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G = D4⋊(C4×D5)  order 320 = 26·5

2nd semidirect product of D4 and C4×D5 acting via C4×D5/D10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D46(C4×D5), (C4×D5).4D4, D42D51C4, C4.156(D4×D5), D4⋊C417D5, C4⋊C4.133D10, C20.105(C2×D4), D4⋊Dic55C2, C10.Q164C2, (C2×C8).168D10, C22.70(D4×D5), Dic1012(C2×C4), (C2×D4).132D10, C2.3(D8⋊D5), C20.41(C22×C4), C20.44D418C2, C10.30(C8⋊C22), (C2×C20).211C23, (C2×C40).182C22, (C2×Dic5).196D4, C2.2(SD16⋊D5), (C22×D5).106D4, (D4×C10).32C22, C52(C23.36D4), C4⋊Dic5.67C22, D10.23(C22⋊C4), C10.28(C8.C22), Dic5.37(C22⋊C4), (C2×Dic10).59C22, C4.6(C2×C4×D5), (D5×C4⋊C4)⋊2C2, (C5×D4)⋊13(C2×C4), (C4×D5).2(C2×C4), (C2×C8⋊D5)⋊16C2, C2.19(D5×C22⋊C4), (C5×D4⋊C4)⋊20C2, (C2×C4×D5).10C22, (C2×D42D5).3C2, (C2×C10).224(C2×D4), (C5×C4⋊C4).14C22, C10.59(C2×C22⋊C4), (C2×C52C8).14C22, (C2×C4).318(C22×D5), SmallGroup(320,398)

Series: Derived Chief Lower central Upper central

C1C20 — D4⋊(C4×D5)
C1C5C10C2×C10C2×C20C2×C4×D5C2×D42D5 — D4⋊(C4×D5)
C5C10C20 — D4⋊(C4×D5)
C1C22C2×C4D4⋊C4

Generators and relations for D4⋊(C4×D5)
 G = < a,b,c,d,e | a4=b2=c4=d5=e2=1, bab=cac-1=a-1, ad=da, ae=ea, cbc-1=a-1b, bd=db, ebe=a2b, cd=dc, ce=ec, ede=d-1 >

Subgroups: 638 in 162 conjugacy classes, 55 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, C23, D5, C10, C10, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, D4⋊C4, D4⋊C4, Q8⋊C4, C2×C4⋊C4, C2×M4(2), C2×C4○D4, C52C8, C40, Dic10, Dic10, C4×D5, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C22×D5, C22×C10, C23.36D4, C8⋊D5, C2×C52C8, C10.D4, C4⋊Dic5, C5×C4⋊C4, C2×C40, C2×Dic10, C2×C4×D5, C2×C4×D5, D42D5, D42D5, C22×Dic5, C2×C5⋊D4, D4×C10, C10.Q16, C20.44D4, D4⋊Dic5, C5×D4⋊C4, D5×C4⋊C4, C2×C8⋊D5, C2×D42D5, D4⋊(C4×D5)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, D10, C2×C22⋊C4, C8⋊C22, C8.C22, C4×D5, C22×D5, C23.36D4, C2×C4×D5, D4×D5, D5×C22⋊C4, D8⋊D5, SD16⋊D5, D4⋊(C4×D5)

Smallest permutation representation of D4⋊(C4×D5)
On 160 points
Generators in S160
(1 16 6 11)(2 17 7 12)(3 18 8 13)(4 19 9 14)(5 20 10 15)(21 36 26 31)(22 37 27 32)(23 38 28 33)(24 39 29 34)(25 40 30 35)(41 51 46 56)(42 52 47 57)(43 53 48 58)(44 54 49 59)(45 55 50 60)(61 71 66 76)(62 72 67 77)(63 73 68 78)(64 74 69 79)(65 75 70 80)(81 91 86 96)(82 92 87 97)(83 93 88 98)(84 94 89 99)(85 95 90 100)(101 111 106 116)(102 112 107 117)(103 113 108 118)(104 114 109 119)(105 115 110 120)(121 136 126 131)(122 137 127 132)(123 138 128 133)(124 139 129 134)(125 140 130 135)(141 156 146 151)(142 157 147 152)(143 158 148 153)(144 159 149 154)(145 160 150 155)
(1 96)(2 97)(3 98)(4 99)(5 100)(6 91)(7 92)(8 93)(9 94)(10 95)(11 81)(12 82)(13 83)(14 84)(15 85)(16 86)(17 87)(18 88)(19 89)(20 90)(21 116)(22 117)(23 118)(24 119)(25 120)(26 111)(27 112)(28 113)(29 114)(30 115)(31 101)(32 102)(33 103)(34 104)(35 105)(36 106)(37 107)(38 108)(39 109)(40 110)(41 131)(42 132)(43 133)(44 134)(45 135)(46 136)(47 137)(48 138)(49 139)(50 140)(51 126)(52 127)(53 128)(54 129)(55 130)(56 121)(57 122)(58 123)(59 124)(60 125)(61 151)(62 152)(63 153)(64 154)(65 155)(66 156)(67 157)(68 158)(69 159)(70 160)(71 146)(72 147)(73 148)(74 149)(75 150)(76 141)(77 142)(78 143)(79 144)(80 145)
(1 61 21 41)(2 62 22 42)(3 63 23 43)(4 64 24 44)(5 65 25 45)(6 66 26 46)(7 67 27 47)(8 68 28 48)(9 69 29 49)(10 70 30 50)(11 71 31 51)(12 72 32 52)(13 73 33 53)(14 74 34 54)(15 75 35 55)(16 76 36 56)(17 77 37 57)(18 78 38 58)(19 79 39 59)(20 80 40 60)(81 151 101 131)(82 152 102 132)(83 153 103 133)(84 154 104 134)(85 155 105 135)(86 156 106 136)(87 157 107 137)(88 158 108 138)(89 159 109 139)(90 160 110 140)(91 146 111 126)(92 147 112 127)(93 148 113 128)(94 149 114 129)(95 150 115 130)(96 141 116 121)(97 142 117 122)(98 143 118 123)(99 144 119 124)(100 145 120 125)
(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 25)(2 24)(3 23)(4 22)(5 21)(6 30)(7 29)(8 28)(9 27)(10 26)(11 35)(12 34)(13 33)(14 32)(15 31)(16 40)(17 39)(18 38)(19 37)(20 36)(41 65)(42 64)(43 63)(44 62)(45 61)(46 70)(47 69)(48 68)(49 67)(50 66)(51 75)(52 74)(53 73)(54 72)(55 71)(56 80)(57 79)(58 78)(59 77)(60 76)(81 110)(82 109)(83 108)(84 107)(85 106)(86 105)(87 104)(88 103)(89 102)(90 101)(91 120)(92 119)(93 118)(94 117)(95 116)(96 115)(97 114)(98 113)(99 112)(100 111)(121 150)(122 149)(123 148)(124 147)(125 146)(126 145)(127 144)(128 143)(129 142)(130 141)(131 160)(132 159)(133 158)(134 157)(135 156)(136 155)(137 154)(138 153)(139 152)(140 151)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J5A5B8A8B8C8D10A···10F10G10H10I10J20A20B20C20D20E20F20G20H40A···40H
order12222222444444444455888810···1010101010202020202020202040···40
size11114410102244101020202020224420202···28888444488884···4

50 irreducible representations

dim11111111122222222444444
type++++++++++++++++-++-
imageC1C2C2C2C2C2C2C2C4D4D4D4D5D10D10D10C4×D5C8⋊C22C8.C22D4×D5D4×D5D8⋊D5SD16⋊D5
kernelD4⋊(C4×D5)C10.Q16C20.44D4D4⋊Dic5C5×D4⋊C4D5×C4⋊C4C2×C8⋊D5C2×D42D5D42D5C4×D5C2×Dic5C22×D5D4⋊C4C4⋊C4C2×C8C2×D4D4C10C10C4C22C2C2
# reps11111111821122228112244

Matrix representation of D4⋊(C4×D5) in GL6(𝔽41)

4000000
0400000
00102824
00011328
001114400
001111040
,
2250000
10190000
0023353315
006183333
0000236
00003518
,
3380000
3880000
0024081
00024338
0094170
0099017
,
100000
010000
0004000
001600
0000640
000010
,
100000
010000
00354000
0035600
0000640
00003535

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,11,11,0,0,0,1,14,11,0,0,28,13,40,0,0,0,24,28,0,40],[22,10,0,0,0,0,5,19,0,0,0,0,0,0,23,6,0,0,0,0,35,18,0,0,0,0,33,33,23,35,0,0,15,33,6,18],[33,38,0,0,0,0,8,8,0,0,0,0,0,0,24,0,9,9,0,0,0,24,4,9,0,0,8,33,17,0,0,0,1,8,0,17],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,40,6,0,0,0,0,0,0,6,1,0,0,0,0,40,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,35,35,0,0,0,0,40,6,0,0,0,0,0,0,6,35,0,0,0,0,40,35] >;

D4⋊(C4×D5) in GAP, Magma, Sage, TeX

D_4\rtimes (C_4\times D_5)
% in TeX

G:=Group("D4:(C4xD5)");
// GroupNames label

G:=SmallGroup(320,398);
// by ID

G=gap.SmallGroup(320,398);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,219,58,851,438,102,12550]);
// Polycyclic

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

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