G = C10.452+ 1+4 order 320 = 26·5
45th non-split extension by C10 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.452+ 1+4, C20⋊Q8⋊21C2, C4⋊D4⋊19D5, C20⋊2D4⋊25C2, C4⋊C4.183D10, (D4×Dic5)⋊23C2, (C2×D4).94D10, (C2×C20).43C23, C22⋊C4.51D10, Dic5⋊D4⋊15C2, C20.204(C4○D4), C20.17D4⋊19C2, C4.97(D4⋊2D5), (C2×C10).160C24, (C22×C4).227D10, C2.47(D4⋊6D10), C23.20(C22×D5), Dic5.39(C4○D4), Dic5.5D4⋊21C2, (D4×C10).126C22, C23.11D10⋊7C2, C4⋊Dic5.373C22, (C2×Dic5).79C23, (C22×D5).67C23, C22.181(C23×D5), D10⋊C4.16C22, C23.21D10⋊27C2, (C22×C20).244C22, (C22×C10).189C23, C5⋊3(C22.49C24), (C4×Dic5).105C22, C10.D4.19C22, C23.D5.112C22, (C2×Dic10).162C22, (C22×Dic5).113C22, (C4×C5⋊D4)⋊20C2, C2.44(D5×C4○D4), C4⋊C4⋊7D5⋊21C2, (C5×C4⋊D4)⋊22C2, (C2×C4×D5).96C22, C10.157(C2×C4○D4), C2.39(C2×D4⋊2D5), (C5×C4⋊C4).148C22, (C2×C4).588(C22×D5), (C2×C5⋊D4).33C22, (C5×C22⋊C4).17C22, SmallGroup(320,1288)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10.452+ 1+4
G = < a,b,c,d,e | a10=b4=c2=e2=1, d2=a5b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=a5c, ede=a5b2d >
Subgroups: 790 in 236 conjugacy classes, 97 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, D5, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, Dic5, Dic5, C20, C20, D10, C2×C10, C2×C10, C42⋊C2, C4×D4, C4⋊D4, C4⋊D4, C4.4D4, C4⋊Q8, Dic10, C4×D5, C2×Dic5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C5×D4, C22×D5, C22×C10, C22×C10, C22.49C24, C4×Dic5, C4×Dic5, C10.D4, C10.D4, C4⋊Dic5, D10⋊C4, D10⋊C4, C23.D5, C23.D5, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C22×Dic5, C2×C5⋊D4, C2×C5⋊D4, C22×C20, D4×C10, D4×C10, C23.11D10, Dic5.5D4, C20⋊Q8, C4⋊C4⋊7D5, C23.21D10, C4×C5⋊D4, D4×Dic5, C20.17D4, C20⋊2D4, Dic5⋊D4, C5×C4⋊D4, C10.452+ 1+4
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2+ 1+4, C22×D5, C22.49C24, D4⋊2D5, C23×D5, C2×D4⋊2D5, D4⋊6D10, D5×C4○D4, C10.452+ 1+4
►On 160 points
Generators in S160
(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)
(1 138 13 128)(2 139 14 129)(3 140 15 130)(4 131 16 121)(5 132 17 122)(6 133 18 123)(7 134 19 124)(8 135 20 125)(9 136 11 126)(10 137 12 127)(21 156 31 146)(22 157 32 147)(23 158 33 148)(24 159 34 149)(25 160 35 150)(26 151 36 141)(27 152 37 142)(28 153 38 143)(29 154 39 144)(30 155 40 145)(41 96 51 86)(42 97 52 87)(43 98 53 88)(44 99 54 89)(45 100 55 90)(46 91 56 81)(47 92 57 82)(48 93 58 83)(49 94 59 84)(50 95 60 85)(61 116 71 106)(62 117 72 107)(63 118 73 108)(64 119 74 109)(65 120 75 110)(66 111 76 101)(67 112 77 102)(68 113 78 103)(69 114 79 104)(70 115 80 105)
(1 83)(2 84)(3 85)(4 86)(5 87)(6 88)(7 89)(8 90)(9 81)(10 82)(11 91)(12 92)(13 93)(14 94)(15 95)(16 96)(17 97)(18 98)(19 99)(20 100)(21 101)(22 102)(23 103)(24 104)(25 105)(26 106)(27 107)(28 108)(29 109)(30 110)(31 111)(32 112)(33 113)(34 114)(35 115)(36 116)(37 117)(38 118)(39 119)(40 120)(41 121)(42 122)(43 123)(44 124)(45 125)(46 126)(47 127)(48 128)(49 129)(50 130)(51 131)(52 132)(53 133)(54 134)(55 135)(56 136)(57 137)(58 138)(59 139)(60 140)(61 141)(62 142)(63 143)(64 144)(65 145)(66 146)(67 147)(68 148)(69 149)(70 150)(71 151)(72 152)(73 153)(74 154)(75 155)(76 156)(77 157)(78 158)(79 159)(80 160)
(1 38 18 23)(2 37 19 22)(3 36 20 21)(4 35 11 30)(5 34 12 29)(6 33 13 28)(7 32 14 27)(8 31 15 26)(9 40 16 25)(10 39 17 24)(41 80 56 65)(42 79 57 64)(43 78 58 63)(44 77 59 62)(45 76 60 61)(46 75 51 70)(47 74 52 69)(48 73 53 68)(49 72 54 67)(50 71 55 66)(81 120 96 105)(82 119 97 104)(83 118 98 103)(84 117 99 102)(85 116 100 101)(86 115 91 110)(87 114 92 109)(88 113 93 108)(89 112 94 107)(90 111 95 106)(121 160 136 145)(122 159 137 144)(123 158 138 143)(124 157 139 142)(125 156 140 141)(126 155 131 150)(127 154 132 149)(128 153 133 148)(129 152 134 147)(130 151 135 146)
(1 23)(2 24)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 21)(10 22)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(41 66)(42 67)(43 68)(44 69)(45 70)(46 61)(47 62)(48 63)(49 64)(50 65)(51 76)(52 77)(53 78)(54 79)(55 80)(56 71)(57 72)(58 73)(59 74)(60 75)(81 106)(82 107)(83 108)(84 109)(85 110)(86 101)(87 102)(88 103)(89 104)(90 105)(91 116)(92 117)(93 118)(94 119)(95 120)(96 111)(97 112)(98 113)(99 114)(100 115)(121 141)(122 142)(123 143)(124 144)(125 145)(126 146)(127 147)(128 148)(129 149)(130 150)(131 151)(132 152)(133 153)(134 154)(135 155)(136 156)(137 157)(138 158)(139 159)(140 160)
G:=sub<Sym(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), (1,138,13,128)(2,139,14,129)(3,140,15,130)(4,131,16,121)(5,132,17,122)(6,133,18,123)(7,134,19,124)(8,135,20,125)(9,136,11,126)(10,137,12,127)(21,156,31,146)(22,157,32,147)(23,158,33,148)(24,159,34,149)(25,160,35,150)(26,151,36,141)(27,152,37,142)(28,153,38,143)(29,154,39,144)(30,155,40,145)(41,96,51,86)(42,97,52,87)(43,98,53,88)(44,99,54,89)(45,100,55,90)(46,91,56,81)(47,92,57,82)(48,93,58,83)(49,94,59,84)(50,95,60,85)(61,116,71,106)(62,117,72,107)(63,118,73,108)(64,119,74,109)(65,120,75,110)(66,111,76,101)(67,112,77,102)(68,113,78,103)(69,114,79,104)(70,115,80,105), (1,83)(2,84)(3,85)(4,86)(5,87)(6,88)(7,89)(8,90)(9,81)(10,82)(11,91)(12,92)(13,93)(14,94)(15,95)(16,96)(17,97)(18,98)(19,99)(20,100)(21,101)(22,102)(23,103)(24,104)(25,105)(26,106)(27,107)(28,108)(29,109)(30,110)(31,111)(32,112)(33,113)(34,114)(35,115)(36,116)(37,117)(38,118)(39,119)(40,120)(41,121)(42,122)(43,123)(44,124)(45,125)(46,126)(47,127)(48,128)(49,129)(50,130)(51,131)(52,132)(53,133)(54,134)(55,135)(56,136)(57,137)(58,138)(59,139)(60,140)(61,141)(62,142)(63,143)(64,144)(65,145)(66,146)(67,147)(68,148)(69,149)(70,150)(71,151)(72,152)(73,153)(74,154)(75,155)(76,156)(77,157)(78,158)(79,159)(80,160), (1,38,18,23)(2,37,19,22)(3,36,20,21)(4,35,11,30)(5,34,12,29)(6,33,13,28)(7,32,14,27)(8,31,15,26)(9,40,16,25)(10,39,17,24)(41,80,56,65)(42,79,57,64)(43,78,58,63)(44,77,59,62)(45,76,60,61)(46,75,51,70)(47,74,52,69)(48,73,53,68)(49,72,54,67)(50,71,55,66)(81,120,96,105)(82,119,97,104)(83,118,98,103)(84,117,99,102)(85,116,100,101)(86,115,91,110)(87,114,92,109)(88,113,93,108)(89,112,94,107)(90,111,95,106)(121,160,136,145)(122,159,137,144)(123,158,138,143)(124,157,139,142)(125,156,140,141)(126,155,131,150)(127,154,132,149)(128,153,133,148)(129,152,134,147)(130,151,135,146), (1,23)(2,24)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,21)(10,22)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(41,66)(42,67)(43,68)(44,69)(45,70)(46,61)(47,62)(48,63)(49,64)(50,65)(51,76)(52,77)(53,78)(54,79)(55,80)(56,71)(57,72)(58,73)(59,74)(60,75)(81,106)(82,107)(83,108)(84,109)(85,110)(86,101)(87,102)(88,103)(89,104)(90,105)(91,116)(92,117)(93,118)(94,119)(95,120)(96,111)(97,112)(98,113)(99,114)(100,115)(121,141)(122,142)(123,143)(124,144)(125,145)(126,146)(127,147)(128,148)(129,149)(130,150)(131,151)(132,152)(133,153)(134,154)(135,155)(136,156)(137,157)(138,158)(139,159)(140,160)>;
G:=Group( (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), (1,138,13,128)(2,139,14,129)(3,140,15,130)(4,131,16,121)(5,132,17,122)(6,133,18,123)(7,134,19,124)(8,135,20,125)(9,136,11,126)(10,137,12,127)(21,156,31,146)(22,157,32,147)(23,158,33,148)(24,159,34,149)(25,160,35,150)(26,151,36,141)(27,152,37,142)(28,153,38,143)(29,154,39,144)(30,155,40,145)(41,96,51,86)(42,97,52,87)(43,98,53,88)(44,99,54,89)(45,100,55,90)(46,91,56,81)(47,92,57,82)(48,93,58,83)(49,94,59,84)(50,95,60,85)(61,116,71,106)(62,117,72,107)(63,118,73,108)(64,119,74,109)(65,120,75,110)(66,111,76,101)(67,112,77,102)(68,113,78,103)(69,114,79,104)(70,115,80,105), (1,83)(2,84)(3,85)(4,86)(5,87)(6,88)(7,89)(8,90)(9,81)(10,82)(11,91)(12,92)(13,93)(14,94)(15,95)(16,96)(17,97)(18,98)(19,99)(20,100)(21,101)(22,102)(23,103)(24,104)(25,105)(26,106)(27,107)(28,108)(29,109)(30,110)(31,111)(32,112)(33,113)(34,114)(35,115)(36,116)(37,117)(38,118)(39,119)(40,120)(41,121)(42,122)(43,123)(44,124)(45,125)(46,126)(47,127)(48,128)(49,129)(50,130)(51,131)(52,132)(53,133)(54,134)(55,135)(56,136)(57,137)(58,138)(59,139)(60,140)(61,141)(62,142)(63,143)(64,144)(65,145)(66,146)(67,147)(68,148)(69,149)(70,150)(71,151)(72,152)(73,153)(74,154)(75,155)(76,156)(77,157)(78,158)(79,159)(80,160), (1,38,18,23)(2,37,19,22)(3,36,20,21)(4,35,11,30)(5,34,12,29)(6,33,13,28)(7,32,14,27)(8,31,15,26)(9,40,16,25)(10,39,17,24)(41,80,56,65)(42,79,57,64)(43,78,58,63)(44,77,59,62)(45,76,60,61)(46,75,51,70)(47,74,52,69)(48,73,53,68)(49,72,54,67)(50,71,55,66)(81,120,96,105)(82,119,97,104)(83,118,98,103)(84,117,99,102)(85,116,100,101)(86,115,91,110)(87,114,92,109)(88,113,93,108)(89,112,94,107)(90,111,95,106)(121,160,136,145)(122,159,137,144)(123,158,138,143)(124,157,139,142)(125,156,140,141)(126,155,131,150)(127,154,132,149)(128,153,133,148)(129,152,134,147)(130,151,135,146), (1,23)(2,24)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,21)(10,22)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(41,66)(42,67)(43,68)(44,69)(45,70)(46,61)(47,62)(48,63)(49,64)(50,65)(51,76)(52,77)(53,78)(54,79)(55,80)(56,71)(57,72)(58,73)(59,74)(60,75)(81,106)(82,107)(83,108)(84,109)(85,110)(86,101)(87,102)(88,103)(89,104)(90,105)(91,116)(92,117)(93,118)(94,119)(95,120)(96,111)(97,112)(98,113)(99,114)(100,115)(121,141)(122,142)(123,143)(124,144)(125,145)(126,146)(127,147)(128,148)(129,149)(130,150)(131,151)(132,152)(133,153)(134,154)(135,155)(136,156)(137,157)(138,158)(139,159)(140,160) );
G=PermutationGroup([[(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)], [(1,138,13,128),(2,139,14,129),(3,140,15,130),(4,131,16,121),(5,132,17,122),(6,133,18,123),(7,134,19,124),(8,135,20,125),(9,136,11,126),(10,137,12,127),(21,156,31,146),(22,157,32,147),(23,158,33,148),(24,159,34,149),(25,160,35,150),(26,151,36,141),(27,152,37,142),(28,153,38,143),(29,154,39,144),(30,155,40,145),(41,96,51,86),(42,97,52,87),(43,98,53,88),(44,99,54,89),(45,100,55,90),(46,91,56,81),(47,92,57,82),(48,93,58,83),(49,94,59,84),(50,95,60,85),(61,116,71,106),(62,117,72,107),(63,118,73,108),(64,119,74,109),(65,120,75,110),(66,111,76,101),(67,112,77,102),(68,113,78,103),(69,114,79,104),(70,115,80,105)], [(1,83),(2,84),(3,85),(4,86),(5,87),(6,88),(7,89),(8,90),(9,81),(10,82),(11,91),(12,92),(13,93),(14,94),(15,95),(16,96),(17,97),(18,98),(19,99),(20,100),(21,101),(22,102),(23,103),(24,104),(25,105),(26,106),(27,107),(28,108),(29,109),(30,110),(31,111),(32,112),(33,113),(34,114),(35,115),(36,116),(37,117),(38,118),(39,119),(40,120),(41,121),(42,122),(43,123),(44,124),(45,125),(46,126),(47,127),(48,128),(49,129),(50,130),(51,131),(52,132),(53,133),(54,134),(55,135),(56,136),(57,137),(58,138),(59,139),(60,140),(61,141),(62,142),(63,143),(64,144),(65,145),(66,146),(67,147),(68,148),(69,149),(70,150),(71,151),(72,152),(73,153),(74,154),(75,155),(76,156),(77,157),(78,158),(79,159),(80,160)], [(1,38,18,23),(2,37,19,22),(3,36,20,21),(4,35,11,30),(5,34,12,29),(6,33,13,28),(7,32,14,27),(8,31,15,26),(9,40,16,25),(10,39,17,24),(41,80,56,65),(42,79,57,64),(43,78,58,63),(44,77,59,62),(45,76,60,61),(46,75,51,70),(47,74,52,69),(48,73,53,68),(49,72,54,67),(50,71,55,66),(81,120,96,105),(82,119,97,104),(83,118,98,103),(84,117,99,102),(85,116,100,101),(86,115,91,110),(87,114,92,109),(88,113,93,108),(89,112,94,107),(90,111,95,106),(121,160,136,145),(122,159,137,144),(123,158,138,143),(124,157,139,142),(125,156,140,141),(126,155,131,150),(127,154,132,149),(128,153,133,148),(129,152,134,147),(130,151,135,146)], [(1,23),(2,24),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30),(9,21),(10,22),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40),(41,66),(42,67),(43,68),(44,69),(45,70),(46,61),(47,62),(48,63),(49,64),(50,65),(51,76),(52,77),(53,78),(54,79),(55,80),(56,71),(57,72),(58,73),(59,74),(60,75),(81,106),(82,107),(83,108),(84,109),(85,110),(86,101),(87,102),(88,103),(89,104),(90,105),(91,116),(92,117),(93,118),(94,119),(95,120),(96,111),(97,112),(98,113),(99,114),(100,115),(121,141),(122,142),(123,143),(124,144),(125,145),(126,146),(127,147),(128,148),(129,149),(130,150),(131,151),(132,152),(133,153),(134,154),(135,155),(136,156),(137,157),(138,158),(139,159),(140,160)]])
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 4O | 4P | 4Q | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 10M | 10N | 20A | ··· | 20H | 20I | 20J | 20K | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | C4○D4 | D10 | D10 | D10 | D10 | 2+ 1+4 | D4⋊2D5 | D4⋊6D10 | D5×C4○D4 |
| kernel | C10.452+ 1+4 | C23.11D10 | Dic5.5D4 | C20⋊Q8 | C4⋊C4⋊7D5 | C23.21D10 | C4×C5⋊D4 | D4×Dic5 | C20.17D4 | C20⋊2D4 | Dic5⋊D4 | C5×C4⋊D4 | C4⋊D4 | Dic5 | C20 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C10 | C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 4 | 4 | 4 | 2 | 2 | 6 | 1 | 4 | 4 | 4 |
Matrix representation of C10.452+ 1+4 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 35 | 0 | 0 |
| 0 | 0 | 7 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 9 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 39 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 36 |
| 0 | 0 | 0 | 0 | 12 | 26 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 35 | 0 | 0 |
| 0 | 0 | 8 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 37 |
| 0 | 0 | 0 | 0 | 26 | 29 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 37 |
| 0 | 0 | 0 | 0 | 5 | 29 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,34,7,0,0,0,0,35,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[9,9,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,39,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,15,12,0,0,0,0,36,26],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,34,8,0,0,0,0,35,7,0,0,0,0,0,0,12,26,0,0,0,0,37,29],[1,1,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,5,0,0,0,0,37,29] >;
C10.452+ 1+4 in GAP, Magma, Sage, TeX
C_{10}._{45}2_+^{1+4} % in TeX
G:=Group("C10.45ES+(2,2)"); // GroupNames label
G:=SmallGroup(320,1288);
// by ID
G=gap.SmallGroup(320,1288);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,758,219,1571,570,297,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^4=c^2=e^2=1,d^2=a^5*b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=a^5*c,e*d*e=a^5*b^2*d>;
// generators/relations