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## G = C2×C4⋊C4⋊7D5order 320 = 26·5

### Direct product of C2 and C4⋊C4⋊7D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C2×C4⋊C4⋊7D5
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — D5×C22×C4 — C2×C4⋊C4⋊7D5
 Lower central C5 — C10 — C2×C4⋊C4⋊7D5
 Upper central C1 — C23 — C2×C4⋊C4

Generators and relations for C2×C4⋊C47D5
G = < a,b,c,d,e | a2=b4=c4=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ece=b2c, ede=d-1 >

Subgroups: 990 in 330 conjugacy classes, 167 normal (21 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×12], C22, C22 [×6], C22 [×16], C5, C2×C4 [×10], C2×C4 [×34], C23, C23 [×10], D5 [×4], C10 [×3], C10 [×4], C42 [×8], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×4], C22×C4, C22×C4 [×2], C22×C4 [×15], C24, Dic5 [×4], Dic5 [×4], C20 [×4], C20 [×4], D10 [×4], D10 [×12], C2×C10, C2×C10 [×6], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4, C2×C4⋊C4, C42⋊C2 [×8], C23×C4, C4×D5 [×16], C2×Dic5 [×10], C2×Dic5 [×4], C2×C20 [×10], C2×C20 [×4], C22×D5 [×6], C22×D5 [×4], C22×C10, C2×C42⋊C2, C4×Dic5 [×8], C4⋊Dic5 [×4], D10⋊C4 [×8], C5×C4⋊C4 [×4], C2×C4×D5 [×12], C22×Dic5, C22×Dic5 [×2], C22×C20, C22×C20 [×2], C23×D5, C4⋊C47D5 [×8], C2×C4×Dic5 [×2], C2×C4⋊Dic5, C2×D10⋊C4 [×2], C10×C4⋊C4, D5×C22×C4, C2×C4⋊C47D5
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D5, C22×C4 [×14], C4○D4 [×4], C24, D10 [×7], C42⋊C2 [×4], C23×C4, C2×C4○D4 [×2], C4×D5 [×4], C22×D5 [×7], C2×C42⋊C2, C2×C4×D5 [×6], D42D5 [×2], Q82D5 [×2], C23×D5, C4⋊C47D5 [×4], D5×C22×C4, C2×D42D5, C2×Q82D5, C2×C4⋊C47D5

Smallest permutation representation of C2×C4⋊C47D5
On 160 points
Generators in S160
(1 96)(2 97)(3 98)(4 99)(5 100)(6 91)(7 92)(8 93)(9 94)(10 95)(11 86)(12 87)(13 88)(14 89)(15 90)(16 81)(17 82)(18 83)(19 84)(20 85)(21 116)(22 117)(23 118)(24 119)(25 120)(26 111)(27 112)(28 113)(29 114)(30 115)(31 106)(32 107)(33 108)(34 109)(35 110)(36 101)(37 102)(38 103)(39 104)(40 105)(41 136)(42 137)(43 138)(44 139)(45 140)(46 131)(47 132)(48 133)(49 134)(50 135)(51 126)(52 127)(53 128)(54 129)(55 130)(56 121)(57 122)(58 123)(59 124)(60 125)(61 156)(62 157)(63 158)(64 159)(65 160)(66 151)(67 152)(68 153)(69 154)(70 155)(71 146)(72 147)(73 148)(74 149)(75 150)(76 141)(77 142)(78 143)(79 144)(80 145)
(1 131 6 136)(2 132 7 137)(3 133 8 138)(4 134 9 139)(5 135 10 140)(11 121 16 126)(12 122 17 127)(13 123 18 128)(14 124 19 129)(15 125 20 130)(21 151 26 156)(22 152 27 157)(23 153 28 158)(24 154 29 159)(25 155 30 160)(31 141 36 146)(32 142 37 147)(33 143 38 148)(34 144 39 149)(35 145 40 150)(41 96 46 91)(42 97 47 92)(43 98 48 93)(44 99 49 94)(45 100 50 95)(51 86 56 81)(52 87 57 82)(53 88 58 83)(54 89 59 84)(55 90 60 85)(61 116 66 111)(62 117 67 112)(63 118 68 113)(64 119 69 114)(65 120 70 115)(71 106 76 101)(72 107 77 102)(73 108 78 103)(74 109 79 104)(75 110 80 105)
(1 71 11 61)(2 72 12 62)(3 73 13 63)(4 74 14 64)(5 75 15 65)(6 76 16 66)(7 77 17 67)(8 78 18 68)(9 79 19 69)(10 80 20 70)(21 41 31 51)(22 42 32 52)(23 43 33 53)(24 44 34 54)(25 45 35 55)(26 46 36 56)(27 47 37 57)(28 48 38 58)(29 49 39 59)(30 50 40 60)(81 151 91 141)(82 152 92 142)(83 153 93 143)(84 154 94 144)(85 155 95 145)(86 156 96 146)(87 157 97 147)(88 158 98 148)(89 159 99 149)(90 160 100 150)(101 121 111 131)(102 122 112 132)(103 123 113 133)(104 124 114 134)(105 125 115 135)(106 126 116 136)(107 127 117 137)(108 128 118 138)(109 129 119 139)(110 130 120 140)
(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 95)(2 94)(3 93)(4 92)(5 91)(6 100)(7 99)(8 98)(9 97)(10 96)(11 85)(12 84)(13 83)(14 82)(15 81)(16 90)(17 89)(18 88)(19 87)(20 86)(21 120)(22 119)(23 118)(24 117)(25 116)(26 115)(27 114)(28 113)(29 112)(30 111)(31 110)(32 109)(33 108)(34 107)(35 106)(36 105)(37 104)(38 103)(39 102)(40 101)(41 135)(42 134)(43 133)(44 132)(45 131)(46 140)(47 139)(48 138)(49 137)(50 136)(51 125)(52 124)(53 123)(54 122)(55 121)(56 130)(57 129)(58 128)(59 127)(60 126)(61 160)(62 159)(63 158)(64 157)(65 156)(66 155)(67 154)(68 153)(69 152)(70 151)(71 150)(72 149)(73 148)(74 147)(75 146)(76 145)(77 144)(78 143)(79 142)(80 141)

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

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

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

80 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4L 4M ··· 4T 4U ··· 4AB 5A 5B 10A ··· 10N 20A ··· 20X order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 ··· 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 10 10 10 10 2 ··· 2 5 ··· 5 10 ··· 10 2 2 2 ··· 2 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C2 C4 D5 C4○D4 D10 D10 C4×D5 D4⋊2D5 Q8⋊2D5 kernel C2×C4⋊C4⋊7D5 C4⋊C4⋊7D5 C2×C4×Dic5 C2×C4⋊Dic5 C2×D10⋊C4 C10×C4⋊C4 D5×C22×C4 C2×C4×D5 C2×C4⋊C4 C2×C10 C4⋊C4 C22×C4 C2×C4 C22 C22 # reps 1 8 2 1 2 1 1 16 2 8 8 6 16 4 4

Matrix representation of C2×C4⋊C47D5 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 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 9 0 0 0 0 0 23 32
,
 32 0 0 0 0 0 0 32 0 0 0 0 0 0 32 0 0 0 0 0 0 32 0 0 0 0 0 0 1 1 0 0 0 0 0 40
,
 35 5 0 0 0 0 1 40 0 0 0 0 0 0 6 40 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 35 0 0 0 0 34 0 0 0 0 0 0 0 35 1 0 0 0 0 6 6 0 0 0 0 0 0 40 0 0 0 0 0 2 1

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,23,0,0,0,0,0,32],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,0,0,0,1,40],[35,1,0,0,0,0,5,40,0,0,0,0,0,0,6,1,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,34,0,0,0,0,35,0,0,0,0,0,0,0,35,6,0,0,0,0,1,6,0,0,0,0,0,0,40,2,0,0,0,0,0,1] >;

C2×C4⋊C47D5 in GAP, Magma, Sage, TeX

C_2\times C_4\rtimes C_4\rtimes_7D_5
% in TeX

G:=Group("C2xC4:C4:7D5");
// GroupNames label

G:=SmallGroup(320,1174);
// by ID

G=gap.SmallGroup(320,1174);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,1123,297,80,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=c^4=d^5=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations

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