G = C10.682- 1+4 order 320 = 26·5
23rd non-split extension by C10 of 2- 1+4 acting via 2- 1+4/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.682- 1+4, C10.322+ 1+4, C4⋊D4⋊5D5, C4⋊C4.89D10, (C2×Dic5)⋊9D4, C20⋊7D4⋊43C2, C22.2(D4×D5), (C2×D4).89D10, C22⋊C4.4D10, D10⋊D4⋊16C2, D10⋊Q8⋊13C2, Dic5⋊D4⋊8C2, Dic5.85(C2×D4), C10.60(C22×D4), C23.8(C22×D5), (C2×C20).172C23, (C2×C10).141C24, (C2×D20).32C22, (C22×C4).217D10, C4⋊Dic5.44C22, C2.34(D4⋊6D10), (D4×C10).115C22, (C22×C10).12C23, (C22×D5).60C23, C22.162(C23×D5), Dic5.14D4⋊15C2, C23.D5.19C22, D10⋊C4.69C22, (C22×C20).310C22, C5⋊2(C22.31C24), (C2×Dic5).234C23, C2.26(D4.10D10), (C2×Dic10).157C22, C10.D4.158C22, (C22×Dic5).102C22, C2.33(C2×D4×D5), (C5×C4⋊D4)⋊6C2, (C2×C10).4(C2×D4), (C2×D4⋊2D5)⋊9C2, (C2×C4×D5).89C22, (C2×C4).35(C22×D5), (C2×C10.D4)⋊28C2, (C5×C4⋊C4).137C22, (C2×C5⋊D4).24C22, (C5×C22⋊C4).6C22, SmallGroup(320,1269)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10.682- 1+4
G = < a,b,c,d,e | a10=b4=e2=1, c2=a5, d2=b2, ab=ba, cac-1=eae=a-1, ad=da, cbc-1=a5b-1, dbd-1=a5b, be=eb, dcd-1=ece=a5c, ede=a5b2d >
Subgroups: 1102 in 294 conjugacy classes, 103 normal (31 characteristic)
C1, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, D5, C10, C10, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic5, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, C2×C4⋊C4, C4⋊D4, C4⋊D4, C22⋊Q8, C2×C4○D4, Dic10, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C5×D4, C22×D5, C22×C10, C22×C10, C22.31C24, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×D20, D4⋊2D5, C22×Dic5, C22×Dic5, C2×C5⋊D4, C22×C20, D4×C10, D4×C10, Dic5.14D4, D10⋊D4, D10⋊Q8, C2×C10.D4, C20⋊7D4, Dic5⋊D4, C5×C4⋊D4, C2×D4⋊2D5, C10.682- 1+4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C22×D4, 2+ 1+4, 2- 1+4, C22×D5, C22.31C24, D4×D5, C23×D5, C2×D4×D5, D4⋊6D10, D4.10D10, C10.682- 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)
(1 136 30 150)(2 137 21 141)(3 138 22 142)(4 139 23 143)(5 140 24 144)(6 131 25 145)(7 132 26 146)(8 133 27 147)(9 134 28 148)(10 135 29 149)(11 31 151 45)(12 32 152 46)(13 33 153 47)(14 34 154 48)(15 35 155 49)(16 36 156 50)(17 37 157 41)(18 38 158 42)(19 39 159 43)(20 40 160 44)(51 91 65 105)(52 92 66 106)(53 93 67 107)(54 94 68 108)(55 95 69 109)(56 96 70 110)(57 97 61 101)(58 98 62 102)(59 99 63 103)(60 100 64 104)(71 111 85 125)(72 112 86 126)(73 113 87 127)(74 114 88 128)(75 115 89 129)(76 116 90 130)(77 117 81 121)(78 118 82 122)(79 119 83 123)(80 120 84 124)
(1 45 6 50)(2 44 7 49)(3 43 8 48)(4 42 9 47)(5 41 10 46)(11 150 16 145)(12 149 17 144)(13 148 18 143)(14 147 19 142)(15 146 20 141)(21 40 26 35)(22 39 27 34)(23 38 28 33)(24 37 29 32)(25 36 30 31)(51 71 56 76)(52 80 57 75)(53 79 58 74)(54 78 59 73)(55 77 60 72)(61 89 66 84)(62 88 67 83)(63 87 68 82)(64 86 69 81)(65 85 70 90)(91 130 96 125)(92 129 97 124)(93 128 98 123)(94 127 99 122)(95 126 100 121)(101 120 106 115)(102 119 107 114)(103 118 108 113)(104 117 109 112)(105 116 110 111)(131 151 136 156)(132 160 137 155)(133 159 138 154)(134 158 139 153)(135 157 140 152)
(1 125 30 111)(2 126 21 112)(3 127 22 113)(4 128 23 114)(5 129 24 115)(6 130 25 116)(7 121 26 117)(8 122 27 118)(9 123 28 119)(10 124 29 120)(11 65 151 51)(12 66 152 52)(13 67 153 53)(14 68 154 54)(15 69 155 55)(16 70 156 56)(17 61 157 57)(18 62 158 58)(19 63 159 59)(20 64 160 60)(31 110 45 96)(32 101 46 97)(33 102 47 98)(34 103 48 99)(35 104 49 100)(36 105 50 91)(37 106 41 92)(38 107 42 93)(39 108 43 94)(40 109 44 95)(71 145 85 131)(72 146 86 132)(73 147 87 133)(74 148 88 134)(75 149 89 135)(76 150 90 136)(77 141 81 137)(78 142 82 138)(79 143 83 139)(80 144 84 140)
(1 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 21)(11 156)(12 155)(13 154)(14 153)(15 152)(16 151)(17 160)(18 159)(19 158)(20 157)(31 50)(32 49)(33 48)(34 47)(35 46)(36 45)(37 44)(38 43)(39 42)(40 41)(52 60)(53 59)(54 58)(55 57)(61 69)(62 68)(63 67)(64 66)(71 76)(72 75)(73 74)(77 80)(78 79)(81 84)(82 83)(85 90)(86 89)(87 88)(92 100)(93 99)(94 98)(95 97)(101 109)(102 108)(103 107)(104 106)(111 116)(112 115)(113 114)(117 120)(118 119)(121 124)(122 123)(125 130)(126 129)(127 128)(131 145)(132 144)(133 143)(134 142)(135 141)(136 150)(137 149)(138 148)(139 147)(140 146)
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), (1,136,30,150)(2,137,21,141)(3,138,22,142)(4,139,23,143)(5,140,24,144)(6,131,25,145)(7,132,26,146)(8,133,27,147)(9,134,28,148)(10,135,29,149)(11,31,151,45)(12,32,152,46)(13,33,153,47)(14,34,154,48)(15,35,155,49)(16,36,156,50)(17,37,157,41)(18,38,158,42)(19,39,159,43)(20,40,160,44)(51,91,65,105)(52,92,66,106)(53,93,67,107)(54,94,68,108)(55,95,69,109)(56,96,70,110)(57,97,61,101)(58,98,62,102)(59,99,63,103)(60,100,64,104)(71,111,85,125)(72,112,86,126)(73,113,87,127)(74,114,88,128)(75,115,89,129)(76,116,90,130)(77,117,81,121)(78,118,82,122)(79,119,83,123)(80,120,84,124), (1,45,6,50)(2,44,7,49)(3,43,8,48)(4,42,9,47)(5,41,10,46)(11,150,16,145)(12,149,17,144)(13,148,18,143)(14,147,19,142)(15,146,20,141)(21,40,26,35)(22,39,27,34)(23,38,28,33)(24,37,29,32)(25,36,30,31)(51,71,56,76)(52,80,57,75)(53,79,58,74)(54,78,59,73)(55,77,60,72)(61,89,66,84)(62,88,67,83)(63,87,68,82)(64,86,69,81)(65,85,70,90)(91,130,96,125)(92,129,97,124)(93,128,98,123)(94,127,99,122)(95,126,100,121)(101,120,106,115)(102,119,107,114)(103,118,108,113)(104,117,109,112)(105,116,110,111)(131,151,136,156)(132,160,137,155)(133,159,138,154)(134,158,139,153)(135,157,140,152), (1,125,30,111)(2,126,21,112)(3,127,22,113)(4,128,23,114)(5,129,24,115)(6,130,25,116)(7,121,26,117)(8,122,27,118)(9,123,28,119)(10,124,29,120)(11,65,151,51)(12,66,152,52)(13,67,153,53)(14,68,154,54)(15,69,155,55)(16,70,156,56)(17,61,157,57)(18,62,158,58)(19,63,159,59)(20,64,160,60)(31,110,45,96)(32,101,46,97)(33,102,47,98)(34,103,48,99)(35,104,49,100)(36,105,50,91)(37,106,41,92)(38,107,42,93)(39,108,43,94)(40,109,44,95)(71,145,85,131)(72,146,86,132)(73,147,87,133)(74,148,88,134)(75,149,89,135)(76,150,90,136)(77,141,81,137)(78,142,82,138)(79,143,83,139)(80,144,84,140), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,156)(12,155)(13,154)(14,153)(15,152)(16,151)(17,160)(18,159)(19,158)(20,157)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)(52,60)(53,59)(54,58)(55,57)(61,69)(62,68)(63,67)(64,66)(71,76)(72,75)(73,74)(77,80)(78,79)(81,84)(82,83)(85,90)(86,89)(87,88)(92,100)(93,99)(94,98)(95,97)(101,109)(102,108)(103,107)(104,106)(111,116)(112,115)(113,114)(117,120)(118,119)(121,124)(122,123)(125,130)(126,129)(127,128)(131,145)(132,144)(133,143)(134,142)(135,141)(136,150)(137,149)(138,148)(139,147)(140,146)>;
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), (1,136,30,150)(2,137,21,141)(3,138,22,142)(4,139,23,143)(5,140,24,144)(6,131,25,145)(7,132,26,146)(8,133,27,147)(9,134,28,148)(10,135,29,149)(11,31,151,45)(12,32,152,46)(13,33,153,47)(14,34,154,48)(15,35,155,49)(16,36,156,50)(17,37,157,41)(18,38,158,42)(19,39,159,43)(20,40,160,44)(51,91,65,105)(52,92,66,106)(53,93,67,107)(54,94,68,108)(55,95,69,109)(56,96,70,110)(57,97,61,101)(58,98,62,102)(59,99,63,103)(60,100,64,104)(71,111,85,125)(72,112,86,126)(73,113,87,127)(74,114,88,128)(75,115,89,129)(76,116,90,130)(77,117,81,121)(78,118,82,122)(79,119,83,123)(80,120,84,124), (1,45,6,50)(2,44,7,49)(3,43,8,48)(4,42,9,47)(5,41,10,46)(11,150,16,145)(12,149,17,144)(13,148,18,143)(14,147,19,142)(15,146,20,141)(21,40,26,35)(22,39,27,34)(23,38,28,33)(24,37,29,32)(25,36,30,31)(51,71,56,76)(52,80,57,75)(53,79,58,74)(54,78,59,73)(55,77,60,72)(61,89,66,84)(62,88,67,83)(63,87,68,82)(64,86,69,81)(65,85,70,90)(91,130,96,125)(92,129,97,124)(93,128,98,123)(94,127,99,122)(95,126,100,121)(101,120,106,115)(102,119,107,114)(103,118,108,113)(104,117,109,112)(105,116,110,111)(131,151,136,156)(132,160,137,155)(133,159,138,154)(134,158,139,153)(135,157,140,152), (1,125,30,111)(2,126,21,112)(3,127,22,113)(4,128,23,114)(5,129,24,115)(6,130,25,116)(7,121,26,117)(8,122,27,118)(9,123,28,119)(10,124,29,120)(11,65,151,51)(12,66,152,52)(13,67,153,53)(14,68,154,54)(15,69,155,55)(16,70,156,56)(17,61,157,57)(18,62,158,58)(19,63,159,59)(20,64,160,60)(31,110,45,96)(32,101,46,97)(33,102,47,98)(34,103,48,99)(35,104,49,100)(36,105,50,91)(37,106,41,92)(38,107,42,93)(39,108,43,94)(40,109,44,95)(71,145,85,131)(72,146,86,132)(73,147,87,133)(74,148,88,134)(75,149,89,135)(76,150,90,136)(77,141,81,137)(78,142,82,138)(79,143,83,139)(80,144,84,140), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,156)(12,155)(13,154)(14,153)(15,152)(16,151)(17,160)(18,159)(19,158)(20,157)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)(52,60)(53,59)(54,58)(55,57)(61,69)(62,68)(63,67)(64,66)(71,76)(72,75)(73,74)(77,80)(78,79)(81,84)(82,83)(85,90)(86,89)(87,88)(92,100)(93,99)(94,98)(95,97)(101,109)(102,108)(103,107)(104,106)(111,116)(112,115)(113,114)(117,120)(118,119)(121,124)(122,123)(125,130)(126,129)(127,128)(131,145)(132,144)(133,143)(134,142)(135,141)(136,150)(137,149)(138,148)(139,147)(140,146) );
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)], [(1,136,30,150),(2,137,21,141),(3,138,22,142),(4,139,23,143),(5,140,24,144),(6,131,25,145),(7,132,26,146),(8,133,27,147),(9,134,28,148),(10,135,29,149),(11,31,151,45),(12,32,152,46),(13,33,153,47),(14,34,154,48),(15,35,155,49),(16,36,156,50),(17,37,157,41),(18,38,158,42),(19,39,159,43),(20,40,160,44),(51,91,65,105),(52,92,66,106),(53,93,67,107),(54,94,68,108),(55,95,69,109),(56,96,70,110),(57,97,61,101),(58,98,62,102),(59,99,63,103),(60,100,64,104),(71,111,85,125),(72,112,86,126),(73,113,87,127),(74,114,88,128),(75,115,89,129),(76,116,90,130),(77,117,81,121),(78,118,82,122),(79,119,83,123),(80,120,84,124)], [(1,45,6,50),(2,44,7,49),(3,43,8,48),(4,42,9,47),(5,41,10,46),(11,150,16,145),(12,149,17,144),(13,148,18,143),(14,147,19,142),(15,146,20,141),(21,40,26,35),(22,39,27,34),(23,38,28,33),(24,37,29,32),(25,36,30,31),(51,71,56,76),(52,80,57,75),(53,79,58,74),(54,78,59,73),(55,77,60,72),(61,89,66,84),(62,88,67,83),(63,87,68,82),(64,86,69,81),(65,85,70,90),(91,130,96,125),(92,129,97,124),(93,128,98,123),(94,127,99,122),(95,126,100,121),(101,120,106,115),(102,119,107,114),(103,118,108,113),(104,117,109,112),(105,116,110,111),(131,151,136,156),(132,160,137,155),(133,159,138,154),(134,158,139,153),(135,157,140,152)], [(1,125,30,111),(2,126,21,112),(3,127,22,113),(4,128,23,114),(5,129,24,115),(6,130,25,116),(7,121,26,117),(8,122,27,118),(9,123,28,119),(10,124,29,120),(11,65,151,51),(12,66,152,52),(13,67,153,53),(14,68,154,54),(15,69,155,55),(16,70,156,56),(17,61,157,57),(18,62,158,58),(19,63,159,59),(20,64,160,60),(31,110,45,96),(32,101,46,97),(33,102,47,98),(34,103,48,99),(35,104,49,100),(36,105,50,91),(37,106,41,92),(38,107,42,93),(39,108,43,94),(40,109,44,95),(71,145,85,131),(72,146,86,132),(73,147,87,133),(74,148,88,134),(75,149,89,135),(76,150,90,136),(77,141,81,137),(78,142,82,138),(79,143,83,139),(80,144,84,140)], [(1,30),(2,29),(3,28),(4,27),(5,26),(6,25),(7,24),(8,23),(9,22),(10,21),(11,156),(12,155),(13,154),(14,153),(15,152),(16,151),(17,160),(18,159),(19,158),(20,157),(31,50),(32,49),(33,48),(34,47),(35,46),(36,45),(37,44),(38,43),(39,42),(40,41),(52,60),(53,59),(54,58),(55,57),(61,69),(62,68),(63,67),(64,66),(71,76),(72,75),(73,74),(77,80),(78,79),(81,84),(82,83),(85,90),(86,89),(87,88),(92,100),(93,99),(94,98),(95,97),(101,109),(102,108),(103,107),(104,106),(111,116),(112,115),(113,114),(117,120),(118,119),(121,124),(122,123),(125,130),(126,129),(127,128),(131,145),(132,144),(133,143),(134,142),(135,141),(136,150),(137,149),(138,148),(139,147),(140,146)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 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 | 2 | 2 | 4 | 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 | 2 | 2 | 4 | 4 | 20 | 20 | 4 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | D10 | D10 | D10 | D10 | 2+ 1+4 | 2- 1+4 | D4×D5 | D4⋊6D10 | D4.10D10 |
| kernel | C10.682- 1+4 | Dic5.14D4 | D10⋊D4 | D10⋊Q8 | C2×C10.D4 | C20⋊7D4 | Dic5⋊D4 | C5×C4⋊D4 | C2×D4⋊2D5 | C2×Dic5 | C4⋊D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C10 | C10 | C22 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 1 | 4 | 1 | 2 | 4 | 2 | 4 | 2 | 2 | 6 | 1 | 1 | 4 | 4 | 4 |
Matrix representation of C10.682- 1+4 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 6 | 0 | 0 |
| 0 | 0 | 35 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 6 |
| 0 | 0 | 0 | 0 | 35 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 39 | 28 |
| 0 | 0 | 0 | 0 | 13 | 2 |
| 0 | 0 | 39 | 28 | 0 | 0 |
| 0 | 0 | 13 | 2 | 0 | 0 |
| 40 | 2 | 0 | 0 | 0 | 0 |
| 40 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 35 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 6 | 40 | 0 | 0 |
| 40 | 2 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 0 | 28 | 18 |
| 0 | 0 | 0 | 34 | 23 | 13 |
| 0 | 0 | 13 | 23 | 7 | 0 |
| 0 | 0 | 18 | 28 | 0 | 7 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 35 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 6 | 40 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,6,35,0,0,0,0,6,1,0,0,0,0,0,0,6,35,0,0,0,0,6,1],[1,1,0,0,0,0,0,40,0,0,0,0,0,0,0,0,39,13,0,0,0,0,28,2,0,0,39,13,0,0,0,0,28,2,0,0],[40,40,0,0,0,0,2,1,0,0,0,0,0,0,0,0,1,6,0,0,0,0,0,40,0,0,40,35,0,0,0,0,0,1,0,0],[40,0,0,0,0,0,2,1,0,0,0,0,0,0,34,0,13,18,0,0,0,34,23,28,0,0,28,23,7,0,0,0,18,13,0,7],[1,1,0,0,0,0,0,40,0,0,0,0,0,0,40,35,0,0,0,0,0,1,0,0,0,0,0,0,1,6,0,0,0,0,0,40] >;
C10.682- 1+4 in GAP, Magma, Sage, TeX
C_{10}._{68}2_-^{1+4} % in TeX
G:=Group("C10.68ES-(2,2)"); // GroupNames label
G:=SmallGroup(320,1269);
// by ID
G=gap.SmallGroup(320,1269);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,387,1123,570,185,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^4=e^2=1,c^2=a^5,d^2=b^2,a*b=b*a,c*a*c^-1=e*a*e=a^-1,a*d=d*a,c*b*c^-1=a^5*b^-1,d*b*d^-1=a^5*b,b*e=e*b,d*c*d^-1=e*c*e=a^5*c,e*d*e=a^5*b^2*d>;
// generators/relations