,152),(79,150,84,153),(80,146,85,154),(86,140,94,145),(87,136,95,141),(88,137,91,142),(89,138,92,143),(90,139,93,144)], [(1,135),(2,131),(3,132),(4,133),(5,134),(6,74),(7,75),(8,71),(9,72),(10,73),(11,62),(12,63),(13,64),(14,65),(15,61),(16,67),(17,68),(18,69),(19,70),(20,66),(21,127),(22,128),(23,129),(24,130),(25,126),(26,125),(27,121),(28,122),(29,123),(30,124),(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),(81,147),(82,148),(83,149),(84,150),(85,146),(86,145),(87,141),(88,142),(89,143),(90,144),(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),(6,140,160,154),(7,136,156,155),(8,137,157,151),(9,138,158,152),(10,139,159,153),(11,148,17,142),(12,149,18,143),(13,150,19,144),(14,146,20,145),(15,147,16,141),(21,47,27,41),(22,48,28,42),(23,49,29,43),(24,50,30,44),(25,46,26,45),(56,76,75,95),(57,77,71,91),(58,78,72,92),(59,79,73,93),(60,80,74,94),(61,81,67,87),(62,82,68,88),(63,83,69,89),(64,84,70,90),(65,85,66,86),(96,135,115,116),(97,131,111,117),(98,132,112,118),(99,133,113,119),(100,134,114,120),(101,127,107,121),(102,128,108,122),(103,129,109,123),(104,130,110,124),(105,126,106,125)], [(1,7,35,156),(2,8,31,157),(3,9,32,158),(4,10,33,159),(5,6,34,160),(11,22,17,28),(12,23,18,29),(13,24,19,30),(14,25,20,26),(15,21,16,27),(36,136,55,155),(37,137,51,151),(38,138,52,152),(39,139,53,153),(40,140,54,154),(41,141,47,147),(42,142,48,148),(43,143,49,149),(44,144,50,150),(45,145,46,146),(56,127,75,121),(57,128,71,122),(58,129,72,123),(59,130,73,124),(60,126,74,125),(61,135,67,116),(62,131,68,117),(63,132,69,118),(64,133,70,119),(65,134,66,120),(76,101,95,107),(77,102,91,108),(78,103,92,109),(79,104,93,110),(80,105,94,106),(81,96,87,115),(82,97,88,111),(83,98,89,112),(84,99,90,113),(85,100,86,114)]])
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 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | C10 | Q8 | C4○D4 | C5×Q8 | C5×C4○D4 | 2+ 1+4 | C5×2+ 1+4 |
| kernel | C5×D4⋊3Q8 | C10×C4⋊C4 | D4×C20 | Q8×C20 | C5×C22⋊Q8 | C5×C42.C2 | C5×C4⋊Q8 | D4⋊3Q8 | C2×C4⋊C4 | C4×D4 | C4×Q8 | C22⋊Q8 | C42.C2 | C4⋊Q8 | C5×D4 | C20 | D4 | C4 | C10 | C2 |
| # reps | 1 | 2 | 3 | 1 | 6 | 2 | 1 | 4 | 8 | 12 | 4 | 24 | 8 | 4 | 4 | 4 | 16 | 16 | 1 | 4 |
Matrix representation of C5×D4⋊3Q8 ►in GL4(𝔽41) generated by
| 18 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 13 | 39 |
| 0 | 0 | 3 | 28 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 13 | 39 |
| 0 | 0 | 2 | 28 |
| 0 | 1 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 20 | 38 | 0 | 0 |
| 38 | 21 | 0 | 0 |
| 0 | 0 | 35 | 23 |
| 0 | 0 | 27 | 6 |
G:=sub<GL(4,GF(41))| [18,0,0,0,0,18,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,13,3,0,0,39,28],[1,0,0,0,0,1,0,0,0,0,13,2,0,0,39,28],[0,40,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[20,38,0,0,38,21,0,0,0,0,35,27,0,0,23,6] >;
C5×D4⋊3Q8 in GAP, Magma, Sage, TeX
C_5\times D_4\rtimes_3Q_8
% in TeX
G:=Group("C5xD4:3Q8"); // GroupNames label
G:=SmallGroup(320,1556);
// by ID
G=gap.SmallGroup(320,1556);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,560,1149,1688,3446,1242,304]);
// 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,e*c*e^-1=b^2*c,e*d*e^-1=d^-1>;
// generators/relations