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 [×3], C2 [×2], C4 [×4], C4 [×2], C22 [×3], C22 [×2], C5, C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×4], C23, C10 [×3], C10 [×2], C4⋊C4 [×2], C2×C8 [×2], M4(2) [×4], M4(2) [×2], C22×C4, C22×C4 [×2], C20 [×4], C20 [×2], C2×C10 [×3], C2×C10 [×2], C2×C4⋊C4, C2×M4(2) [×2], C40 [×4], C2×C20 [×2], C2×C20 [×4], C2×C20 [×4], C22×C10, C22.C42, C5×C4⋊C4 [×2], C2×C40 [×2], C5×M4(2) [×4], C5×M4(2) [×2], C22×C20, C22×C20 [×2], C10×C4⋊C4, C10×M4(2) [×2], C5×C22.C42
Quotients: C1, C2 [×3], C4 [×6], C22, C5, C2×C4 [×3], D4 [×3], Q8, C10 [×3], C42, C22⋊C4 [×3], C4⋊C4 [×3], C20 [×6], C2×C10, C2.C42, C4.D4, C4.10D4, C2×C20 [×3], C5×D4 [×3], C5×Q8, C22.C42, C4×C20, C5×C22⋊C4 [×3], C5×C4⋊C4 [×3], C5×C2.C42, C5×C4.D4, C5×C4.10D4, C5×C22.C42
►On 160 points
Generators in S160
(1 105 25 97 17)(2 106 26 98 18)(3 107 27 99 19)(4 108 28 100 20)(5 109 29 101 21)(6 110 30 102 22)(7 111 31 103 23)(8 112 32 104 24)(9 46 118 38 90)(10 47 119 39 91)(11 48 120 40 92)(12 41 113 33 93)(13 42 114 34 94)(14 43 115 35 95)(15 44 116 36 96)(16 45 117 37 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)(9 156)(10 153)(11 158)(12 155)(13 160)(14 157)(15 154)(16 159)(17 55)(18 52)(19 49)(20 54)(21 51)(22 56)(23 53)(24 50)(25 63)(26 60)(27 57)(28 62)(29 59)(30 64)(31 61)(32 58)(33 67)(34 72)(35 69)(36 66)(37 71)(38 68)(39 65)(40 70)(41 75)(42 80)(43 77)(44 74)(45 79)(46 76)(47 73)(48 78)(81 93)(82 90)(83 95)(84 92)(85 89)(86 94)(87 91)(88 96)(97 135)(98 132)(99 129)(100 134)(101 131)(102 136)(103 133)(104 130)(105 143)(106 140)(107 137)(108 142)(109 139)(110 144)(111 141)(112 138)(113 147)(114 152)(115 149)(116 146)(117 151)(118 148)(119 145)(120 150)
(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 33)(2 38 128 72)(3 65 121 39)(4 36 122 70)(5 71 123 37)(6 34 124 68)(7 69 125 35)(8 40 126 66)(9 64 160 26)(10 27 153 57)(11 62 154 32)(12 25 155 63)(13 60 156 30)(14 31 157 61)(15 58 158 28)(16 29 159 59)(17 147 55 113)(18 118 56 152)(19 145 49 119)(20 116 50 150)(21 151 51 117)(22 114 52 148)(23 149 53 115)(24 120 54 146)(41 97 75 135)(42 132 76 102)(43 103 77 133)(44 130 78 100)(45 101 79 131)(46 136 80 98)(47 99 73 129)(48 134 74 104)(81 143 93 105)(82 110 94 140)(83 141 95 111)(84 108 96 138)(85 139 89 109)(86 106 90 144)(87 137 91 107)(88 112 92 142)
G:=sub<Sym(160)| (1,105,25,97,17)(2,106,26,98,18)(3,107,27,99,19)(4,108,28,100,20)(5,109,29,101,21)(6,110,30,102,22)(7,111,31,103,23)(8,112,32,104,24)(9,46,118,38,90)(10,47,119,39,91)(11,48,120,40,92)(12,41,113,33,93)(13,42,114,34,94)(14,43,115,35,95)(15,44,116,36,96)(16,45,117,37,89)(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)(9,156)(10,153)(11,158)(12,155)(13,160)(14,157)(15,154)(16,159)(17,55)(18,52)(19,49)(20,54)(21,51)(22,56)(23,53)(24,50)(25,63)(26,60)(27,57)(28,62)(29,59)(30,64)(31,61)(32,58)(33,67)(34,72)(35,69)(36,66)(37,71)(38,68)(39,65)(40,70)(41,75)(42,80)(43,77)(44,74)(45,79)(46,76)(47,73)(48,78)(81,93)(82,90)(83,95)(84,92)(85,89)(86,94)(87,91)(88,96)(97,135)(98,132)(99,129)(100,134)(101,131)(102,136)(103,133)(104,130)(105,143)(106,140)(107,137)(108,142)(109,139)(110,144)(111,141)(112,138)(113,147)(114,152)(115,149)(116,146)(117,151)(118,148)(119,145)(120,150), (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), (1,67,127,33)(2,38,128,72)(3,65,121,39)(4,36,122,70)(5,71,123,37)(6,34,124,68)(7,69,125,35)(8,40,126,66)(9,64,160,26)(10,27,153,57)(11,62,154,32)(12,25,155,63)(13,60,156,30)(14,31,157,61)(15,58,158,28)(16,29,159,59)(17,147,55,113)(18,118,56,152)(19,145,49,119)(20,116,50,150)(21,151,51,117)(22,114,52,148)(23,149,53,115)(24,120,54,146)(41,97,75,135)(42,132,76,102)(43,103,77,133)(44,130,78,100)(45,101,79,131)(46,136,80,98)(47,99,73,129)(48,134,74,104)(81,143,93,105)(82,110,94,140)(83,141,95,111)(84,108,96,138)(85,139,89,109)(86,106,90,144)(87,137,91,107)(88,112,92,142)>;
G:=Group( (1,105,25,97,17)(2,106,26,98,18)(3,107,27,99,19)(4,108,28,100,20)(5,109,29,101,21)(6,110,30,102,22)(7,111,31,103,23)(8,112,32,104,24)(9,46,118,38,90)(10,47,119,39,91)(11,48,120,40,92)(12,41,113,33,93)(13,42,114,34,94)(14,43,115,35,95)(15,44,116,36,96)(16,45,117,37,89)(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)(9,156)(10,153)(11,158)(12,155)(13,160)(14,157)(15,154)(16,159)(17,55)(18,52)(19,49)(20,54)(21,51)(22,56)(23,53)(24,50)(25,63)(26,60)(27,57)(28,62)(29,59)(30,64)(31,61)(32,58)(33,67)(34,72)(35,69)(36,66)(37,71)(38,68)(39,65)(40,70)(41,75)(42,80)(43,77)(44,74)(45,79)(46,76)(47,73)(48,78)(81,93)(82,90)(83,95)(84,92)(85,89)(86,94)(87,91)(88,96)(97,135)(98,132)(99,129)(100,134)(101,131)(102,136)(103,133)(104,130)(105,143)(106,140)(107,137)(108,142)(109,139)(110,144)(111,141)(112,138)(113,147)(114,152)(115,149)(116,146)(117,151)(118,148)(119,145)(120,150), (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), (1,67,127,33)(2,38,128,72)(3,65,121,39)(4,36,122,70)(5,71,123,37)(6,34,124,68)(7,69,125,35)(8,40,126,66)(9,64,160,26)(10,27,153,57)(11,62,154,32)(12,25,155,63)(13,60,156,30)(14,31,157,61)(15,58,158,28)(16,29,159,59)(17,147,55,113)(18,118,56,152)(19,145,49,119)(20,116,50,150)(21,151,51,117)(22,114,52,148)(23,149,53,115)(24,120,54,146)(41,97,75,135)(42,132,76,102)(43,103,77,133)(44,130,78,100)(45,101,79,131)(46,136,80,98)(47,99,73,129)(48,134,74,104)(81,143,93,105)(82,110,94,140)(83,141,95,111)(84,108,96,138)(85,139,89,109)(86,106,90,144)(87,137,91,107)(88,112,92,142) );
G=PermutationGroup([(1,105,25,97,17),(2,106,26,98,18),(3,107,27,99,19),(4,108,28,100,20),(5,109,29,101,21),(6,110,30,102,22),(7,111,31,103,23),(8,112,32,104,24),(9,46,118,38,90),(10,47,119,39,91),(11,48,120,40,92),(12,41,113,33,93),(13,42,114,34,94),(14,43,115,35,95),(15,44,116,36,96),(16,45,117,37,89),(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),(9,156),(10,153),(11,158),(12,155),(13,160),(14,157),(15,154),(16,159),(17,55),(18,52),(19,49),(20,54),(21,51),(22,56),(23,53),(24,50),(25,63),(26,60),(27,57),(28,62),(29,59),(30,64),(31,61),(32,58),(33,67),(34,72),(35,69),(36,66),(37,71),(38,68),(39,65),(40,70),(41,75),(42,80),(43,77),(44,74),(45,79),(46,76),(47,73),(48,78),(81,93),(82,90),(83,95),(84,92),(85,89),(86,94),(87,91),(88,96),(97,135),(98,132),(99,129),(100,134),(101,131),(102,136),(103,133),(104,130),(105,143),(106,140),(107,137),(108,142),(109,139),(110,144),(111,141),(112,138),(113,147),(114,152),(115,149),(116,146),(117,151),(118,148),(119,145),(120,150)], [(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)], [(1,67,127,33),(2,38,128,72),(3,65,121,39),(4,36,122,70),(5,71,123,37),(6,34,124,68),(7,69,125,35),(8,40,126,66),(9,64,160,26),(10,27,153,57),(11,62,154,32),(12,25,155,63),(13,60,156,30),(14,31,157,61),(15,58,158,28),(16,29,159,59),(17,147,55,113),(18,118,56,152),(19,145,49,119),(20,116,50,150),(21,151,51,117),(22,114,52,148),(23,149,53,115),(24,120,54,146),(41,97,75,135),(42,132,76,102),(43,103,77,133),(44,130,78,100),(45,101,79,131),(46,136,80,98),(47,99,73,129),(48,134,74,104),(81,143,93,105),(82,110,94,140),(83,141,95,111),(84,108,96,138),(85,139,89,109),(86,106,90,144),(87,137,91,107),(88,112,92,142)])
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