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## G = C2×D4.F5order 320 = 26·5

### Direct product of C2 and D4.F5

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 [×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 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 5 8A ··· 8H 8I ··· 8T 10A 10B 10C 10D 10E 10F 10G 20A 20B order 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 5 8 ··· 8 8 ··· 8 10 10 10 10 10 10 10 20 20 size 1 1 1 1 2 2 2 2 10 10 2 2 5 5 5 5 10 10 10 10 4 5 ··· 5 10 ··· 10 4 4 4 8 8 8 8 8 8

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 4 4 4 4 8 type + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 C8○D4 F5 C2×F5 C2×F5 C2×F5 D4.F5 kernel C2×D4.F5 C2×D5⋊C8 C2×C4.F5 D4.F5 C22×C5⋊C8 C2×C22.F5 C2×D4⋊2D5 C2×Dic10 D4⋊2D5 C2×C5⋊D4 D4×C10 C10 C2×D4 C2×C4 D4 C23 C2 # reps 1 1 1 8 2 2 1 2 8 4 2 8 1 1 4 2 2

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

 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 0 0 0 0 0 0 0 32 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 32 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 9 23 0 0 0 0 0 0 9 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 40 40 40 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 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 0 8 8 0 0 0 0 8 8 0 36 0 0 0 0 33 28 33 0 0 0 0 0 5 13 13 5

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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