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G = C24.64D10order 320 = 26·5

4th non-split extension by C24 of D10 acting via D10/C10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.64D10, C23.27D20, C23.16Dic10, (C23×C4).8D5, (C22×C20)⋊25C4, (C2×C20).475D4, C42(C23.D5), (C22×C4)⋊8Dic5, C2010(C22⋊C4), C2.4(C207D4), (C23×C20).11C2, C222(C4⋊Dic5), C22.60(C2×D20), (C22×C10).26Q8, C10.79(C4⋊D4), C55(C23.7Q8), (C22×C4).433D10, (C22×C10).143D4, C10.68(C22⋊Q8), C23.31(C2×Dic5), C2.5(C20.48D4), C22.63(C4○D20), (C23×C10).99C22, C22.32(C2×Dic10), C23.303(C22×D5), C10.10C4224C2, C10.68(C42⋊C2), (C22×C20).484C22, (C22×C10).363C23, C22.50(C22×Dic5), (C22×Dic5).66C22, C2.12(C23.21D10), C10.79(C2×C4⋊C4), (C2×C10)⋊10(C4⋊C4), (C2×C4⋊Dic5)⋊16C2, C2.16(C2×C4⋊Dic5), (C2×C10).44(C2×Q8), (C2×C20).455(C2×C4), C2.6(C2×C23.D5), (C2×C10).549(C2×D4), (C2×C4).85(C2×Dic5), C22.87(C2×C5⋊D4), (C2×C10).91(C4○D4), (C2×C4).260(C5⋊D4), C10.111(C2×C22⋊C4), (C2×C23.D5).18C2, (C22×C10).204(C2×C4), (C2×C10).294(C22×C4), SmallGroup(320,839)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C24.64D10
C1C5C10C2×C10C22×C10C22×Dic5C2×C4⋊Dic5 — C24.64D10
C5C2×C10 — C24.64D10
C1C23C23×C4

Generators and relations for C24.64D10
 G = < a,b,c,d,e | a2=b2=c2=d20=1, e2=b, ab=ba, eae-1=ac=ca, ad=da, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 638 in 234 conjugacy classes, 103 normal (25 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×6], C22 [×3], C22 [×8], C22 [×12], C5, C2×C4 [×8], C2×C4 [×22], C23, C23 [×6], C23 [×4], C10 [×3], C10 [×4], C10 [×4], C22⋊C4 [×4], C4⋊C4 [×4], C22×C4 [×2], C22×C4 [×4], C22×C4 [×8], C24, Dic5 [×4], C20 [×4], C20 [×2], C2×C10 [×3], C2×C10 [×8], C2×C10 [×12], C2.C42 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C23×C4, C2×Dic5 [×12], C2×C20 [×8], C2×C20 [×10], C22×C10, C22×C10 [×6], C22×C10 [×4], C23.7Q8, C4⋊Dic5 [×4], C23.D5 [×4], C22×Dic5 [×4], C22×C20 [×2], C22×C20 [×4], C22×C20 [×4], C23×C10, C10.10C42 [×2], C2×C4⋊Dic5 [×2], C2×C23.D5 [×2], C23×C20, C24.64D10
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×2], C23, D5, C22⋊C4 [×4], C4⋊C4 [×4], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], Dic5 [×4], D10 [×3], C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4 [×2], C22⋊Q8 [×2], Dic10 [×2], D20 [×2], C2×Dic5 [×6], C5⋊D4 [×4], C22×D5, C23.7Q8, C4⋊Dic5 [×4], C23.D5 [×4], C2×Dic10, C2×D20, C4○D20 [×2], C22×Dic5, C2×C5⋊D4 [×2], C20.48D4 [×2], C2×C4⋊Dic5, C23.21D10, C207D4 [×2], C2×C23.D5, C24.64D10

Smallest permutation representation of C24.64D10
On 160 points
Generators in S160
(1 110)(2 111)(3 112)(4 113)(5 114)(6 115)(7 116)(8 117)(9 118)(10 119)(11 120)(12 101)(13 102)(14 103)(15 104)(16 105)(17 106)(18 107)(19 108)(20 109)(21 158)(22 159)(23 160)(24 141)(25 142)(26 143)(27 144)(28 145)(29 146)(30 147)(31 148)(32 149)(33 150)(34 151)(35 152)(36 153)(37 154)(38 155)(39 156)(40 157)(41 72)(42 73)(43 74)(44 75)(45 76)(46 77)(47 78)(48 79)(49 80)(50 61)(51 62)(52 63)(53 64)(54 65)(55 66)(56 67)(57 68)(58 69)(59 70)(60 71)(81 128)(82 129)(83 130)(84 131)(85 132)(86 133)(87 134)(88 135)(89 136)(90 137)(91 138)(92 139)(93 140)(94 121)(95 122)(96 123)(97 124)(98 125)(99 126)(100 127)
(1 110)(2 111)(3 112)(4 113)(5 114)(6 115)(7 116)(8 117)(9 118)(10 119)(11 120)(12 101)(13 102)(14 103)(15 104)(16 105)(17 106)(18 107)(19 108)(20 109)(21 78)(22 79)(23 80)(24 61)(25 62)(26 63)(27 64)(28 65)(29 66)(30 67)(31 68)(32 69)(33 70)(34 71)(35 72)(36 73)(37 74)(38 75)(39 76)(40 77)(41 152)(42 153)(43 154)(44 155)(45 156)(46 157)(47 158)(48 159)(49 160)(50 141)(51 142)(52 143)(53 144)(54 145)(55 146)(56 147)(57 148)(58 149)(59 150)(60 151)(81 128)(82 129)(83 130)(84 131)(85 132)(86 133)(87 134)(88 135)(89 136)(90 137)(91 138)(92 139)(93 140)(94 121)(95 122)(96 123)(97 124)(98 125)(99 126)(100 127)
(1 136)(2 137)(3 138)(4 139)(5 140)(6 121)(7 122)(8 123)(9 124)(10 125)(11 126)(12 127)(13 128)(14 129)(15 130)(16 131)(17 132)(18 133)(19 134)(20 135)(21 47)(22 48)(23 49)(24 50)(25 51)(26 52)(27 53)(28 54)(29 55)(30 56)(31 57)(32 58)(33 59)(34 60)(35 41)(36 42)(37 43)(38 44)(39 45)(40 46)(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)(81 102)(82 103)(83 104)(84 105)(85 106)(86 107)(87 108)(88 109)(89 110)(90 111)(91 112)(92 113)(93 114)(94 115)(95 116)(96 117)(97 118)(98 119)(99 120)(100 101)
(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)
(1 147 110 56)(2 146 111 55)(3 145 112 54)(4 144 113 53)(5 143 114 52)(6 142 115 51)(7 141 116 50)(8 160 117 49)(9 159 118 48)(10 158 119 47)(11 157 120 46)(12 156 101 45)(13 155 102 44)(14 154 103 43)(15 153 104 42)(16 152 105 41)(17 151 106 60)(18 150 107 59)(19 149 108 58)(20 148 109 57)(21 125 78 98)(22 124 79 97)(23 123 80 96)(24 122 61 95)(25 121 62 94)(26 140 63 93)(27 139 64 92)(28 138 65 91)(29 137 66 90)(30 136 67 89)(31 135 68 88)(32 134 69 87)(33 133 70 86)(34 132 71 85)(35 131 72 84)(36 130 73 83)(37 129 74 82)(38 128 75 81)(39 127 76 100)(40 126 77 99)

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

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

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

92 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4P5A5B10A···10AD20A···20AF
order12···222224···44···45510···1020···20
size11···122222···220···20222···22···2

92 irreducible representations

dim111111222222222222
type+++++++-+-++-+
imageC1C2C2C2C2C4D4D4Q8D5C4○D4Dic5D10D10C5⋊D4Dic10D20C4○D20
kernelC24.64D10C10.10C42C2×C4⋊Dic5C2×C23.D5C23×C20C22×C20C2×C20C22×C10C22×C10C23×C4C2×C10C22×C4C22×C4C24C2×C4C23C23C22
# reps12221842224842168816

Matrix representation of C24.64D10 in GL5(𝔽41)

10000
01000
004000
00010
000040
,
400000
01000
00100
00010
00001
,
10000
040000
004000
000400
000040
,
10000
09000
003200
000180
000016
,
90000
003200
09000
000016
000180

G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,40],[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],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,9,0,0,0,0,0,32,0,0,0,0,0,18,0,0,0,0,0,16],[9,0,0,0,0,0,0,9,0,0,0,32,0,0,0,0,0,0,0,18,0,0,0,16,0] >;

C24.64D10 in GAP, Magma, Sage, TeX

C_2^4._{64}D_{10}
% in TeX

G:=Group("C2^4.64D10");
// GroupNames label

G:=SmallGroup(320,839);
// by ID

G=gap.SmallGroup(320,839);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,477,232,422,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^20=1,e^2=b,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

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