direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4×C22.F5, C20⋊9M4(2), Dic5⋊8M4(2), Dic5.14C42, C5⋊3(C4×M4(2)), C22.5(C4×F5), C23.39(C2×F5), (C22×C4).11F5, (C2×C10).21C42, (C22×C20).33C4, C10.18(C2×C42), (C4×Dic5).48C4, C2.6(D5⋊M4(2)), C10.24(C2×M4(2)), C10.C42⋊17C2, C22.44(C22×F5), (C22×Dic5).30C4, Dic5.37(C22×C4), (C2×Dic5).341C23, (C4×Dic5).325C22, (C22×Dic5).269C22, C5⋊C8⋊3(C2×C4), (C4×C5⋊C8)⋊17C2, C2.18(C2×C4×F5), (C2×C5⋊C8).36C22, (C2×C4).104(C2×F5), (C2×C4×Dic5).51C2, (C2×C20).104(C2×C4), C2.2(C2×C22.F5), (C2×C22.F5).7C2, (C22×C10).57(C2×C4), (C2×C10).57(C22×C4), (C2×Dic5).128(C2×C4), SmallGroup(320,1088)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×C22.F5
G = < a,b,c,d,e | a4=b2=c2=d5=1, e4=c, ab=ba, ac=ca, ad=da, ae=ea, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d3 >
Subgroups: 378 in 142 conjugacy classes, 74 normal (30 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C23, C10, C10, C42, C2×C8, M4(2), C22×C4, C22×C4, Dic5, Dic5, C20, C20, C2×C10, C2×C10, C2×C10, C4×C8, C8⋊C4, C2×C42, C2×M4(2), C5⋊C8, C2×Dic5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C4×M4(2), C4×Dic5, C2×C5⋊C8, C22.F5, C22×Dic5, C22×C20, C4×C5⋊C8, C10.C42, C2×C4×Dic5, C2×C22.F5, C4×C22.F5
Quotients: C1, C2, C4, C22, C2×C4, C23, C42, M4(2), C22×C4, F5, C2×C42, C2×M4(2), C2×F5, C4×M4(2), C4×F5, C22.F5, C22×F5, D5⋊M4(2), C2×C4×F5, C2×C22.F5, C4×C22.F5
►On 160 points
Generators in S160
(1 78 88 108)(2 79 81 109)(3 80 82 110)(4 73 83 111)(5 74 84 112)(6 75 85 105)(7 76 86 106)(8 77 87 107)(9 18 132 128)(10 19 133 121)(11 20 134 122)(12 21 135 123)(13 22 136 124)(14 23 129 125)(15 24 130 126)(16 17 131 127)(25 69 150 46)(26 70 151 47)(27 71 152 48)(28 72 145 41)(29 65 146 42)(30 66 147 43)(31 67 148 44)(32 68 149 45)(33 96 140 61)(34 89 141 62)(35 90 142 63)(36 91 143 64)(37 92 144 57)(38 93 137 58)(39 94 138 59)(40 95 139 60)(49 98 115 159)(50 99 116 160)(51 100 117 153)(52 101 118 154)(53 102 119 155)(54 103 120 156)(55 104 113 157)(56 97 114 158)
(1 88)(2 85)(3 82)(4 87)(5 84)(6 81)(7 86)(8 83)(9 132)(10 129)(11 134)(12 131)(13 136)(14 133)(15 130)(16 135)(17 123)(18 128)(19 125)(20 122)(21 127)(22 124)(23 121)(24 126)(25 150)(26 147)(27 152)(28 149)(29 146)(30 151)(31 148)(32 145)(33 144)(34 141)(35 138)(36 143)(37 140)(38 137)(39 142)(40 139)(41 68)(42 65)(43 70)(44 67)(45 72)(46 69)(47 66)(48 71)(49 115)(50 120)(51 117)(52 114)(53 119)(54 116)(55 113)(56 118)(57 96)(58 93)(59 90)(60 95)(61 92)(62 89)(63 94)(64 91)(73 107)(74 112)(75 109)(76 106)(77 111)(78 108)(79 105)(80 110)(97 154)(98 159)(99 156)(100 153)(101 158)(102 155)(103 160)(104 157)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)(65 69)(66 70)(67 71)(68 72)(73 77)(74 78)(75 79)(76 80)(81 85)(82 86)(83 87)(84 88)(89 93)(90 94)(91 95)(92 96)(97 101)(98 102)(99 103)(100 104)(105 109)(106 110)(107 111)(108 112)(113 117)(114 118)(115 119)(116 120)(121 125)(122 126)(123 127)(124 128)(129 133)(130 134)(131 135)(132 136)(137 141)(138 142)(139 143)(140 144)(145 149)(146 150)(147 151)(148 152)(153 157)(154 158)(155 159)(156 160)
(1 159 36 9 65)(2 10 160 66 37)(3 67 11 38 153)(4 39 68 154 12)(5 155 40 13 69)(6 14 156 70 33)(7 71 15 34 157)(8 35 72 158 16)(17 77 90 145 56)(18 146 78 49 91)(19 50 147 92 79)(20 93 51 80 148)(21 73 94 149 52)(22 150 74 53 95)(23 54 151 96 75)(24 89 55 76 152)(25 112 119 60 124)(26 61 105 125 120)(27 126 62 113 106)(28 114 127 107 63)(29 108 115 64 128)(30 57 109 121 116)(31 122 58 117 110)(32 118 123 111 59)(41 97 131 87 142)(42 88 98 143 132)(43 144 81 133 99)(44 134 137 100 82)(45 101 135 83 138)(46 84 102 139 136)(47 140 85 129 103)(48 130 141 104 86)
(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)| (1,78,88,108)(2,79,81,109)(3,80,82,110)(4,73,83,111)(5,74,84,112)(6,75,85,105)(7,76,86,106)(8,77,87,107)(9,18,132,128)(10,19,133,121)(11,20,134,122)(12,21,135,123)(13,22,136,124)(14,23,129,125)(15,24,130,126)(16,17,131,127)(25,69,150,46)(26,70,151,47)(27,71,152,48)(28,72,145,41)(29,65,146,42)(30,66,147,43)(31,67,148,44)(32,68,149,45)(33,96,140,61)(34,89,141,62)(35,90,142,63)(36,91,143,64)(37,92,144,57)(38,93,137,58)(39,94,138,59)(40,95,139,60)(49,98,115,159)(50,99,116,160)(51,100,117,153)(52,101,118,154)(53,102,119,155)(54,103,120,156)(55,104,113,157)(56,97,114,158), (1,88)(2,85)(3,82)(4,87)(5,84)(6,81)(7,86)(8,83)(9,132)(10,129)(11,134)(12,131)(13,136)(14,133)(15,130)(16,135)(17,123)(18,128)(19,125)(20,122)(21,127)(22,124)(23,121)(24,126)(25,150)(26,147)(27,152)(28,149)(29,146)(30,151)(31,148)(32,145)(33,144)(34,141)(35,138)(36,143)(37,140)(38,137)(39,142)(40,139)(41,68)(42,65)(43,70)(44,67)(45,72)(46,69)(47,66)(48,71)(49,115)(50,120)(51,117)(52,114)(53,119)(54,116)(55,113)(56,118)(57,96)(58,93)(59,90)(60,95)(61,92)(62,89)(63,94)(64,91)(73,107)(74,112)(75,109)(76,106)(77,111)(78,108)(79,105)(80,110)(97,154)(98,159)(99,156)(100,153)(101,158)(102,155)(103,160)(104,157), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(105,109)(106,110)(107,111)(108,112)(113,117)(114,118)(115,119)(116,120)(121,125)(122,126)(123,127)(124,128)(129,133)(130,134)(131,135)(132,136)(137,141)(138,142)(139,143)(140,144)(145,149)(146,150)(147,151)(148,152)(153,157)(154,158)(155,159)(156,160), (1,159,36,9,65)(2,10,160,66,37)(3,67,11,38,153)(4,39,68,154,12)(5,155,40,13,69)(6,14,156,70,33)(7,71,15,34,157)(8,35,72,158,16)(17,77,90,145,56)(18,146,78,49,91)(19,50,147,92,79)(20,93,51,80,148)(21,73,94,149,52)(22,150,74,53,95)(23,54,151,96,75)(24,89,55,76,152)(25,112,119,60,124)(26,61,105,125,120)(27,126,62,113,106)(28,114,127,107,63)(29,108,115,64,128)(30,57,109,121,116)(31,122,58,117,110)(32,118,123,111,59)(41,97,131,87,142)(42,88,98,143,132)(43,144,81,133,99)(44,134,137,100,82)(45,101,135,83,138)(46,84,102,139,136)(47,140,85,129,103)(48,130,141,104,86), (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( (1,78,88,108)(2,79,81,109)(3,80,82,110)(4,73,83,111)(5,74,84,112)(6,75,85,105)(7,76,86,106)(8,77,87,107)(9,18,132,128)(10,19,133,121)(11,20,134,122)(12,21,135,123)(13,22,136,124)(14,23,129,125)(15,24,130,126)(16,17,131,127)(25,69,150,46)(26,70,151,47)(27,71,152,48)(28,72,145,41)(29,65,146,42)(30,66,147,43)(31,67,148,44)(32,68,149,45)(33,96,140,61)(34,89,141,62)(35,90,142,63)(36,91,143,64)(37,92,144,57)(38,93,137,58)(39,94,138,59)(40,95,139,60)(49,98,115,159)(50,99,116,160)(51,100,117,153)(52,101,118,154)(53,102,119,155)(54,103,120,156)(55,104,113,157)(56,97,114,158), (1,88)(2,85)(3,82)(4,87)(5,84)(6,81)(7,86)(8,83)(9,132)(10,129)(11,134)(12,131)(13,136)(14,133)(15,130)(16,135)(17,123)(18,128)(19,125)(20,122)(21,127)(22,124)(23,121)(24,126)(25,150)(26,147)(27,152)(28,149)(29,146)(30,151)(31,148)(32,145)(33,144)(34,141)(35,138)(36,143)(37,140)(38,137)(39,142)(40,139)(41,68)(42,65)(43,70)(44,67)(45,72)(46,69)(47,66)(48,71)(49,115)(50,120)(51,117)(52,114)(53,119)(54,116)(55,113)(56,118)(57,96)(58,93)(59,90)(60,95)(61,92)(62,89)(63,94)(64,91)(73,107)(74,112)(75,109)(76,106)(77,111)(78,108)(79,105)(80,110)(97,154)(98,159)(99,156)(100,153)(101,158)(102,155)(103,160)(104,157), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(105,109)(106,110)(107,111)(108,112)(113,117)(114,118)(115,119)(116,120)(121,125)(122,126)(123,127)(124,128)(129,133)(130,134)(131,135)(132,136)(137,141)(138,142)(139,143)(140,144)(145,149)(146,150)(147,151)(148,152)(153,157)(154,158)(155,159)(156,160), (1,159,36,9,65)(2,10,160,66,37)(3,67,11,38,153)(4,39,68,154,12)(5,155,40,13,69)(6,14,156,70,33)(7,71,15,34,157)(8,35,72,158,16)(17,77,90,145,56)(18,146,78,49,91)(19,50,147,92,79)(20,93,51,80,148)(21,73,94,149,52)(22,150,74,53,95)(23,54,151,96,75)(24,89,55,76,152)(25,112,119,60,124)(26,61,105,125,120)(27,126,62,113,106)(28,114,127,107,63)(29,108,115,64,128)(30,57,109,121,116)(31,122,58,117,110)(32,118,123,111,59)(41,97,131,87,142)(42,88,98,143,132)(43,144,81,133,99)(44,134,137,100,82)(45,101,135,83,138)(46,84,102,139,136)(47,140,85,129,103)(48,130,141,104,86), (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([[(1,78,88,108),(2,79,81,109),(3,80,82,110),(4,73,83,111),(5,74,84,112),(6,75,85,105),(7,76,86,106),(8,77,87,107),(9,18,132,128),(10,19,133,121),(11,20,134,122),(12,21,135,123),(13,22,136,124),(14,23,129,125),(15,24,130,126),(16,17,131,127),(25,69,150,46),(26,70,151,47),(27,71,152,48),(28,72,145,41),(29,65,146,42),(30,66,147,43),(31,67,148,44),(32,68,149,45),(33,96,140,61),(34,89,141,62),(35,90,142,63),(36,91,143,64),(37,92,144,57),(38,93,137,58),(39,94,138,59),(40,95,139,60),(49,98,115,159),(50,99,116,160),(51,100,117,153),(52,101,118,154),(53,102,119,155),(54,103,120,156),(55,104,113,157),(56,97,114,158)], [(1,88),(2,85),(3,82),(4,87),(5,84),(6,81),(7,86),(8,83),(9,132),(10,129),(11,134),(12,131),(13,136),(14,133),(15,130),(16,135),(17,123),(18,128),(19,125),(20,122),(21,127),(22,124),(23,121),(24,126),(25,150),(26,147),(27,152),(28,149),(29,146),(30,151),(31,148),(32,145),(33,144),(34,141),(35,138),(36,143),(37,140),(38,137),(39,142),(40,139),(41,68),(42,65),(43,70),(44,67),(45,72),(46,69),(47,66),(48,71),(49,115),(50,120),(51,117),(52,114),(53,119),(54,116),(55,113),(56,118),(57,96),(58,93),(59,90),(60,95),(61,92),(62,89),(63,94),(64,91),(73,107),(74,112),(75,109),(76,106),(77,111),(78,108),(79,105),(80,110),(97,154),(98,159),(99,156),(100,153),(101,158),(102,155),(103,160),(104,157)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(33,37),(34,38),(35,39),(36,40),(41,45),(42,46),(43,47),(44,48),(49,53),(50,54),(51,55),(52,56),(57,61),(58,62),(59,63),(60,64),(65,69),(66,70),(67,71),(68,72),(73,77),(74,78),(75,79),(76,80),(81,85),(82,86),(83,87),(84,88),(89,93),(90,94),(91,95),(92,96),(97,101),(98,102),(99,103),(100,104),(105,109),(106,110),(107,111),(108,112),(113,117),(114,118),(115,119),(116,120),(121,125),(122,126),(123,127),(124,128),(129,133),(130,134),(131,135),(132,136),(137,141),(138,142),(139,143),(140,144),(145,149),(146,150),(147,151),(148,152),(153,157),(154,158),(155,159),(156,160)], [(1,159,36,9,65),(2,10,160,66,37),(3,67,11,38,153),(4,39,68,154,12),(5,155,40,13,69),(6,14,156,70,33),(7,71,15,34,157),(8,35,72,158,16),(17,77,90,145,56),(18,146,78,49,91),(19,50,147,92,79),(20,93,51,80,148),(21,73,94,149,52),(22,150,74,53,95),(23,54,151,96,75),(24,89,55,76,152),(25,112,119,60,124),(26,61,105,125,120),(27,126,62,113,106),(28,114,127,107,63),(29,108,115,64,128),(30,57,109,121,116),(31,122,58,117,110),(32,118,123,111,59),(41,97,131,87,142),(42,88,98,143,132),(43,144,81,133,99),(44,134,137,100,82),(45,101,135,83,138),(46,84,102,139,136),(47,140,85,129,103),(48,130,141,104,86)], [(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)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 4O | 4P | 4Q | 4R | 5 | 8A | ··· | 8P | 10A | ··· | 10G | 20A | ··· | 20H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 5 | ··· | 5 | 10 | 10 | 10 | 10 | 4 | 10 | ··· | 10 | 4 | ··· | 4 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | M4(2) | M4(2) | F5 | C2×F5 | C2×F5 | C22.F5 | C4×F5 | D5⋊M4(2) |
| kernel | C4×C22.F5 | C4×C5⋊C8 | C10.C42 | C2×C4×Dic5 | C2×C22.F5 | C4×Dic5 | C22.F5 | C22×Dic5 | C22×C20 | Dic5 | C20 | C22×C4 | C2×C4 | C23 | C4 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 2 | 4 | 16 | 2 | 2 | 4 | 4 | 1 | 2 | 1 | 4 | 4 | 4 |
Matrix representation of C4×C22.F5 ►in GL8(𝔽41)
| 32 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 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 | 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 |
| 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 | 16 | 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 |
| 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 | 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 |
| 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 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 8 | 40 | 34 |
| 0 | 0 | 0 | 0 | 8 | 32 | 7 | 7 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 32 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 39 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 16 | 1 | 0 |
| 0 | 0 | 0 | 0 | 17 | 16 | 0 | 1 |
| 0 | 0 | 0 | 0 | 10 | 31 | 24 | 25 |
| 0 | 0 | 0 | 0 | 19 | 8 | 24 | 25 |
G:=sub<GL(8,GF(41))| [32,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,16,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,3,8,0,0,0,0,1,34,8,32,0,0,0,0,0,0,40,7,0,0,0,0,0,0,34,7],[0,32,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,16,30,0,0,0,0,0,0,39,25,0,0,0,0,0,0,0,0,17,17,10,19,0,0,0,0,16,16,31,8,0,0,0,0,1,0,24,24,0,0,0,0,0,1,25,25] >;
C4×C22.F5 in GAP, Magma, Sage, TeX
C_4\times C_2^2.F_5
% in TeX
G:=Group("C4xC2^2.F5"); // GroupNames label
G:=SmallGroup(320,1088);
// by ID
G=gap.SmallGroup(320,1088);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,477,100,136,6278,1595]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^2=c^2=d^5=1,e^4=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,e*b*e^-1=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^3>;
// generators/relations