direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×Q8⋊6D4, C10.1622+ 1+4, Q8⋊6(C5×D4), (C5×Q8)⋊24D4, (C4×D4)⋊18C10, (D4×C20)⋊47C2, C4⋊1D4⋊8C10, (Q8×C20)⋊33C2, (C4×Q8)⋊13C10, C4.44(D4×C10), C20⋊19(C4○D4), C4⋊D4⋊14C10, C20.405(C2×D4), C42.45(C2×C10), (C2×C10).370C24, (C4×C20).286C22, (C2×C20).677C23, C10.198(C22×D4), (D4×C10).220C22, C22.44(C23×C10), C23.17(C22×C10), (Q8×C10).286C22, C2.14(C5×2+ 1+4), (C22×C10).101C23, (C22×C20).456C22, C4⋊3(C5×C4○D4), C2.22(D4×C2×C10), (C2×C4○D4)⋊9C10, (C10×C4○D4)⋊25C2, (C5×C4⋊1D4)⋊19C2, C4⋊C4.72(C2×C10), (C5×C4⋊D4)⋊41C2, C2.23(C10×C4○D4), (C2×D4).68(C2×C10), C10.242(C2×C4○D4), (C2×Q8).74(C2×C10), (C5×C4⋊C4).398C22, C22⋊C4.21(C2×C10), (C22×C4).68(C2×C10), (C2×C4).137(C22×C10), (C5×C22⋊C4).154C22, SmallGroup(320,1552)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×Q8⋊6D4
G = < a,b,c,d,e | a5=b4=d4=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=dbd-1=b-1, be=eb, dcd-1=ece=b2c, ede=d-1 >
Subgroups: 506 in 312 conjugacy classes, 166 normal (20 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C20, C20, C2×C10, C2×C10, C4×D4, C4×Q8, C4⋊D4, C4⋊1D4, C2×C4○D4, C2×C20, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, Q8⋊6D4, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C22×C20, D4×C10, Q8×C10, C5×C4○D4, D4×C20, Q8×C20, C5×C4⋊D4, C5×C4⋊1D4, C10×C4○D4, C5×Q8⋊6D4
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C4○D4, C24, C2×C10, C22×D4, C2×C4○D4, 2+ 1+4, C5×D4, C22×C10, Q8⋊6D4, D4×C10, C5×C4○D4, C23×C10, D4×C2×C10, C10×C4○D4, C5×2+ 1+4, C5×Q8⋊6D4
►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 47 35 41)(2 48 31 42)(3 49 32 43)(4 50 33 44)(5 46 34 45)(6 146 160 145)(7 147 156 141)(8 148 157 142)(9 149 158 143)(10 150 159 144)(11 137 17 151)(12 138 18 152)(13 139 19 153)(14 140 20 154)(15 136 16 155)(21 55 27 36)(22 51 28 37)(23 52 29 38)(24 53 30 39)(25 54 26 40)(56 87 75 81)(57 88 71 82)(58 89 72 83)(59 90 73 84)(60 86 74 85)(61 95 67 76)(62 91 68 77)(63 92 69 78)(64 93 70 79)(65 94 66 80)(96 127 115 121)(97 128 111 122)(98 129 112 123)(99 130 113 124)(100 126 114 125)(101 135 107 116)(102 131 108 117)(103 132 109 118)(104 133 110 119)(105 134 106 120)
(1 81 35 87)(2 82 31 88)(3 83 32 89)(4 84 33 90)(5 85 34 86)(6 106 160 105)(7 107 156 101)(8 108 157 102)(9 109 158 103)(10 110 159 104)(11 97 17 111)(12 98 18 112)(13 99 19 113)(14 100 20 114)(15 96 16 115)(21 76 27 95)(22 77 28 91)(23 78 29 92)(24 79 30 93)(25 80 26 94)(36 61 55 67)(37 62 51 68)(38 63 52 69)(39 64 53 70)(40 65 54 66)(41 56 47 75)(42 57 48 71)(43 58 49 72)(44 59 50 73)(45 60 46 74)(116 147 135 141)(117 148 131 142)(118 149 132 143)(119 150 133 144)(120 146 134 145)(121 155 127 136)(122 151 128 137)(123 152 129 138)(124 153 130 139)(125 154 126 140)
(1 135 27 121)(2 131 28 122)(3 132 29 123)(4 133 30 124)(5 134 26 125)(6 66 14 60)(7 67 15 56)(8 68 11 57)(9 69 12 58)(10 70 13 59)(16 75 156 61)(17 71 157 62)(18 72 158 63)(19 73 159 64)(20 74 160 65)(21 127 35 116)(22 128 31 117)(23 129 32 118)(24 130 33 119)(25 126 34 120)(36 115 47 101)(37 111 48 102)(38 112 49 103)(39 113 50 104)(40 114 46 105)(41 107 55 96)(42 108 51 97)(43 109 52 98)(44 110 53 99)(45 106 54 100)(76 155 87 141)(77 151 88 142)(78 152 89 143)(79 153 90 144)(80 154 86 145)(81 147 95 136)(82 148 91 137)(83 149 92 138)(84 150 93 139)(85 146 94 140)
(1 96)(2 97)(3 98)(4 99)(5 100)(6 94)(7 95)(8 91)(9 92)(10 93)(11 82)(12 83)(13 84)(14 85)(15 81)(16 87)(17 88)(18 89)(19 90)(20 86)(21 101)(22 102)(23 103)(24 104)(25 105)(26 106)(27 107)(28 108)(29 109)(30 110)(31 111)(32 112)(33 113)(34 114)(35 115)(36 116)(37 117)(38 118)(39 119)(40 120)(41 121)(42 122)(43 123)(44 124)(45 125)(46 126)(47 127)(48 128)(49 129)(50 130)(51 131)(52 132)(53 133)(54 134)(55 135)(56 136)(57 137)(58 138)(59 139)(60 140)(61 141)(62 142)(63 143)(64 144)(65 145)(66 146)(67 147)(68 148)(69 149)(70 150)(71 151)(72 152)(73 153)(74 154)(75 155)(76 156)(77 157)(78 158)(79 159)(80 160)
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), (1,47,35,41)(2,48,31,42)(3,49,32,43)(4,50,33,44)(5,46,34,45)(6,146,160,145)(7,147,156,141)(8,148,157,142)(9,149,158,143)(10,150,159,144)(11,137,17,151)(12,138,18,152)(13,139,19,153)(14,140,20,154)(15,136,16,155)(21,55,27,36)(22,51,28,37)(23,52,29,38)(24,53,30,39)(25,54,26,40)(56,87,75,81)(57,88,71,82)(58,89,72,83)(59,90,73,84)(60,86,74,85)(61,95,67,76)(62,91,68,77)(63,92,69,78)(64,93,70,79)(65,94,66,80)(96,127,115,121)(97,128,111,122)(98,129,112,123)(99,130,113,124)(100,126,114,125)(101,135,107,116)(102,131,108,117)(103,132,109,118)(104,133,110,119)(105,134,106,120), (1,81,35,87)(2,82,31,88)(3,83,32,89)(4,84,33,90)(5,85,34,86)(6,106,160,105)(7,107,156,101)(8,108,157,102)(9,109,158,103)(10,110,159,104)(11,97,17,111)(12,98,18,112)(13,99,19,113)(14,100,20,114)(15,96,16,115)(21,76,27,95)(22,77,28,91)(23,78,29,92)(24,79,30,93)(25,80,26,94)(36,61,55,67)(37,62,51,68)(38,63,52,69)(39,64,53,70)(40,65,54,66)(41,56,47,75)(42,57,48,71)(43,58,49,72)(44,59,50,73)(45,60,46,74)(116,147,135,141)(117,148,131,142)(118,149,132,143)(119,150,133,144)(120,146,134,145)(121,155,127,136)(122,151,128,137)(123,152,129,138)(124,153,130,139)(125,154,126,140), (1,135,27,121)(2,131,28,122)(3,132,29,123)(4,133,30,124)(5,134,26,125)(6,66,14,60)(7,67,15,56)(8,68,11,57)(9,69,12,58)(10,70,13,59)(16,75,156,61)(17,71,157,62)(18,72,158,63)(19,73,159,64)(20,74,160,65)(21,127,35,116)(22,128,31,117)(23,129,32,118)(24,130,33,119)(25,126,34,120)(36,115,47,101)(37,111,48,102)(38,112,49,103)(39,113,50,104)(40,114,46,105)(41,107,55,96)(42,108,51,97)(43,109,52,98)(44,110,53,99)(45,106,54,100)(76,155,87,141)(77,151,88,142)(78,152,89,143)(79,153,90,144)(80,154,86,145)(81,147,95,136)(82,148,91,137)(83,149,92,138)(84,150,93,139)(85,146,94,140), (1,96)(2,97)(3,98)(4,99)(5,100)(6,94)(7,95)(8,91)(9,92)(10,93)(11,82)(12,83)(13,84)(14,85)(15,81)(16,87)(17,88)(18,89)(19,90)(20,86)(21,101)(22,102)(23,103)(24,104)(25,105)(26,106)(27,107)(28,108)(29,109)(30,110)(31,111)(32,112)(33,113)(34,114)(35,115)(36,116)(37,117)(38,118)(39,119)(40,120)(41,121)(42,122)(43,123)(44,124)(45,125)(46,126)(47,127)(48,128)(49,129)(50,130)(51,131)(52,132)(53,133)(54,134)(55,135)(56,136)(57,137)(58,138)(59,139)(60,140)(61,141)(62,142)(63,143)(64,144)(65,145)(66,146)(67,147)(68,148)(69,149)(70,150)(71,151)(72,152)(73,153)(74,154)(75,155)(76,156)(77,157)(78,158)(79,159)(80,160)>;
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), (1,47,35,41)(2,48,31,42)(3,49,32,43)(4,50,33,44)(5,46,34,45)(6,146,160,145)(7,147,156,141)(8,148,157,142)(9,149,158,143)(10,150,159,144)(11,137,17,151)(12,138,18,152)(13,139,19,153)(14,140,20,154)(15,136,16,155)(21,55,27,36)(22,51,28,37)(23,52,29,38)(24,53,30,39)(25,54,26,40)(56,87,75,81)(57,88,71,82)(58,89,72,83)(59,90,73,84)(60,86,74,85)(61,95,67,76)(62,91,68,77)(63,92,69,78)(64,93,70,79)(65,94,66,80)(96,127,115,121)(97,128,111,122)(98,129,112,123)(99,130,113,124)(100,126,114,125)(101,135,107,116)(102,131,108,117)(103,132,109,118)(104,133,110,119)(105,134,106,120), (1,81,35,87)(2,82,31,88)(3,83,32,89)(4,84,33,90)(5,85,34,86)(6,106,160,105)(7,107,156,101)(8,108,157,102)(9,109,158,103)(10,110,159,104)(11,97,17,111)(12,98,18,112)(13,99,19,113)(14,100,20,114)(15,96,16,115)(21,76,27,95)(22,77,28,91)(23,78,29,92)(24,79,30,93)(25,80,26,94)(36,61,55,67)(37,62,51,68)(38,63,52,69)(39,64,53,70)(40,65,54,66)(41,56,47,75)(42,57,48,71)(43,58,49,72)(44,59,50,73)(45,60,46,74)(116,147,135,141)(117,148,131,142)(118,149,132,143)(119,150,133,144)(120,146,134,145)(121,155,127,136)(122,151,128,137)(123,152,129,138)(124,153,130,139)(125,154,126,140), (1,135,27,121)(2,131,28,122)(3,132,29,123)(4,133,30,124)(5,134,26,125)(6,66,14,60)(7,67,15,56)(8,68,11,57)(9,69,12,58)(10,70,13,59)(16,75,156,61)(17,71,157,62)(18,72,158,63)(19,73,159,64)(20,74,160,65)(21,127,35,116)(22,128,31,117)(23,129,32,118)(24,130,33,119)(25,126,34,120)(36,115,47,101)(37,111,48,102)(38,112,49,103)(39,113,50,104)(40,114,46,105)(41,107,55,96)(42,108,51,97)(43,109,52,98)(44,110,53,99)(45,106,54,100)(76,155,87,141)(77,151,88,142)(78,152,89,143)(79,153,90,144)(80,154,86,145)(81,147,95,136)(82,148,91,137)(83,149,92,138)(84,150,93,139)(85,146,94,140), (1,96)(2,97)(3,98)(4,99)(5,100)(6,94)(7,95)(8,91)(9,92)(10,93)(11,82)(12,83)(13,84)(14,85)(15,81)(16,87)(17,88)(18,89)(19,90)(20,86)(21,101)(22,102)(23,103)(24,104)(25,105)(26,106)(27,107)(28,108)(29,109)(30,110)(31,111)(32,112)(33,113)(34,114)(35,115)(36,116)(37,117)(38,118)(39,119)(40,120)(41,121)(42,122)(43,123)(44,124)(45,125)(46,126)(47,127)(48,128)(49,129)(50,130)(51,131)(52,132)(53,133)(54,134)(55,135)(56,136)(57,137)(58,138)(59,139)(60,140)(61,141)(62,142)(63,143)(64,144)(65,145)(66,146)(67,147)(68,148)(69,149)(70,150)(71,151)(72,152)(73,153)(74,154)(75,155)(76,156)(77,157)(78,158)(79,159)(80,160) );
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)], [(1,47,35,41),(2,48,31,42),(3,49,32,43),(4,50,33,44),(5,46,34,45),(6,146,160,145),(7,147,156,141),(8,148,157,142),(9,149,158,143),(10,150,159,144),(11,137,17,151),(12,138,18,152),(13,139,19,153),(14,140,20,154),(15,136,16,155),(21,55,27,36),(22,51,28,37),(23,52,29,38),(24,53,30,39),(25,54,26,40),(56,87,75,81),(57,88,71,82),(58,89,72,83),(59,90,73,84),(60,86,74,85),(61,95,67,76),(62,91,68,77),(63,92,69,78),(64,93,70,79),(65,94,66,80),(96,127,115,121),(97,128,111,122),(98,129,112,123),(99,130,113,124),(100,126,114,125),(101,135,107,116),(102,131,108,117),(103,132,109,118),(104,133,110,119),(105,134,106,120)], [(1,81,35,87),(2,82,31,88),(3,83,32,89),(4,84,33,90),(5,85,34,86),(6,106,160,105),(7,107,156,101),(8,108,157,102),(9,109,158,103),(10,110,159,104),(11,97,17,111),(12,98,18,112),(13,99,19,113),(14,100,20,114),(15,96,16,115),(21,76,27,95),(22,77,28,91),(23,78,29,92),(24,79,30,93),(25,80,26,94),(36,61,55,67),(37,62,51,68),(38,63,52,69),(39,64,53,70),(40,65,54,66),(41,56,47,75),(42,57,48,71),(43,58,49,72),(44,59,50,73),(45,60,46,74),(116,147,135,141),(117,148,131,142),(118,149,132,143),(119,150,133,144),(120,146,134,145),(121,155,127,136),(122,151,128,137),(123,152,129,138),(124,153,130,139),(125,154,126,140)], [(1,135,27,121),(2,131,28,122),(3,132,29,123),(4,133,30,124),(5,134,26,125),(6,66,14,60),(7,67,15,56),(8,68,11,57),(9,69,12,58),(10,70,13,59),(16,75,156,61),(17,71,157,62),(18,72,158,63),(19,73,159,64),(20,74,160,65),(21,127,35,116),(22,128,31,117),(23,129,32,118),(24,130,33,119),(25,126,34,120),(36,115,47,101),(37,111,48,102),(38,112,49,103),(39,113,50,104),(40,114,46,105),(41,107,55,96),(42,108,51,97),(43,109,52,98),(44,110,53,99),(45,106,54,100),(76,155,87,141),(77,151,88,142),(78,152,89,143),(79,153,90,144),(80,154,86,145),(81,147,95,136),(82,148,91,137),(83,149,92,138),(84,150,93,139),(85,146,94,140)], [(1,96),(2,97),(3,98),(4,99),(5,100),(6,94),(7,95),(8,91),(9,92),(10,93),(11,82),(12,83),(13,84),(14,85),(15,81),(16,87),(17,88),(18,89),(19,90),(20,86),(21,101),(22,102),(23,103),(24,104),(25,105),(26,106),(27,107),(28,108),(29,109),(30,110),(31,111),(32,112),(33,113),(34,114),(35,115),(36,116),(37,117),(38,118),(39,119),(40,120),(41,121),(42,122),(43,123),(44,124),(45,125),(46,126),(47,127),(48,128),(49,129),(50,130),(51,131),(52,132),(53,133),(54,134),(55,135),(56,136),(57,137),(58,138),(59,139),(60,140),(61,141),(62,142),(63,143),(64,144),(65,145),(66,146),(67,147),(68,148),(69,149),(70,150),(71,151),(72,152),(73,153),(74,154),(75,155),(76,156),(77,157),(78,158),(79,159),(80,160)]])
125 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 4A | ··· | 4L | 4M | 4N | 4O | 5A | 5B | 5C | 5D | 10A | ··· | 10L | 10M | ··· | 10AJ | 20A | ··· | 20AV | 20AW | ··· | 20BH |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 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 | D4 | C4○D4 | C5×D4 | C5×C4○D4 | 2+ 1+4 | C5×2+ 1+4 |
| kernel | C5×Q8⋊6D4 | D4×C20 | Q8×C20 | C5×C4⋊D4 | C5×C4⋊1D4 | C10×C4○D4 | Q8⋊6D4 | C4×D4 | C4×Q8 | C4⋊D4 | C4⋊1D4 | C2×C4○D4 | C5×Q8 | C20 | Q8 | C4 | 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×Q8⋊6D4 ►in GL4(𝔽41) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 18 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 9 |
| 0 | 0 | 9 | 0 |
| 5 | 39 | 0 | 0 |
| 13 | 36 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 32 |
| 36 | 2 | 0 | 0 |
| 29 | 5 | 0 | 0 |
| 0 | 0 | 0 | 32 |
| 0 | 0 | 9 | 0 |
G:=sub<GL(4,GF(41))| [16,0,0,0,0,16,0,0,0,0,18,0,0,0,0,18],[40,0,0,0,0,40,0,0,0,0,0,40,0,0,1,0],[40,0,0,0,0,40,0,0,0,0,0,9,0,0,9,0],[5,13,0,0,39,36,0,0,0,0,9,0,0,0,0,32],[36,29,0,0,2,5,0,0,0,0,0,9,0,0,32,0] >;
C5×Q8⋊6D4 in GAP, Magma, Sage, TeX
C_5\times Q_8\rtimes_6D_4
% in TeX
G:=Group("C5xQ8:6D4"); // GroupNames label
G:=SmallGroup(320,1552);
// by ID
G=gap.SmallGroup(320,1552);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1149,568,3446,436,1242,304]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^4=d^4=e^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations