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G = C4.(D4×D5)  order 320 = 26·5

28th non-split extension by C4 of D4×D5 acting via D4×D5/C22×D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C5⋊2C8⋊5D4, C5⋊4(C8⋊D4), C4⋊C4.62D10, C4⋊D4.7D5, C4.173(D4×D5), (C2×C20).74D4, (C2×D4).42D10, C20.151(C2×D4), C10.D8⋊37C2, D4⋊Dic5⋊18C2, C10.Q16⋊36C2, (C22×C10).88D4, C20.186(C4○D4), C4.62(D4⋊2D5), C20.48D4⋊25C2, C10.96(C4⋊D4), C10.92(C8⋊C22), (C2×C20).361C23, (D4×C10).58C22, (C22×C4).123D10, C23.25(C5⋊D4), C4⋊Dic5.145C22, C2.17(Dic5⋊D4), C2.13(D4.D10), C2.13(D4.9D10), C10.115(C8.C22), (C22×C20).165C22, (C2×Dic10).108C22, (C2×D4.D5)⋊11C2, (C5×C4⋊D4).6C2, (C2×C10).492(C2×D4), (C2×C4).52(C5⋊D4), (C2×C4.Dic5)⋊12C2, (C5×C4⋊C4).109C22, (C2×C4).461(C22×D5), C22.167(C2×C5⋊D4), (C2×C5⋊2C8).111C22, SmallGroup(320,669)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C20 — C4.(D4×D5)
Chief series
C1 — C5 — C10 — C20 — C2×C20 — C2×Dic10 — C20.48D4 — C4.(D4×D5)
Lower central
C5 — C10 — C2×C20 — C4.(D4×D5)
Upper central
C1 — C22 — C22×C4 — C4⋊D4

Generators and relations for C4.(D4×D5)
 G = < a,b,c,d,e | a4=b4=c2=d5=1, e2=a, bab-1=a-1, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe-1=ab, cd=dc, ece-1=a2c, ede-1=d-1 >

Subgroups: 406 in 120 conjugacy classes, 41 normal (39 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C10, C10, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), SD16, C22×C4, C2×D4, C2×D4, C2×Q8, Dic5, C20, C20, C2×C10, C2×C10, D4⋊C4, Q8⋊C4, C2.D8, C4⋊D4, C22⋊Q8, C2×M4(2), C2×SD16, C5⋊2C8, C5⋊2C8, Dic10, C2×Dic5, C2×C20, C2×C20, C5×D4, C22×C10, C22×C10, C8⋊D4, C2×C5⋊2C8, C4.Dic5, C10.D4, C4⋊Dic5, D4.D5, C23.D5, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C22×C20, D4×C10, D4×C10, C10.D8, C10.Q16, D4⋊Dic5, C2×C4.Dic5, C20.48D4, C2×D4.D5, C5×C4⋊D4, C4.(D4×D5)
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C8⋊C22, C8.C22, C5⋊D4, C22×D5, C8⋊D4, D4×D5, D4⋊2D5, C2×C5⋊D4, D4.D10, Dic5⋊D4, D4.9D10, C4.(D4×D5)

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

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

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F5A5B8A8B8C8D10A···10F10G10H10I10J10K10L10M10N20A···20H20I20J20K20L
order12222244444455888810···10101010101010101020···2020202020
size1111482248404022202020202···2444488884···48888

44 irreducible representations

dim111111112222222222444444
type++++++++++++++++-+--
imageC1C2C2C2C2C2C2C2D4D4D4D5C4○D4D10D10D10C5⋊D4C5⋊D4C8⋊C22C8.C22D4×D5D4⋊2D5D4.D10D4.9D10
kernelC4.(D4×D5)C10.D8C10.Q16D4⋊Dic5C2×C4.Dic5C20.48D4C2×D4.D5C5×C4⋊D4C5⋊2C8C2×C20C22×C10C4⋊D4C20C4⋊C4C22×C4C2×D4C2×C4C23C10C10C4C4C2C2
# reps111111112112222244112244

Matrix representation of C4.(D4×D5) ►in GL8(𝔽41)

400000000
040000000
004000000
000400000
00000010
00000001
000040000
000004000
,
10900000
00010000
1804000000
040000000
000000228
0000001339
000022800
0000133900
,
10900000
00010000
004000000
01000000
0000003913
000000282
000022800
0000133900
,
370000000
2310000000
003700000
2302100000
000004000
00001600
000000040
00000016
,
21025110000
12026200000
02420170000
1220000000
000016172524
00003253816
000016171617
0000325325

G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[1,0,18,0,0,0,0,0,0,0,0,40,0,0,0,0,9,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,2,13,0,0,0,0,0,0,28,39,0,0,0,0,2,13,0,0,0,0,0,0,28,39,0,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,9,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,2,13,0,0,0,0,0,0,28,39,0,0,0,0,39,28,0,0,0,0,0,0,13,2,0,0],[37,23,0,23,0,0,0,0,0,10,0,0,0,0,0,0,0,0,37,2,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,6,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,6],[21,12,0,12,0,0,0,0,0,0,24,20,0,0,0,0,25,26,20,0,0,0,0,0,11,20,17,0,0,0,0,0,0,0,0,0,16,3,16,3,0,0,0,0,17,25,17,25,0,0,0,0,25,38,16,3,0,0,0,0,24,16,17,25] >;
 

C4.(D4×D5) in GAP, Magma, Sage, TeX 

C_4.(D_4\times D_5)
 
% in TeX
 
G:=Group("C4.(D4xD5)");
 
// GroupNames label
 
G:=SmallGroup(320,669);
 
// by ID
 
G=gap.SmallGroup(320,669);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,254,555,1123,297,136,12550]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^4=c^2=d^5=1,e^2=a,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,e*b*e^-1=a*b,c*d=d*c,e*c*e^-1=a^2*c,e*d*e^-1=d^-1>;
 
// generators/relations
 

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