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, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, Q8, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C20, C20, C2×C10, C2×C10, C2×C10, C4×D4, C4×Q8, C22⋊Q8, C4⋊Q8, C22×Q8, C2×C20, C2×C20, C2×C20, C5×D4, C5×Q8, C5×Q8, C22×C10, D4×Q8, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C22×C20, D4×C10, Q8×C10, Q8×C10, Q8×C10, D4×C20, Q8×C20, C5×C22⋊Q8, C5×C4⋊Q8, Q8×C2×C10, C5×D4×Q8
Quotients: C1, C2, C22, C5, D4, Q8, C23, C10, C2×D4, C2×Q8, C24, C2×C10, C22×D4, C22×Q8, 2- 1+4, C5×D4, C5×Q8, C22×C10, D4×Q8, D4×C10, Q8×C10, 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)
(1 131 21 126)(2 132 22 127)(3 133 23 128)(4 134 24 129)(5 135 25 130)(6 61 156 56)(7 62 157 57)(8 63 158 58)(9 64 159 59)(10 65 160 60)(11 66 16 71)(12 67 17 72)(13 68 18 73)(14 69 19 74)(15 70 20 75)(26 121 31 116)(27 122 32 117)(28 123 33 118)(29 124 34 119)(30 125 35 120)(36 101 41 96)(37 102 42 97)(38 103 43 98)(39 104 44 99)(40 105 45 100)(46 111 51 106)(47 112 52 107)(48 113 53 108)(49 114 54 109)(50 115 55 110)(76 151 81 146)(77 152 82 147)(78 153 83 148)(79 154 84 149)(80 155 85 150)(86 141 91 136)(87 142 92 137)(88 143 93 138)(89 144 94 139)(90 145 95 140)
(1 126)(2 127)(3 128)(4 129)(5 130)(6 61)(7 62)(8 63)(9 64)(10 65)(11 66)(12 67)(13 68)(14 69)(15 70)(16 71)(17 72)(18 73)(19 74)(20 75)(21 131)(22 132)(23 133)(24 134)(25 135)(26 116)(27 117)(28 118)(29 119)(30 120)(31 121)(32 122)(33 123)(34 124)(35 125)(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)(76 146)(77 147)(78 148)(79 149)(80 150)(81 151)(82 152)(83 153)(84 154)(85 155)(86 136)(87 137)(88 138)(89 139)(90 140)(91 141)(92 142)(93 143)(94 144)(95 145)
(1 46 26 36)(2 47 27 37)(3 48 28 38)(4 49 29 39)(5 50 30 40)(6 151 16 141)(7 152 17 142)(8 153 18 143)(9 154 19 144)(10 155 20 145)(11 136 156 146)(12 137 157 147)(13 138 158 148)(14 139 159 149)(15 140 160 150)(21 51 31 41)(22 52 32 42)(23 53 33 43)(24 54 34 44)(25 55 35 45)(56 76 66 86)(57 77 67 87)(58 78 68 88)(59 79 69 89)(60 80 70 90)(61 81 71 91)(62 82 72 92)(63 83 73 93)(64 84 74 94)(65 85 75 95)(96 126 106 116)(97 127 107 117)(98 128 108 118)(99 129 109 119)(100 130 110 120)(101 131 111 121)(102 132 112 122)(103 133 113 123)(104 134 114 124)(105 135 115 125)
(1 66 26 56)(2 67 27 57)(3 68 28 58)(4 69 29 59)(5 70 30 60)(6 131 16 121)(7 132 17 122)(8 133 18 123)(9 134 19 124)(10 135 20 125)(11 116 156 126)(12 117 157 127)(13 118 158 128)(14 119 159 129)(15 120 160 130)(21 71 31 61)(22 72 32 62)(23 73 33 63)(24 74 34 64)(25 75 35 65)(36 86 46 76)(37 87 47 77)(38 88 48 78)(39 89 49 79)(40 90 50 80)(41 91 51 81)(42 92 52 82)(43 93 53 83)(44 94 54 84)(45 95 55 85)(96 136 106 146)(97 137 107 147)(98 138 108 148)(99 139 109 149)(100 140 110 150)(101 141 111 151)(102 142 112 152)(103 143 113 153)(104 144 114 154)(105 145 115 155)
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), (1,131,21,126)(2,132,22,127)(3,133,23,128)(4,134,24,129)(5,135,25,130)(6,61,156,56)(7,62,157,57)(8,63,158,58)(9,64,159,59)(10,65,160,60)(11,66,16,71)(12,67,17,72)(13,68,18,73)(14,69,19,74)(15,70,20,75)(26,121,31,116)(27,122,32,117)(28,123,33,118)(29,124,34,119)(30,125,35,120)(36,101,41,96)(37,102,42,97)(38,103,43,98)(39,104,44,99)(40,105,45,100)(46,111,51,106)(47,112,52,107)(48,113,53,108)(49,114,54,109)(50,115,55,110)(76,151,81,146)(77,152,82,147)(78,153,83,148)(79,154,84,149)(80,155,85,150)(86,141,91,136)(87,142,92,137)(88,143,93,138)(89,144,94,139)(90,145,95,140), (1,126)(2,127)(3,128)(4,129)(5,130)(6,61)(7,62)(8,63)(9,64)(10,65)(11,66)(12,67)(13,68)(14,69)(15,70)(16,71)(17,72)(18,73)(19,74)(20,75)(21,131)(22,132)(23,133)(24,134)(25,135)(26,116)(27,117)(28,118)(29,119)(30,120)(31,121)(32,122)(33,123)(34,124)(35,125)(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)(76,146)(77,147)(78,148)(79,149)(80,150)(81,151)(82,152)(83,153)(84,154)(85,155)(86,136)(87,137)(88,138)(89,139)(90,140)(91,141)(92,142)(93,143)(94,144)(95,145), (1,46,26,36)(2,47,27,37)(3,48,28,38)(4,49,29,39)(5,50,30,40)(6,151,16,141)(7,152,17,142)(8,153,18,143)(9,154,19,144)(10,155,20,145)(11,136,156,146)(12,137,157,147)(13,138,158,148)(14,139,159,149)(15,140,160,150)(21,51,31,41)(22,52,32,42)(23,53,33,43)(24,54,34,44)(25,55,35,45)(56,76,66,86)(57,77,67,87)(58,78,68,88)(59,79,69,89)(60,80,70,90)(61,81,71,91)(62,82,72,92)(63,83,73,93)(64,84,74,94)(65,85,75,95)(96,126,106,116)(97,127,107,117)(98,128,108,118)(99,129,109,119)(100,130,110,120)(101,131,111,121)(102,132,112,122)(103,133,113,123)(104,134,114,124)(105,135,115,125), (1,66,26,56)(2,67,27,57)(3,68,28,58)(4,69,29,59)(5,70,30,60)(6,131,16,121)(7,132,17,122)(8,133,18,123)(9,134,19,124)(10,135,20,125)(11,116,156,126)(12,117,157,127)(13,118,158,128)(14,119,159,129)(15,120,160,130)(21,71,31,61)(22,72,32,62)(23,73,33,63)(24,74,34,64)(25,75,35,65)(36,86,46,76)(37,87,47,77)(38,88,48,78)(39,89,49,79)(40,90,50,80)(41,91,51,81)(42,92,52,82)(43,93,53,83)(44,94,54,84)(45,95,55,85)(96,136,106,146)(97,137,107,147)(98,138,108,148)(99,139,109,149)(100,140,110,150)(101,141,111,151)(102,142,112,152)(103,143,113,153)(104,144,114,154)(105,145,115,155)>;
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), (1,131,21,126)(2,132,22,127)(3,133,23,128)(4,134,24,129)(5,135,25,130)(6,61,156,56)(7,62,157,57)(8,63,158,58)(9,64,159,59)(10,65,160,60)(11,66,16,71)(12,67,17,72)(13,68,18,73)(14,69,19,74)(15,70,20,75)(26,121,31,116)(27,122,32,117)(28,123,33,118)(29,124,34,119)(30,125,35,120)(36,101,41,96)(37,102,42,97)(38,103,43,98)(39,104,44,99)(40,105,45,100)(46,111,51,106)(47,112,52,107)(48,113,53,108)(49,114,54,109)(50,115,55,110)(76,151,81,146)(77,152,82,147)(78,153,83,148)(79,154,84,149)(80,155,85,150)(86,141,91,136)(87,142,92,137)(88,143,93,138)(89,144,94,139)(90,145,95,140), (1,126)(2,127)(3,128)(4,129)(5,130)(6,61)(7,62)(8,63)(9,64)(10,65)(11,66)(12,67)(13,68)(14,69)(15,70)(16,71)(17,72)(18,73)(19,74)(20,75)(21,131)(22,132)(23,133)(24,134)(25,135)(26,116)(27,117)(28,118)(29,119)(30,120)(31,121)(32,122)(33,123)(34,124)(35,125)(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)(76,146)(77,147)(78,148)(79,149)(80,150)(81,151)(82,152)(83,153)(84,154)(85,155)(86,136)(87,137)(88,138)(89,139)(90,140)(91,141)(92,142)(93,143)(94,144)(95,145), (1,46,26,36)(2,47,27,37)(3,48,28,38)(4,49,29,39)(5,50,30,40)(6,151,16,141)(7,152,17,142)(8,153,18,143)(9,154,19,144)(10,155,20,145)(11,136,156,146)(12,137,157,147)(13,138,158,148)(14,139,159,149)(15,140,160,150)(21,51,31,41)(22,52,32,42)(23,53,33,43)(24,54,34,44)(25,55,35,45)(56,76,66,86)(57,77,67,87)(58,78,68,88)(59,79,69,89)(60,80,70,90)(61,81,71,91)(62,82,72,92)(63,83,73,93)(64,84,74,94)(65,85,75,95)(96,126,106,116)(97,127,107,117)(98,128,108,118)(99,129,109,119)(100,130,110,120)(101,131,111,121)(102,132,112,122)(103,133,113,123)(104,134,114,124)(105,135,115,125), (1,66,26,56)(2,67,27,57)(3,68,28,58)(4,69,29,59)(5,70,30,60)(6,131,16,121)(7,132,17,122)(8,133,18,123)(9,134,19,124)(10,135,20,125)(11,116,156,126)(12,117,157,127)(13,118,158,128)(14,119,159,129)(15,120,160,130)(21,71,31,61)(22,72,32,62)(23,73,33,63)(24,74,34,64)(25,75,35,65)(36,86,46,76)(37,87,47,77)(38,88,48,78)(39,89,49,79)(40,90,50,80)(41,91,51,81)(42,92,52,82)(43,93,53,83)(44,94,54,84)(45,95,55,85)(96,136,106,146)(97,137,107,147)(98,138,108,148)(99,139,109,149)(100,140,110,150)(101,141,111,151)(102,142,112,152)(103,143,113,153)(104,144,114,154)(105,145,115,155) );
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)], [(1,131,21,126),(2,132,22,127),(3,133,23,128),(4,134,24,129),(5,135,25,130),(6,61,156,56),(7,62,157,57),(8,63,158,58),(9,64,159,59),(10,65,160,60),(11,66,16,71),(12,67,17,72),(13,68,18,73),(14,69,19,74),(15,70,20,75),(26,121,31,116),(27,122,32,117),(28,123,33,118),(29,124,34,119),(30,125,35,120),(36,101,41,96),(37,102,42,97),(38,103,43,98),(39,104,44,99),(40,105,45,100),(46,111,51,106),(47,112,52,107),(48,113,53,108),(49,114,54,109),(50,115,55,110),(76,151,81,146),(77,152,82,147),(78,153,83,148),(79,154,84,149),(80,155,85,150),(86,141,91,136),(87,142,92,137),(88,143,93,138),(89,144,94,139),(90,145,95,140)], [(1,126),(2,127),(3,128),(4,129),(5,130),(6,61),(7,62),(8,63),(9,64),(10,65),(11,66),(12,67),(13,68),(14,69),(15,70),(16,71),(17,72),(18,73),(19,74),(20,75),(21,131),(22,132),(23,133),(24,134),(25,135),(26,116),(27,117),(28,118),(29,119),(30,120),(31,121),(32,122),(33,123),(34,124),(35,125),(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),(76,146),(77,147),(78,148),(79,149),(80,150),(81,151),(82,152),(83,153),(84,154),(85,155),(86,136),(87,137),(88,138),(89,139),(90,140),(91,141),(92,142),(93,143),(94,144),(95,145)], [(1,46,26,36),(2,47,27,37),(3,48,28,38),(4,49,29,39),(5,50,30,40),(6,151,16,141),(7,152,17,142),(8,153,18,143),(9,154,19,144),(10,155,20,145),(11,136,156,146),(12,137,157,147),(13,138,158,148),(14,139,159,149),(15,140,160,150),(21,51,31,41),(22,52,32,42),(23,53,33,43),(24,54,34,44),(25,55,35,45),(56,76,66,86),(57,77,67,87),(58,78,68,88),(59,79,69,89),(60,80,70,90),(61,81,71,91),(62,82,72,92),(63,83,73,93),(64,84,74,94),(65,85,75,95),(96,126,106,116),(97,127,107,117),(98,128,108,118),(99,129,109,119),(100,130,110,120),(101,131,111,121),(102,132,112,122),(103,133,113,123),(104,134,114,124),(105,135,115,125)], [(1,66,26,56),(2,67,27,57),(3,68,28,58),(4,69,29,59),(5,70,30,60),(6,131,16,121),(7,132,17,122),(8,133,18,123),(9,134,19,124),(10,135,20,125),(11,116,156,126),(12,117,157,127),(13,118,158,128),(14,119,159,129),(15,120,160,130),(21,71,31,61),(22,72,32,62),(23,73,33,63),(24,74,34,64),(25,75,35,65),(36,86,46,76),(37,87,47,77),(38,88,48,78),(39,89,49,79),(40,90,50,80),(41,91,51,81),(42,92,52,82),(43,93,53,83),(44,94,54,84),(45,95,55,85),(96,136,106,146),(97,137,107,147),(98,138,108,148),(99,139,109,149),(100,140,110,150),(101,141,111,151),(102,142,112,152),(103,143,113,153),(104,144,114,154),(105,145,115,155)]])
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