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G = C2×D4.F5  order 320 = 26·5

Direct product of C2 and D4.F5

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×D4.F5, Dic5.17C24, C5⋊C8.1C23, C10⋊1(C8○D4), D5⋊C8⋊6C22, D4.11(C2×F5), (C2×D4).13F5, C4.F5⋊7C22, C2.8(C23×F5), D4⋊2D5.2C4, (D4×C10).12C4, C10.7(C23×C4), C4.25(C22×F5), C23.30(C2×F5), C20.25(C22×C4), (C4×D5).47C23, D10.1(C22×C4), C22.F5⋊3C22, Dic10.11(C2×C4), (C2×Dic10).15C4, C22.1(C22×F5), Dic5.1(C22×C4), D4⋊2D5.15C22, (C2×Dic5).177C23, (C22×Dic5).191C22, C5⋊1(C2×C8○D4), (C2×D5⋊C8)⋊5C2, (C22×C5⋊C8)⋊9C2, (C2×C4.F5)⋊6C2, (C2×C5⋊C8)⋊11C22, C5⋊D4.1(C2×C4), (C2×C4).91(C2×F5), (C2×C20).70(C2×C4), (C5×D4).11(C2×C4), (C4×D5).31(C2×C4), (C2×C5⋊D4).10C4, (C2×C10).1(C22×C4), (C2×C4×D5).214C22, (C2×C22.F5)⋊10C2, (C2×D4⋊2D5).17C2, (C22×C10).31(C2×C4), (C2×Dic5).80(C2×C4), (C22×D5).60(C2×C4), SmallGroup(320,1593)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C10 — C2×D4.F5
Chief series
C1 — C5 — C10 — Dic5 — C5⋊C8 — C2×C5⋊C8 — C22×C5⋊C8 — C2×D4.F5
Lower central
C5 — C10 — C2×D4.F5
Upper central
C1 — C22 — C2×D4

Generators and relations for C2×D4.F5
 G = < a,b,c,d,e | a2=b4=c2=d5=1, e4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d3 >

Subgroups: 730 in 266 conjugacy classes, 140 normal (22 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C23, D5, C10, C10, C10, C2×C8, M4(2), C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic5, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C10, C22×C8, C2×M4(2), C8○D4, C2×C4○D4, C5⋊C8, Dic10, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×C10, C2×C8○D4, D5⋊C8, C4.F5, C2×C5⋊C8, C2×C5⋊C8, C22.F5, C2×Dic10, C2×C4×D5, D4⋊2D5, C22×Dic5, C2×C5⋊D4, D4×C10, C2×D5⋊C8, C2×C4.F5, D4.F5, C22×C5⋊C8, C2×C22.F5, C2×D4⋊2D5, C2×D4.F5
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C24, F5, C8○D4, C23×C4, C2×F5, C2×C8○D4, C22×F5, D4.F5, C23×F5, C2×D4.F5

Smallest permutation representation of C2×D4.F5
►On 160 points
Generators in S160
(1 159)(2 160)(3 153)(4 154)(5 155)(6 156)(7 157)(8 158)(9 71)(10 72)(11 65)(12 66)(13 67)(14 68)(15 69)(16 70)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(25 118)(26 119)(27 120)(28 113)(29 114)(30 115)(31 116)(32 117)(33 128)(34 121)(35 122)(36 123)(37 124)(38 125)(39 126)(40 127)(41 102)(42 103)(43 104)(44 97)(45 98)(46 99)(47 100)(48 101)(49 106)(50 107)(51 108)(52 109)(53 110)(54 111)(55 112)(56 105)(73 129)(74 130)(75 131)(76 132)(77 133)(78 134)(79 135)(80 136)(81 142)(82 143)(83 144)(84 137)(85 138)(86 139)(87 140)(88 141)(89 146)(90 147)(91 148)(92 149)(93 150)(94 151)(95 152)(96 145)

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

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

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

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

50 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J 5 8A···8H8I···8T10A10B10C10D10E10F10G20A20B
order1222222222444444444458···88···8101010101010102020
size1111222210102255551010101045···510···10444888888

50 irreducible representations

dim11111111111244448
type+++++++++++-
imageC1C2C2C2C2C2C2C4C4C4C4C8○D4F5C2×F5C2×F5C2×F5D4.F5
kernelC2×D4.F5C2×D5⋊C8C2×C4.F5D4.F5C22×C5⋊C8C2×C22.F5C2×D4⋊2D5C2×Dic10D4⋊2D5C2×C5⋊D4D4×C10C10C2×D4C2×C4D4C23C2
# reps11182212842811422

Matrix representation of C2×D4.F5 ►in GL8(𝔽41)

10000000
01000000
004000000
000400000
00001000
00000100
00000010
00000001
,
90000000
032000000
003200000
003290000
00001000
00000100
00000010
00000001
,
032000000
90000000
009230000
009320000
000040000
000004000
000000400
000000040
,
10000000
01000000
00100000
00010000
000040404040
00001000
00000100
00000010
,
270000000
027000000
003800000
000380000
000036088
000088036
00003328330
0000513135

G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[9,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,32,32,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,9,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,9,9,0,0,0,0,0,0,23,32,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,1,0,0,0,0,0,0,40,0,1,0,0,0,0,0,40,0,0,1,0,0,0,0,40,0,0,0],[27,0,0,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,38,0,0,0,0,0,0,0,0,38,0,0,0,0,0,0,0,0,36,8,33,5,0,0,0,0,0,8,28,13,0,0,0,0,8,0,33,13,0,0,0,0,8,36,0,5] >;
 

C2×D4.F5 in GAP, Magma, Sage, TeX 

C_2\times D_4.F_5
 
% in TeX
 
G:=Group("C2xD4.F5");
 
// GroupNames label
 
G:=SmallGroup(320,1593);
 
// by ID
 
G=gap.SmallGroup(320,1593);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,297,102,6278,818]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^5=1,e^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^3>;
 
// generators/relations
 

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