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G = C2×D102Q8order 320 = 26·5

Direct product of C2 and D102Q8

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×D102Q8, C4⋊C440D10, D102(C2×Q8), C4.68(C2×D20), (C22×D5)⋊5Q8, C20.221(C2×D4), (C2×C4).156D20, (C2×C20).201D4, C103(C22⋊Q8), C22.34(Q8×D5), (C2×C10).53C24, C4⋊Dic553C22, C10.10(C22×D4), C2.12(C22×D20), C22.68(C2×D20), C10.25(C22×Q8), (C2×C20).488C23, (C22×C4).361D10, C22.87(C23×D5), (C22×Dic10)⋊14C2, (C2×Dic10)⋊60C22, (C2×Dic5).15C23, C23.330(C22×D5), D10⋊C4.93C22, C22.74(D42D5), (C22×C20).218C22, (C22×C10).402C23, (C22×D5).169C23, (C23×D5).114C22, (C22×Dic5).82C22, C2.8(C2×Q8×D5), (C2×C4⋊C4)⋊18D5, C53(C2×C22⋊Q8), (C10×C4⋊C4)⋊15C2, (D5×C22×C4).5C2, (C5×C4⋊C4)⋊48C22, (C2×C4⋊Dic5)⋊21C2, C10.72(C2×C4○D4), (C2×C10).94(C2×Q8), C2.15(C2×D42D5), (C2×C10).175(C2×D4), (C2×C4×D5).314C22, (C2×C4).143(C22×D5), (C2×D10⋊C4).21C2, (C2×C10).172(C4○D4), SmallGroup(320,1181)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C2×D102Q8
C1C5C10C2×C10C22×D5C23×D5D5×C22×C4 — C2×D102Q8
C5C2×C10 — C2×D102Q8
C1C23C2×C4⋊C4

Generators and relations for C2×D102Q8
 G = < a,b,c,d,e | a2=b10=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=b-1, be=eb, dcd-1=b3c, ce=ec, ede-1=d-1 >

Subgroups: 1134 in 322 conjugacy classes, 135 normal (21 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×10], C22, C22 [×6], C22 [×16], C5, C2×C4 [×10], C2×C4 [×24], Q8 [×8], C23, C23 [×10], D5 [×4], C10 [×3], C10 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×8], C22×C4, C22×C4 [×2], C22×C4 [×11], C2×Q8 [×8], C24, Dic5 [×6], C20 [×4], C20 [×4], D10 [×4], D10 [×12], C2×C10, C2×C10 [×6], C2×C22⋊C4 [×2], C2×C4⋊C4, C2×C4⋊C4 [×2], C22⋊Q8 [×8], C23×C4, C22×Q8, Dic10 [×8], C4×D5 [×8], C2×Dic5 [×6], C2×Dic5 [×6], C2×C20 [×10], C2×C20 [×4], C22×D5 [×6], C22×D5 [×4], C22×C10, C2×C22⋊Q8, C4⋊Dic5 [×8], D10⋊C4 [×8], C5×C4⋊C4 [×4], C2×Dic10 [×4], C2×Dic10 [×4], C2×C4×D5 [×4], C2×C4×D5 [×4], C22×Dic5, C22×Dic5 [×2], C22×C20, C22×C20 [×2], C23×D5, D102Q8 [×8], C2×C4⋊Dic5 [×2], C2×D10⋊C4 [×2], C10×C4⋊C4, C22×Dic10, D5×C22×C4, C2×D102Q8
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], D5, C2×D4 [×6], C2×Q8 [×6], C4○D4 [×2], C24, D10 [×7], C22⋊Q8 [×4], C22×D4, C22×Q8, C2×C4○D4, D20 [×4], C22×D5 [×7], C2×C22⋊Q8, C2×D20 [×6], D42D5 [×2], Q8×D5 [×2], C23×D5, D102Q8 [×4], C22×D20, C2×D42D5, C2×Q8×D5, C2×D102Q8

Smallest permutation representation of C2×D102Q8
On 160 points
Generators in S160
(1 105)(2 106)(3 107)(4 108)(5 109)(6 110)(7 101)(8 102)(9 103)(10 104)(11 80)(12 71)(13 72)(14 73)(15 74)(16 75)(17 76)(18 77)(19 78)(20 79)(21 96)(22 97)(23 98)(24 99)(25 100)(26 91)(27 92)(28 93)(29 94)(30 95)(31 126)(32 127)(33 128)(34 129)(35 130)(36 121)(37 122)(38 123)(39 124)(40 125)(41 116)(42 117)(43 118)(44 119)(45 120)(46 111)(47 112)(48 113)(49 114)(50 115)(51 146)(52 147)(53 148)(54 149)(55 150)(56 141)(57 142)(58 143)(59 144)(60 145)(61 136)(62 137)(63 138)(64 139)(65 140)(66 131)(67 132)(68 133)(69 134)(70 135)(81 156)(82 157)(83 158)(84 159)(85 160)(86 151)(87 152)(88 153)(89 154)(90 155)
(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)
(1 104)(2 103)(3 102)(4 101)(5 110)(6 109)(7 108)(8 107)(9 106)(10 105)(11 74)(12 73)(13 72)(14 71)(15 80)(16 79)(17 78)(18 77)(19 76)(20 75)(21 93)(22 92)(23 91)(24 100)(25 99)(26 98)(27 97)(28 96)(29 95)(30 94)(31 128)(32 127)(33 126)(34 125)(35 124)(36 123)(37 122)(38 121)(39 130)(40 129)(41 118)(42 117)(43 116)(44 115)(45 114)(46 113)(47 112)(48 111)(49 120)(50 119)(51 143)(52 142)(53 141)(54 150)(55 149)(56 148)(57 147)(58 146)(59 145)(60 144)(61 133)(62 132)(63 131)(64 140)(65 139)(66 138)(67 137)(68 136)(69 135)(70 134)(81 158)(82 157)(83 156)(84 155)(85 154)(86 153)(87 152)(88 151)(89 160)(90 159)
(1 115 25 130)(2 114 26 129)(3 113 27 128)(4 112 28 127)(5 111 29 126)(6 120 30 125)(7 119 21 124)(8 118 22 123)(9 117 23 122)(10 116 24 121)(11 70 160 55)(12 69 151 54)(13 68 152 53)(14 67 153 52)(15 66 154 51)(16 65 155 60)(17 64 156 59)(18 63 157 58)(19 62 158 57)(20 61 159 56)(31 109 46 94)(32 108 47 93)(33 107 48 92)(34 106 49 91)(35 105 50 100)(36 104 41 99)(37 103 42 98)(38 102 43 97)(39 101 44 96)(40 110 45 95)(71 134 86 149)(72 133 87 148)(73 132 88 147)(74 131 89 146)(75 140 90 145)(76 139 81 144)(77 138 82 143)(78 137 83 142)(79 136 84 141)(80 135 85 150)
(1 65 25 60)(2 66 26 51)(3 67 27 52)(4 68 28 53)(5 69 29 54)(6 70 30 55)(7 61 21 56)(8 62 22 57)(9 63 23 58)(10 64 24 59)(11 125 160 120)(12 126 151 111)(13 127 152 112)(14 128 153 113)(15 129 154 114)(16 130 155 115)(17 121 156 116)(18 122 157 117)(19 123 158 118)(20 124 159 119)(31 86 46 71)(32 87 47 72)(33 88 48 73)(34 89 49 74)(35 90 50 75)(36 81 41 76)(37 82 42 77)(38 83 43 78)(39 84 44 79)(40 85 45 80)(91 146 106 131)(92 147 107 132)(93 148 108 133)(94 149 109 134)(95 150 110 135)(96 141 101 136)(97 142 102 137)(98 143 103 138)(99 144 104 139)(100 145 105 140)

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

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

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

68 conjugacy classes

class 1 2A···2G2H2I2J2K4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P5A5B10A···10N20A···20X
order12···2222244444444444444445510···1020···20
size11···110101010222244441010101020202020222···24···4

68 irreducible representations

dim1111111222222244
type++++++++-++++--
imageC1C2C2C2C2C2C2D4Q8D5C4○D4D10D10D20D42D5Q8×D5
kernelC2×D102Q8D102Q8C2×C4⋊Dic5C2×D10⋊C4C10×C4⋊C4C22×Dic10D5×C22×C4C2×C20C22×D5C2×C4⋊C4C2×C10C4⋊C4C22×C4C2×C4C22C22
# reps18221114424861644

Matrix representation of C2×D102Q8 in GL5(𝔽41)

400000
01000
00100
00010
00001
,
10000
040000
004000
00007
0003535
,
10000
040000
015100
000400
000361
,
400000
017500
0242400
0002514
0001416
,
400000
032000
012900
000400
000040

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,0,35,0,0,0,7,35],[1,0,0,0,0,0,40,15,0,0,0,0,1,0,0,0,0,0,40,36,0,0,0,0,1],[40,0,0,0,0,0,17,24,0,0,0,5,24,0,0,0,0,0,25,14,0,0,0,14,16],[40,0,0,0,0,0,32,12,0,0,0,0,9,0,0,0,0,0,40,0,0,0,0,0,40] >;

C2×D102Q8 in GAP, Magma, Sage, TeX

C_2\times D_{10}\rtimes_2Q_8
% in TeX

G:=Group("C2xD10:2Q8");
// GroupNames label

G:=SmallGroup(320,1181);
// by ID

G=gap.SmallGroup(320,1181);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,675,297,192,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^10=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=d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=b^3*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

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