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G = C5×D4⋊D4order 320 = 26·5

Direct product of C5 and D4⋊D4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C5×D4⋊D4, D43(C5×D4), Q83(C5×D4), (C2×D8)⋊2C10, (C5×D4)⋊21D4, (C5×Q8)⋊21D4, (C10×D8)⋊16C2, C4⋊D42C10, C22⋊C85C10, C4.23(D4×C10), D4⋊C48C10, Q8⋊C44C10, (C2×SD16)⋊8C10, C20.384(C2×D4), (C2×C20).459D4, C23.13(C5×D4), C10.96C22≀C2, (C10×SD16)⋊25C2, C22.79(D4×C10), (C22×C10).31D4, C10.119(C4○D8), (C2×C20).914C23, (C2×C40).298C22, C10.132(C8⋊C22), (D4×C10).294C22, (Q8×C10).259C22, (C22×C20).421C22, C2.6(C5×C4○D8), (C2×C4○D4)⋊1C10, C4⋊C4.2(C2×C10), (C2×C8).1(C2×C10), C2.7(C5×C8⋊C22), (C10×C4○D4)⋊17C2, (C5×C4⋊D4)⋊29C2, (C5×C22⋊C8)⋊22C2, (C2×C4).105(C5×D4), (C5×D4⋊C4)⋊32C2, (C5×Q8⋊C4)⋊27C2, C2.10(C5×C22≀C2), (C2×D4).52(C2×C10), (C2×C10).635(C2×D4), (C2×Q8).44(C2×C10), (C5×C4⋊C4).224C22, (C22×C4).39(C2×C10), (C2×C4).89(C22×C10), SmallGroup(320,950)

Series: Derived Chief Lower central Upper central

C1C2×C4 — C5×D4⋊D4
C1C2C22C2×C4C2×C20D4×C10C10×D8 — C5×D4⋊D4
C1C2C2×C4 — C5×D4⋊D4
C1C2×C10C22×C20 — C5×D4⋊D4

Generators and relations for C5×D4⋊D4
 G = < a,b,c,d,e | a5=b4=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=ebe=b-1, dcd-1=b-1c, ece=bc, ede=d-1 >

Subgroups: 322 in 162 conjugacy classes, 58 normal (50 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×4], C22, C22 [×10], C5, C8 [×2], C2×C4 [×2], C2×C4 [×8], D4 [×2], D4 [×9], Q8 [×2], Q8, C23, C23 [×2], C10 [×3], C10 [×4], C22⋊C4, C4⋊C4, C2×C8 [×2], D8 [×2], SD16 [×2], C22×C4, C22×C4, C2×D4 [×2], C2×D4 [×2], C2×Q8, C4○D4 [×4], C20 [×2], C20 [×4], C2×C10, C2×C10 [×10], C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊D4, C2×D8, C2×SD16, C2×C4○D4, C40 [×2], C2×C20 [×2], C2×C20 [×8], C5×D4 [×2], C5×D4 [×9], C5×Q8 [×2], C5×Q8, C22×C10, C22×C10 [×2], D4⋊D4, C5×C22⋊C4, C5×C4⋊C4, C2×C40 [×2], C5×D8 [×2], C5×SD16 [×2], C22×C20, C22×C20, D4×C10 [×2], D4×C10 [×2], Q8×C10, C5×C4○D4 [×4], C5×C22⋊C8, C5×D4⋊C4, C5×Q8⋊C4, C5×C4⋊D4, C10×D8, C10×SD16, C10×C4○D4, C5×D4⋊D4
Quotients: C1, C2 [×7], C22 [×7], C5, D4 [×6], C23, C10 [×7], C2×D4 [×3], C2×C10 [×7], C22≀C2, C4○D8, C8⋊C22, C5×D4 [×6], C22×C10, D4⋊D4, D4×C10 [×3], C5×C22≀C2, C5×C4○D8, C5×C8⋊C22, C5×D4⋊D4

Smallest permutation representation of C5×D4⋊D4
On 160 points
Generators in S160
(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 41 12 30)(2 42 13 26)(3 43 14 27)(4 44 15 28)(5 45 11 29)(6 151 160 35)(7 152 156 31)(8 153 157 32)(9 154 158 33)(10 155 159 34)(16 145 130 24)(17 141 126 25)(18 142 127 21)(19 143 128 22)(20 144 129 23)(36 54 70 59)(37 55 66 60)(38 51 67 56)(39 52 68 57)(40 53 69 58)(46 61 91 83)(47 62 92 84)(48 63 93 85)(49 64 94 81)(50 65 95 82)(71 109 98 80)(72 110 99 76)(73 106 100 77)(74 107 96 78)(75 108 97 79)(86 123 131 101)(87 124 132 102)(88 125 133 103)(89 121 134 104)(90 122 135 105)(111 120 138 149)(112 116 139 150)(113 117 140 146)(114 118 136 147)(115 119 137 148)
(1 140)(2 136)(3 137)(4 138)(5 139)(6 62)(7 63)(8 64)(9 65)(10 61)(11 112)(12 113)(13 114)(14 115)(15 111)(16 74)(17 75)(18 71)(19 72)(20 73)(21 109)(22 110)(23 106)(24 107)(25 108)(26 147)(27 148)(28 149)(29 150)(30 146)(31 93)(32 94)(33 95)(34 91)(35 92)(36 103)(37 104)(38 105)(39 101)(40 102)(41 117)(42 118)(43 119)(44 120)(45 116)(46 155)(47 151)(48 152)(49 153)(50 154)(51 135)(52 131)(53 132)(54 133)(55 134)(56 90)(57 86)(58 87)(59 88)(60 89)(66 121)(67 122)(68 123)(69 124)(70 125)(76 143)(77 144)(78 145)(79 141)(80 142)(81 157)(82 158)(83 159)(84 160)(85 156)(96 130)(97 126)(98 127)(99 128)(100 129)
(1 64 37 100)(2 65 38 96)(3 61 39 97)(4 62 40 98)(5 63 36 99)(6 87 127 149)(7 88 128 150)(8 89 129 146)(9 90 130 147)(10 86 126 148)(11 85 70 72)(12 81 66 73)(13 82 67 74)(14 83 68 75)(15 84 69 71)(16 118 158 135)(17 119 159 131)(18 120 160 132)(19 116 156 133)(20 117 157 134)(21 138 151 102)(22 139 152 103)(23 140 153 104)(24 136 154 105)(25 137 155 101)(26 95 56 78)(27 91 57 79)(28 92 58 80)(29 93 59 76)(30 94 60 77)(31 125 143 112)(32 121 144 113)(33 122 145 114)(34 123 141 115)(35 124 142 111)(41 49 55 106)(42 50 51 107)(43 46 52 108)(44 47 53 109)(45 48 54 110)
(6 142)(7 143)(8 144)(9 145)(10 141)(16 154)(17 155)(18 151)(19 152)(20 153)(21 160)(22 156)(23 157)(24 158)(25 159)(26 42)(27 43)(28 44)(29 45)(30 41)(31 128)(32 129)(33 130)(34 126)(35 127)(46 79)(47 80)(48 76)(49 77)(50 78)(51 56)(52 57)(53 58)(54 59)(55 60)(61 97)(62 98)(63 99)(64 100)(65 96)(71 84)(72 85)(73 81)(74 82)(75 83)(86 123)(87 124)(88 125)(89 121)(90 122)(91 108)(92 109)(93 110)(94 106)(95 107)(101 131)(102 132)(103 133)(104 134)(105 135)(111 149)(112 150)(113 146)(114 147)(115 148)(116 139)(117 140)(118 136)(119 137)(120 138)

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

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

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

95 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G5A5B5C5D8A8B8C8D10A···10L10M···10X10Y10Z10AA10AB20A···20P20Q···20X20Y20Z20AA20AB40A···40P
order1222222244444445555888810···1010···101010101020···2020···202020202040···40
size111144482222448111144441···14···488882···24···488884···4

95 irreducible representations

dim1111111111111111222222222244
type+++++++++++++
imageC1C2C2C2C2C2C2C2C5C10C10C10C10C10C10C10D4D4D4D4C4○D8C5×D4C5×D4C5×D4C5×D4C5×C4○D8C8⋊C22C5×C8⋊C22
kernelC5×D4⋊D4C5×C22⋊C8C5×D4⋊C4C5×Q8⋊C4C5×C4⋊D4C10×D8C10×SD16C10×C4○D4D4⋊D4C22⋊C8D4⋊C4Q8⋊C4C4⋊D4C2×D8C2×SD16C2×C4○D4C2×C20C5×D4C5×Q8C22×C10C10C2×C4D4Q8C23C2C10C2
# reps11111111444444441221448841614

Matrix representation of C5×D4⋊D4 in GL6(𝔽41)

1000000
0100000
001000
000100
000010
000001
,
4000000
0400000
000100
0040000
0000400
0000040
,
3440000
2970000
00122900
00292900
0000402
000001
,
100000
24400000
000900
009000
0000139
0000140
,
100000
24400000
001000
0004000
000010
0000140

G:=sub<GL(6,GF(41))| [10,0,0,0,0,0,0,10,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[34,29,0,0,0,0,4,7,0,0,0,0,0,0,12,29,0,0,0,0,29,29,0,0,0,0,0,0,40,0,0,0,0,0,2,1],[1,24,0,0,0,0,0,40,0,0,0,0,0,0,0,9,0,0,0,0,9,0,0,0,0,0,0,0,1,1,0,0,0,0,39,40],[1,24,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,1,0,0,0,0,0,40] >;

C5×D4⋊D4 in GAP, Magma, Sage, TeX

C_5\times D_4\rtimes D_4
% in TeX

G:=Group("C5xD4:D4");
// GroupNames label

G:=SmallGroup(320,950);
// by ID

G=gap.SmallGroup(320,950);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,589,1766,856,7004,3511,172]);
// Polycyclic

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

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