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G = C2×D4.F5order 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, C101(C8○D4), D5⋊C86C22, D4.11(C2×F5), (C2×D4).13F5, C4.F57C22, C2.8(C23×F5), D42D5.2C4, (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.F53C22, Dic10.11(C2×C4), (C2×Dic10).15C4, C22.1(C22×F5), Dic5.1(C22×C4), D42D5.15C22, (C2×Dic5).177C23, (C22×Dic5).191C22, C51(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×D42D5).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

C1C10 — C2×D4.F5
C1C5C10Dic5C5⋊C8C2×C5⋊C8C22×C5⋊C8 — C2×D4.F5
C5C10 — C2×D4.F5
C1C22C2×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 [×2], C2 [×6], C4 [×2], C4 [×6], C22, C22 [×4], C22 [×8], C5, C8 [×8], C2×C4, C2×C4 [×15], D4 [×4], D4 [×8], Q8 [×4], C23 [×2], C23, D5 [×2], C10, C10 [×2], C10 [×4], C2×C8 [×16], M4(2) [×12], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×8], Dic5 [×2], Dic5 [×4], C20 [×2], D10 [×2], D10 [×2], C2×C10, C2×C10 [×4], C2×C10 [×4], C22×C8 [×3], C2×M4(2) [×3], C8○D4 [×8], C2×C4○D4, C5⋊C8 [×8], Dic10 [×4], C4×D5 [×4], C2×Dic5, C2×Dic5 [×10], C5⋊D4 [×8], C2×C20, C5×D4 [×4], C22×D5, C22×C10 [×2], C2×C8○D4, D5⋊C8 [×4], C4.F5 [×4], C2×C5⋊C8 [×2], C2×C5⋊C8 [×10], C22.F5 [×8], C2×Dic10, C2×C4×D5, D42D5 [×8], C22×Dic5 [×2], C2×C5⋊D4 [×2], D4×C10, C2×D5⋊C8, C2×C4.F5, D4.F5 [×8], C22×C5⋊C8 [×2], C2×C22.F5 [×2], C2×D42D5, C2×D4.F5
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C24, F5, C8○D4 [×2], C23×C4, C2×F5 [×7], C2×C8○D4, C22×F5 [×7], D4.F5 [×2], 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)(17 115)(18 116)(19 117)(20 118)(21 119)(22 120)(23 113)(24 114)(25 121)(26 122)(27 123)(28 124)(29 125)(30 126)(31 127)(32 128)(33 63)(34 64)(35 57)(36 58)(37 59)(38 60)(39 61)(40 62)(41 97)(42 98)(43 99)(44 100)(45 101)(46 102)(47 103)(48 104)(49 111)(50 112)(51 105)(52 106)(53 107)(54 108)(55 109)(56 110)(73 129)(74 130)(75 131)(76 132)(77 133)(78 134)(79 135)(80 136)(81 137)(82 138)(83 139)(84 140)(85 141)(86 142)(87 143)(88 144)(89 151)(90 152)(91 145)(92 146)(93 147)(94 148)(95 149)(96 150)
(1 7 5 3)(2 8 6 4)(9 95 13 91)(10 96 14 92)(11 89 15 93)(12 90 16 94)(17 101 21 97)(18 102 22 98)(19 103 23 99)(20 104 24 100)(25 139 29 143)(26 140 30 144)(27 141 31 137)(28 142 32 138)(33 129 37 133)(34 130 38 134)(35 131 39 135)(36 132 40 136)(41 115 45 119)(42 116 46 120)(43 117 47 113)(44 118 48 114)(49 51 53 55)(50 52 54 56)(57 75 61 79)(58 76 62 80)(59 77 63 73)(60 78 64 74)(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)
(1 111)(2 112)(3 105)(4 106)(5 107)(6 108)(7 109)(8 110)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(17 145)(18 146)(19 147)(20 148)(21 149)(22 150)(23 151)(24 152)(25 39)(26 40)(27 33)(28 34)(29 35)(30 36)(31 37)(32 38)(49 159)(50 160)(51 153)(52 154)(53 155)(54 156)(55 157)(56 158)(57 125)(58 126)(59 127)(60 128)(61 121)(62 122)(63 123)(64 124)(65 99)(66 100)(67 101)(68 102)(69 103)(70 104)(71 97)(72 98)(73 81)(74 82)(75 83)(76 84)(77 85)(78 86)(79 87)(80 88)(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 35 133 19)(2 134 98 20 36)(3 21 135 37 99)(4 38 22 100 136)(5 101 39 129 23)(6 130 102 24 40)(7 17 131 33 103)(8 34 18 104 132)(9 125 85 93 49)(10 94 126 50 86)(11 51 95 87 127)(12 88 52 128 96)(13 121 81 89 53)(14 90 122 54 82)(15 55 91 83 123)(16 84 56 124 92)(25 137 151 107 67)(26 108 138 68 152)(27 69 109 145 139)(28 146 70 140 110)(29 141 147 111 71)(30 112 142 72 148)(31 65 105 149 143)(32 150 66 144 106)(41 57 77 117 159)(42 118 58 160 78)(43 153 119 79 59)(44 80 154 60 120)(45 61 73 113 155)(46 114 62 156 74)(47 157 115 75 63)(48 76 158 64 116)
(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)(17,115)(18,116)(19,117)(20,118)(21,119)(22,120)(23,113)(24,114)(25,121)(26,122)(27,123)(28,124)(29,125)(30,126)(31,127)(32,128)(33,63)(34,64)(35,57)(36,58)(37,59)(38,60)(39,61)(40,62)(41,97)(42,98)(43,99)(44,100)(45,101)(46,102)(47,103)(48,104)(49,111)(50,112)(51,105)(52,106)(53,107)(54,108)(55,109)(56,110)(73,129)(74,130)(75,131)(76,132)(77,133)(78,134)(79,135)(80,136)(81,137)(82,138)(83,139)(84,140)(85,141)(86,142)(87,143)(88,144)(89,151)(90,152)(91,145)(92,146)(93,147)(94,148)(95,149)(96,150), (1,7,5,3)(2,8,6,4)(9,95,13,91)(10,96,14,92)(11,89,15,93)(12,90,16,94)(17,101,21,97)(18,102,22,98)(19,103,23,99)(20,104,24,100)(25,139,29,143)(26,140,30,144)(27,141,31,137)(28,142,32,138)(33,129,37,133)(34,130,38,134)(35,131,39,135)(36,132,40,136)(41,115,45,119)(42,116,46,120)(43,117,47,113)(44,118,48,114)(49,51,53,55)(50,52,54,56)(57,75,61,79)(58,76,62,80)(59,77,63,73)(60,78,64,74)(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), (1,111)(2,112)(3,105)(4,106)(5,107)(6,108)(7,109)(8,110)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,145)(18,146)(19,147)(20,148)(21,149)(22,150)(23,151)(24,152)(25,39)(26,40)(27,33)(28,34)(29,35)(30,36)(31,37)(32,38)(49,159)(50,160)(51,153)(52,154)(53,155)(54,156)(55,157)(56,158)(57,125)(58,126)(59,127)(60,128)(61,121)(62,122)(63,123)(64,124)(65,99)(66,100)(67,101)(68,102)(69,103)(70,104)(71,97)(72,98)(73,81)(74,82)(75,83)(76,84)(77,85)(78,86)(79,87)(80,88)(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), (1,97,35,133,19)(2,134,98,20,36)(3,21,135,37,99)(4,38,22,100,136)(5,101,39,129,23)(6,130,102,24,40)(7,17,131,33,103)(8,34,18,104,132)(9,125,85,93,49)(10,94,126,50,86)(11,51,95,87,127)(12,88,52,128,96)(13,121,81,89,53)(14,90,122,54,82)(15,55,91,83,123)(16,84,56,124,92)(25,137,151,107,67)(26,108,138,68,152)(27,69,109,145,139)(28,146,70,140,110)(29,141,147,111,71)(30,112,142,72,148)(31,65,105,149,143)(32,150,66,144,106)(41,57,77,117,159)(42,118,58,160,78)(43,153,119,79,59)(44,80,154,60,120)(45,61,73,113,155)(46,114,62,156,74)(47,157,115,75,63)(48,76,158,64,116), (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)(17,115)(18,116)(19,117)(20,118)(21,119)(22,120)(23,113)(24,114)(25,121)(26,122)(27,123)(28,124)(29,125)(30,126)(31,127)(32,128)(33,63)(34,64)(35,57)(36,58)(37,59)(38,60)(39,61)(40,62)(41,97)(42,98)(43,99)(44,100)(45,101)(46,102)(47,103)(48,104)(49,111)(50,112)(51,105)(52,106)(53,107)(54,108)(55,109)(56,110)(73,129)(74,130)(75,131)(76,132)(77,133)(78,134)(79,135)(80,136)(81,137)(82,138)(83,139)(84,140)(85,141)(86,142)(87,143)(88,144)(89,151)(90,152)(91,145)(92,146)(93,147)(94,148)(95,149)(96,150), (1,7,5,3)(2,8,6,4)(9,95,13,91)(10,96,14,92)(11,89,15,93)(12,90,16,94)(17,101,21,97)(18,102,22,98)(19,103,23,99)(20,104,24,100)(25,139,29,143)(26,140,30,144)(27,141,31,137)(28,142,32,138)(33,129,37,133)(34,130,38,134)(35,131,39,135)(36,132,40,136)(41,115,45,119)(42,116,46,120)(43,117,47,113)(44,118,48,114)(49,51,53,55)(50,52,54,56)(57,75,61,79)(58,76,62,80)(59,77,63,73)(60,78,64,74)(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), (1,111)(2,112)(3,105)(4,106)(5,107)(6,108)(7,109)(8,110)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,145)(18,146)(19,147)(20,148)(21,149)(22,150)(23,151)(24,152)(25,39)(26,40)(27,33)(28,34)(29,35)(30,36)(31,37)(32,38)(49,159)(50,160)(51,153)(52,154)(53,155)(54,156)(55,157)(56,158)(57,125)(58,126)(59,127)(60,128)(61,121)(62,122)(63,123)(64,124)(65,99)(66,100)(67,101)(68,102)(69,103)(70,104)(71,97)(72,98)(73,81)(74,82)(75,83)(76,84)(77,85)(78,86)(79,87)(80,88)(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), (1,97,35,133,19)(2,134,98,20,36)(3,21,135,37,99)(4,38,22,100,136)(5,101,39,129,23)(6,130,102,24,40)(7,17,131,33,103)(8,34,18,104,132)(9,125,85,93,49)(10,94,126,50,86)(11,51,95,87,127)(12,88,52,128,96)(13,121,81,89,53)(14,90,122,54,82)(15,55,91,83,123)(16,84,56,124,92)(25,137,151,107,67)(26,108,138,68,152)(27,69,109,145,139)(28,146,70,140,110)(29,141,147,111,71)(30,112,142,72,148)(31,65,105,149,143)(32,150,66,144,106)(41,57,77,117,159)(42,118,58,160,78)(43,153,119,79,59)(44,80,154,60,120)(45,61,73,113,155)(46,114,62,156,74)(47,157,115,75,63)(48,76,158,64,116), (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),(17,115),(18,116),(19,117),(20,118),(21,119),(22,120),(23,113),(24,114),(25,121),(26,122),(27,123),(28,124),(29,125),(30,126),(31,127),(32,128),(33,63),(34,64),(35,57),(36,58),(37,59),(38,60),(39,61),(40,62),(41,97),(42,98),(43,99),(44,100),(45,101),(46,102),(47,103),(48,104),(49,111),(50,112),(51,105),(52,106),(53,107),(54,108),(55,109),(56,110),(73,129),(74,130),(75,131),(76,132),(77,133),(78,134),(79,135),(80,136),(81,137),(82,138),(83,139),(84,140),(85,141),(86,142),(87,143),(88,144),(89,151),(90,152),(91,145),(92,146),(93,147),(94,148),(95,149),(96,150)], [(1,7,5,3),(2,8,6,4),(9,95,13,91),(10,96,14,92),(11,89,15,93),(12,90,16,94),(17,101,21,97),(18,102,22,98),(19,103,23,99),(20,104,24,100),(25,139,29,143),(26,140,30,144),(27,141,31,137),(28,142,32,138),(33,129,37,133),(34,130,38,134),(35,131,39,135),(36,132,40,136),(41,115,45,119),(42,116,46,120),(43,117,47,113),(44,118,48,114),(49,51,53,55),(50,52,54,56),(57,75,61,79),(58,76,62,80),(59,77,63,73),(60,78,64,74),(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)], [(1,111),(2,112),(3,105),(4,106),(5,107),(6,108),(7,109),(8,110),(9,41),(10,42),(11,43),(12,44),(13,45),(14,46),(15,47),(16,48),(17,145),(18,146),(19,147),(20,148),(21,149),(22,150),(23,151),(24,152),(25,39),(26,40),(27,33),(28,34),(29,35),(30,36),(31,37),(32,38),(49,159),(50,160),(51,153),(52,154),(53,155),(54,156),(55,157),(56,158),(57,125),(58,126),(59,127),(60,128),(61,121),(62,122),(63,123),(64,124),(65,99),(66,100),(67,101),(68,102),(69,103),(70,104),(71,97),(72,98),(73,81),(74,82),(75,83),(76,84),(77,85),(78,86),(79,87),(80,88),(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)], [(1,97,35,133,19),(2,134,98,20,36),(3,21,135,37,99),(4,38,22,100,136),(5,101,39,129,23),(6,130,102,24,40),(7,17,131,33,103),(8,34,18,104,132),(9,125,85,93,49),(10,94,126,50,86),(11,51,95,87,127),(12,88,52,128,96),(13,121,81,89,53),(14,90,122,54,82),(15,55,91,83,123),(16,84,56,124,92),(25,137,151,107,67),(26,108,138,68,152),(27,69,109,145,139),(28,146,70,140,110),(29,141,147,111,71),(30,112,142,72,148),(31,65,105,149,143),(32,150,66,144,106),(41,57,77,117,159),(42,118,58,160,78),(43,153,119,79,59),(44,80,154,60,120),(45,61,73,113,155),(46,114,62,156,74),(47,157,115,75,63),(48,76,158,64,116)], [(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×D42D5C2×Dic10D42D5C2×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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