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## G = C2×D10⋊3Q8order 320 = 26·5

### Direct product of C2 and D10⋊3Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C2×D10⋊3Q8
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — D5×C22×C4 — C2×D10⋊3Q8
 Lower central C5 — C2×C10 — C2×D10⋊3Q8
 Upper central C1 — C23 — C22×Q8

Generators and relations for C2×D103Q8
G = < a,b,c,d,e | a2=b10=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=b5c, ce=ec, ede-1=d-1 >

Subgroups: 1038 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 [×12], C22×C4, C22×C4 [×2], C22×C4 [×11], C2×Q8 [×4], C2×Q8 [×4], C24, Dic5 [×6], C20 [×4], C20 [×4], D10 [×4], D10 [×12], C2×C10, C2×C10 [×6], C2×C22⋊C4 [×2], C2×C4⋊C4 [×3], C22⋊Q8 [×8], C23×C4, C22×Q8, C4×D5 [×8], C2×Dic5 [×6], C2×Dic5 [×6], C2×C20 [×10], C2×C20 [×4], C5×Q8 [×8], C22×D5 [×6], C22×D5 [×4], C22×C10, C2×C22⋊Q8, C10.D4 [×8], C4⋊Dic5 [×4], D10⋊C4 [×8], C2×C4×D5 [×4], C2×C4×D5 [×4], C22×Dic5, C22×Dic5 [×2], C22×C20, C22×C20 [×2], Q8×C10 [×4], Q8×C10 [×4], C23×D5, C2×C10.D4 [×2], C2×C4⋊Dic5, C2×D10⋊C4 [×2], D103Q8 [×8], D5×C22×C4, Q8×C2×C10, C2×D103Q8
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, C5⋊D4 [×4], C22×D5 [×7], C2×C22⋊Q8, Q8×D5 [×2], Q82D5 [×2], C2×C5⋊D4 [×6], C23×D5, D103Q8 [×4], C2×Q8×D5, C2×Q82D5, C22×C5⋊D4, C2×D103Q8

Smallest permutation representation of C2×D103Q8
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 74)(12 75)(13 76)(14 77)(15 78)(16 79)(17 80)(18 71)(19 72)(20 73)(21 100)(22 91)(23 92)(24 93)(25 94)(26 95)(27 96)(28 97)(29 98)(30 99)(31 122)(32 123)(33 124)(34 125)(35 126)(36 127)(37 128)(38 129)(39 130)(40 121)(41 120)(42 111)(43 112)(44 113)(45 114)(46 115)(47 116)(48 117)(49 118)(50 119)(51 142)(52 143)(53 144)(54 145)(55 146)(56 147)(57 148)(58 149)(59 150)(60 141)(61 140)(62 131)(63 132)(64 133)(65 134)(66 135)(67 136)(68 137)(69 138)(70 139)(81 160)(82 151)(83 152)(84 153)(85 154)(86 155)(87 156)(88 157)(89 158)(90 159)
(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 78)(12 77)(13 76)(14 75)(15 74)(16 73)(17 72)(18 71)(19 80)(20 79)(21 97)(22 96)(23 95)(24 94)(25 93)(26 92)(27 91)(28 100)(29 99)(30 98)(31 122)(32 121)(33 130)(34 129)(35 128)(36 127)(37 126)(38 125)(39 124)(40 123)(41 112)(42 111)(43 120)(44 119)(45 118)(46 117)(47 116)(48 115)(49 114)(50 113)(51 147)(52 146)(53 145)(54 144)(55 143)(56 142)(57 141)(58 150)(59 149)(60 148)(61 137)(62 136)(63 135)(64 134)(65 133)(66 132)(67 131)(68 140)(69 139)(70 138)(81 152)(82 151)(83 160)(84 159)(85 158)(86 157)(87 156)(88 155)(89 154)(90 153)
(1 45 25 34)(2 46 26 35)(3 47 27 36)(4 48 28 37)(5 49 29 38)(6 50 30 39)(7 41 21 40)(8 42 22 31)(9 43 23 32)(10 44 24 33)(11 134 154 145)(12 135 155 146)(13 136 156 147)(14 137 157 148)(15 138 158 149)(16 139 159 150)(17 140 160 141)(18 131 151 142)(19 132 152 143)(20 133 153 144)(51 71 62 82)(52 72 63 83)(53 73 64 84)(54 74 65 85)(55 75 66 86)(56 76 67 87)(57 77 68 88)(58 78 69 89)(59 79 70 90)(60 80 61 81)(91 122 102 111)(92 123 103 112)(93 124 104 113)(94 125 105 114)(95 126 106 115)(96 127 107 116)(97 128 108 117)(98 129 109 118)(99 130 110 119)(100 121 101 120)
(1 65 25 54)(2 66 26 55)(3 67 27 56)(4 68 28 57)(5 69 29 58)(6 70 30 59)(7 61 21 60)(8 62 22 51)(9 63 23 52)(10 64 24 53)(11 125 154 114)(12 126 155 115)(13 127 156 116)(14 128 157 117)(15 129 158 118)(16 130 159 119)(17 121 160 120)(18 122 151 111)(19 123 152 112)(20 124 153 113)(31 82 42 71)(32 83 43 72)(33 84 44 73)(34 85 45 74)(35 86 46 75)(36 87 47 76)(37 88 48 77)(38 89 49 78)(39 90 50 79)(40 81 41 80)(91 142 102 131)(92 143 103 132)(93 144 104 133)(94 145 105 134)(95 146 106 135)(96 147 107 136)(97 148 108 137)(98 149 109 138)(99 150 110 139)(100 141 101 140)

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

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

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

68 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 4P 5A 5B 10A ··· 10N 20A ··· 20X order 1 2 ··· 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 10 10 10 10 2 2 2 2 4 4 4 4 10 10 10 10 20 20 20 20 2 2 2 ··· 2 4 ··· 4

68 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + - + + + - + image C1 C2 C2 C2 C2 C2 C2 D4 Q8 D5 C4○D4 D10 D10 C5⋊D4 Q8×D5 Q8⋊2D5 kernel C2×D10⋊3Q8 C2×C10.D4 C2×C4⋊Dic5 C2×D10⋊C4 D10⋊3Q8 D5×C22×C4 Q8×C2×C10 C2×C20 C22×D5 C22×Q8 C2×C10 C22×C4 C2×Q8 C2×C4 C22 C22 # reps 1 2 1 2 8 1 1 4 4 2 4 6 8 16 4 4

Matrix representation of C2×D103Q8 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 1 7 0 0 0 0 34 34 0 0 0 0 0 0 0 7 0 0 0 0 35 35 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 40 34 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 36 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 24 1 0 0 0 0 40 17 0 0 0 0 0 0 23 40 0 0 0 0 36 18 0 0 0 0 0 0 0 1 0 0 0 0 40 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 32 0 0 0 0 32 0

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,34,0,0,0,0,7,34,0,0,0,0,0,0,0,35,0,0,0,0,7,35,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,34,1,0,0,0,0,0,0,40,36,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[24,40,0,0,0,0,1,17,0,0,0,0,0,0,23,36,0,0,0,0,40,18,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,32,0,0,0,0,32,0] >;

C2×D103Q8 in GAP, Magma, Sage, TeX

C_2\times D_{10}\rtimes_3Q_8
% in TeX

G:=Group("C2xD10:3Q8");
// GroupNames label

G:=SmallGroup(320,1485);
// by ID

G=gap.SmallGroup(320,1485);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,675,297,136,12550]);
// 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=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=b^5*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

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