direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×C22.C42, M4(2)⋊3C20, C20.84(C4⋊C4), (C2×C20).37Q8, (C2×C20).509D4, (C22×C20).6C4, (C22×C4).3C20, C22.2(C4×C20), (C5×M4(2))⋊15C4, (C2×C10).31C42, C23.23(C2×C20), (C2×M4(2)).8C10, C20.152(C22⋊C4), C10.22(C4.D4), (C10×M4(2)).20C2, C10.18(C4.10D4), (C22×C20).388C22, C10.46(C2.C42), C4.4(C5×C4⋊C4), (C2×C4⋊C4).3C10, (C2×C4).2(C5×Q8), C22.5(C5×C4⋊C4), (C10×C4⋊C4).30C2, (C2×C4).14(C2×C20), (C2×C4).114(C5×D4), C4.20(C5×C22⋊C4), C2.2(C5×C4.D4), (C2×C10).50(C4⋊C4), (C2×C20).354(C2×C4), C2.2(C5×C4.10D4), (C22×C4).18(C2×C10), C22.30(C5×C22⋊C4), C2.8(C5×C2.C42), (C22×C10).176(C2×C4), (C2×C10).196(C22⋊C4), SmallGroup(320,148)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C22.C42
G = < a,b,c,d,e | a5=b2=c2=e4=1, d4=c, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=bcd >
Subgroups: 154 in 98 conjugacy classes, 58 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, C23, C10, C10, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C20, C20, C2×C10, C2×C10, C2×C4⋊C4, C2×M4(2), C40, C2×C20, C2×C20, C2×C20, C22×C10, C22.C42, C5×C4⋊C4, C2×C40, C5×M4(2), C5×M4(2), C22×C20, C22×C20, C10×C4⋊C4, C10×M4(2), C5×C22.C42
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, Q8, C10, C42, C22⋊C4, C4⋊C4, C20, C2×C10, C2.C42, C4.D4, C4.10D4, C2×C20, C5×D4, C5×Q8, C22.C42, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C5×C2.C42, C5×C4.D4, C5×C4.10D4, C5×C22.C42
►On 160 points
Generators in S160
(1 37 111 31 103)(2 38 112 32 104)(3 39 105 25 97)(4 40 106 26 98)(5 33 107 27 99)(6 34 108 28 100)(7 35 109 29 101)(8 36 110 30 102)(9 21 45 117 90)(10 22 46 118 91)(11 23 47 119 92)(12 24 48 120 93)(13 17 41 113 94)(14 18 42 114 95)(15 19 43 115 96)(16 20 44 116 89)(49 121 137 57 129)(50 122 138 58 130)(51 123 139 59 131)(52 124 140 60 132)(53 125 141 61 133)(54 126 142 62 134)(55 127 143 63 135)(56 128 144 64 136)(65 87 153 73 145)(66 88 154 74 146)(67 81 155 75 147)(68 82 156 76 148)(69 83 157 77 149)(70 84 158 78 150)(71 85 159 79 151)(72 86 160 80 152)
(1 127)(2 124)(3 121)(4 126)(5 123)(6 128)(7 125)(8 122)(9 155)(10 160)(11 157)(12 154)(13 159)(14 156)(15 153)(16 158)(17 79)(18 76)(19 73)(20 78)(21 75)(22 80)(23 77)(24 74)(25 129)(26 134)(27 131)(28 136)(29 133)(30 130)(31 135)(32 132)(33 139)(34 144)(35 141)(36 138)(37 143)(38 140)(39 137)(40 142)(41 151)(42 148)(43 145)(44 150)(45 147)(46 152)(47 149)(48 146)(49 97)(50 102)(51 99)(52 104)(53 101)(54 98)(55 103)(56 100)(57 105)(58 110)(59 107)(60 112)(61 109)(62 106)(63 111)(64 108)(65 115)(66 120)(67 117)(68 114)(69 119)(70 116)(71 113)(72 118)(81 90)(82 95)(83 92)(84 89)(85 94)(86 91)(87 96)(88 93)
(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 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)
(1 67 127 117)(2 114 128 72)(3 65 121 115)(4 120 122 70)(5 71 123 113)(6 118 124 68)(7 69 125 119)(8 116 126 66)(9 111 155 63)(10 60 156 108)(11 109 157 61)(12 58 158 106)(13 107 159 59)(14 64 160 112)(15 105 153 57)(16 62 154 110)(17 27 79 131)(18 136 80 32)(19 25 73 129)(20 134 74 30)(21 31 75 135)(22 132 76 28)(23 29 77 133)(24 130 78 26)(33 85 139 94)(34 91 140 82)(35 83 141 92)(36 89 142 88)(37 81 143 90)(38 95 144 86)(39 87 137 96)(40 93 138 84)(41 99 151 51)(42 56 152 104)(43 97 145 49)(44 54 146 102)(45 103 147 55)(46 52 148 100)(47 101 149 53)(48 50 150 98)
G:=sub<Sym(160)| (1,37,111,31,103)(2,38,112,32,104)(3,39,105,25,97)(4,40,106,26,98)(5,33,107,27,99)(6,34,108,28,100)(7,35,109,29,101)(8,36,110,30,102)(9,21,45,117,90)(10,22,46,118,91)(11,23,47,119,92)(12,24,48,120,93)(13,17,41,113,94)(14,18,42,114,95)(15,19,43,115,96)(16,20,44,116,89)(49,121,137,57,129)(50,122,138,58,130)(51,123,139,59,131)(52,124,140,60,132)(53,125,141,61,133)(54,126,142,62,134)(55,127,143,63,135)(56,128,144,64,136)(65,87,153,73,145)(66,88,154,74,146)(67,81,155,75,147)(68,82,156,76,148)(69,83,157,77,149)(70,84,158,78,150)(71,85,159,79,151)(72,86,160,80,152), (1,127)(2,124)(3,121)(4,126)(5,123)(6,128)(7,125)(8,122)(9,155)(10,160)(11,157)(12,154)(13,159)(14,156)(15,153)(16,158)(17,79)(18,76)(19,73)(20,78)(21,75)(22,80)(23,77)(24,74)(25,129)(26,134)(27,131)(28,136)(29,133)(30,130)(31,135)(32,132)(33,139)(34,144)(35,141)(36,138)(37,143)(38,140)(39,137)(40,142)(41,151)(42,148)(43,145)(44,150)(45,147)(46,152)(47,149)(48,146)(49,97)(50,102)(51,99)(52,104)(53,101)(54,98)(55,103)(56,100)(57,105)(58,110)(59,107)(60,112)(61,109)(62,106)(63,111)(64,108)(65,115)(66,120)(67,117)(68,114)(69,119)(70,116)(71,113)(72,118)(81,90)(82,95)(83,92)(84,89)(85,94)(86,91)(87,96)(88,93), (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), (1,2,3,4,5,6,7,8)(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), (1,67,127,117)(2,114,128,72)(3,65,121,115)(4,120,122,70)(5,71,123,113)(6,118,124,68)(7,69,125,119)(8,116,126,66)(9,111,155,63)(10,60,156,108)(11,109,157,61)(12,58,158,106)(13,107,159,59)(14,64,160,112)(15,105,153,57)(16,62,154,110)(17,27,79,131)(18,136,80,32)(19,25,73,129)(20,134,74,30)(21,31,75,135)(22,132,76,28)(23,29,77,133)(24,130,78,26)(33,85,139,94)(34,91,140,82)(35,83,141,92)(36,89,142,88)(37,81,143,90)(38,95,144,86)(39,87,137,96)(40,93,138,84)(41,99,151,51)(42,56,152,104)(43,97,145,49)(44,54,146,102)(45,103,147,55)(46,52,148,100)(47,101,149,53)(48,50,150,98)>;
G:=Group( (1,37,111,31,103)(2,38,112,32,104)(3,39,105,25,97)(4,40,106,26,98)(5,33,107,27,99)(6,34,108,28,100)(7,35,109,29,101)(8,36,110,30,102)(9,21,45,117,90)(10,22,46,118,91)(11,23,47,119,92)(12,24,48,120,93)(13,17,41,113,94)(14,18,42,114,95)(15,19,43,115,96)(16,20,44,116,89)(49,121,137,57,129)(50,122,138,58,130)(51,123,139,59,131)(52,124,140,60,132)(53,125,141,61,133)(54,126,142,62,134)(55,127,143,63,135)(56,128,144,64,136)(65,87,153,73,145)(66,88,154,74,146)(67,81,155,75,147)(68,82,156,76,148)(69,83,157,77,149)(70,84,158,78,150)(71,85,159,79,151)(72,86,160,80,152), (1,127)(2,124)(3,121)(4,126)(5,123)(6,128)(7,125)(8,122)(9,155)(10,160)(11,157)(12,154)(13,159)(14,156)(15,153)(16,158)(17,79)(18,76)(19,73)(20,78)(21,75)(22,80)(23,77)(24,74)(25,129)(26,134)(27,131)(28,136)(29,133)(30,130)(31,135)(32,132)(33,139)(34,144)(35,141)(36,138)(37,143)(38,140)(39,137)(40,142)(41,151)(42,148)(43,145)(44,150)(45,147)(46,152)(47,149)(48,146)(49,97)(50,102)(51,99)(52,104)(53,101)(54,98)(55,103)(56,100)(57,105)(58,110)(59,107)(60,112)(61,109)(62,106)(63,111)(64,108)(65,115)(66,120)(67,117)(68,114)(69,119)(70,116)(71,113)(72,118)(81,90)(82,95)(83,92)(84,89)(85,94)(86,91)(87,96)(88,93), (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), (1,2,3,4,5,6,7,8)(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), (1,67,127,117)(2,114,128,72)(3,65,121,115)(4,120,122,70)(5,71,123,113)(6,118,124,68)(7,69,125,119)(8,116,126,66)(9,111,155,63)(10,60,156,108)(11,109,157,61)(12,58,158,106)(13,107,159,59)(14,64,160,112)(15,105,153,57)(16,62,154,110)(17,27,79,131)(18,136,80,32)(19,25,73,129)(20,134,74,30)(21,31,75,135)(22,132,76,28)(23,29,77,133)(24,130,78,26)(33,85,139,94)(34,91,140,82)(35,83,141,92)(36,89,142,88)(37,81,143,90)(38,95,144,86)(39,87,137,96)(40,93,138,84)(41,99,151,51)(42,56,152,104)(43,97,145,49)(44,54,146,102)(45,103,147,55)(46,52,148,100)(47,101,149,53)(48,50,150,98) );
G=PermutationGroup([[(1,37,111,31,103),(2,38,112,32,104),(3,39,105,25,97),(4,40,106,26,98),(5,33,107,27,99),(6,34,108,28,100),(7,35,109,29,101),(8,36,110,30,102),(9,21,45,117,90),(10,22,46,118,91),(11,23,47,119,92),(12,24,48,120,93),(13,17,41,113,94),(14,18,42,114,95),(15,19,43,115,96),(16,20,44,116,89),(49,121,137,57,129),(50,122,138,58,130),(51,123,139,59,131),(52,124,140,60,132),(53,125,141,61,133),(54,126,142,62,134),(55,127,143,63,135),(56,128,144,64,136),(65,87,153,73,145),(66,88,154,74,146),(67,81,155,75,147),(68,82,156,76,148),(69,83,157,77,149),(70,84,158,78,150),(71,85,159,79,151),(72,86,160,80,152)], [(1,127),(2,124),(3,121),(4,126),(5,123),(6,128),(7,125),(8,122),(9,155),(10,160),(11,157),(12,154),(13,159),(14,156),(15,153),(16,158),(17,79),(18,76),(19,73),(20,78),(21,75),(22,80),(23,77),(24,74),(25,129),(26,134),(27,131),(28,136),(29,133),(30,130),(31,135),(32,132),(33,139),(34,144),(35,141),(36,138),(37,143),(38,140),(39,137),(40,142),(41,151),(42,148),(43,145),(44,150),(45,147),(46,152),(47,149),(48,146),(49,97),(50,102),(51,99),(52,104),(53,101),(54,98),(55,103),(56,100),(57,105),(58,110),(59,107),(60,112),(61,109),(62,106),(63,111),(64,108),(65,115),(66,120),(67,117),(68,114),(69,119),(70,116),(71,113),(72,118),(81,90),(82,95),(83,92),(84,89),(85,94),(86,91),(87,96),(88,93)], [(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)], [(1,2,3,4,5,6,7,8),(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)], [(1,67,127,117),(2,114,128,72),(3,65,121,115),(4,120,122,70),(5,71,123,113),(6,118,124,68),(7,69,125,119),(8,116,126,66),(9,111,155,63),(10,60,156,108),(11,109,157,61),(12,58,158,106),(13,107,159,59),(14,64,160,112),(15,105,153,57),(16,62,154,110),(17,27,79,131),(18,136,80,32),(19,25,73,129),(20,134,74,30),(21,31,75,135),(22,132,76,28),(23,29,77,133),(24,130,78,26),(33,85,139,94),(34,91,140,82),(35,83,141,92),(36,89,142,88),(37,81,143,90),(38,95,144,86),(39,87,137,96),(40,93,138,84),(41,99,151,51),(42,56,152,104),(43,97,145,49),(44,54,146,102),(45,103,147,55),(46,52,148,100),(47,101,149,53),(48,50,150,98)]])
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 10A | ··· | 10L | 10M | ··· | 10T | 20A | ··· | 20P | 20Q | ··· | 20AF | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | - | |||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C5 | C10 | C10 | C20 | C20 | D4 | Q8 | C5×D4 | C5×Q8 | C4.D4 | C4.10D4 | C5×C4.D4 | C5×C4.10D4 |
| kernel | C5×C22.C42 | C10×C4⋊C4 | C10×M4(2) | C5×M4(2) | C22×C20 | C22.C42 | C2×C4⋊C4 | C2×M4(2) | M4(2) | C22×C4 | C2×C20 | C2×C20 | C2×C4 | C2×C4 | C10 | C10 | C2 | C2 |
| # reps | 1 | 1 | 2 | 8 | 4 | 4 | 4 | 8 | 32 | 16 | 3 | 1 | 12 | 4 | 1 | 1 | 4 | 4 |
Matrix representation of C5×C22.C42 ►in GL8(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 10 | 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 | 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 |
| 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 | 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 |
| 30 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 40 | 11 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 40 | 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 | 11 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 30 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 30 | 40 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 11 |
G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[30,40,0,0,0,0,0,0,40,11,0,0,0,0,0,0,0,0,30,13,0,0,0,0,0,0,16,11,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,9,23,0,0,0,0,0,0,9,32,0,0,0,0,0,0,0,0,11,1,0,0,0,0,0,0,1,30,0,0,0,0,0,0,0,0,30,40,0,0,0,0,0,0,40,11] >;
C5×C22.C42 in GAP, Magma, Sage, TeX
C_5\times C_2^2.C_4^2
% in TeX
G:=Group("C5xC2^2.C4^2"); // GroupNames label
G:=SmallGroup(320,148);
// by ID
G=gap.SmallGroup(320,148);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,568,5043,3511,172]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^2=c^2=e^4=1,d^4=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*c*d>;
// generators/relations