G = C10.1062- 1+4 order 320 = 26·5
61st non-split extension by C10 of 2- 1+4 acting via 2- 1+4/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.1062- 1+4, C10.1442+ 1+4, C4○D4⋊5Dic5, D4⋊8(C2×Dic5), Q8⋊7(C2×Dic5), (Q8×Dic5)⋊28C2, (D4×Dic5)⋊41C2, (C2×D4).253D10, C10.72(C23×C4), (C2×Q8).209D10, C2.5(D4⋊8D10), (C2×C10).313C24, (C2×C20).560C23, C20.159(C22×C4), (C22×C4).286D10, C4.22(C22×Dic5), C2.13(C23×Dic5), C22.49(C23×D5), (D4×C10).275C22, C4⋊Dic5.392C22, (Q8×C10).242C22, C23.210(C22×D5), C2.5(D4.10D10), C23.21D10⋊37C2, C22.4(C22×Dic5), (C22×C20).295C22, (C22×C10).239C23, C5⋊6(C23.33C23), (C4×Dic5).183C22, (C2×Dic5).302C23, C23.D5.135C22, (C22×Dic5).167C22, (C2×C20)⋊30(C2×C4), (C5×C4○D4)⋊12C4, (C5×D4)⋊32(C2×C4), (C5×Q8)⋊29(C2×C4), (C2×C4)⋊5(C2×Dic5), (C2×C4⋊Dic5)⋊47C2, (C2×C4○D4).12D5, (C10×C4○D4).14C2, (C2×C4).638(C22×D5), (C2×C10).132(C22×C4), SmallGroup(320,1499)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10.1062- 1+4
G = < a,b,c,d,e | a10=b4=c2=1, d2=b2, e2=a5, ab=ba, ac=ca, ad=da, eae-1=a-1, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=b2d >
Subgroups: 734 in 294 conjugacy classes, 191 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, C20, C2×C10, C2×C10, C2×C10, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C2×C4○D4, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C23.33C23, C4×Dic5, C4⋊Dic5, C4⋊Dic5, C23.D5, C22×Dic5, C22×C20, D4×C10, Q8×C10, C5×C4○D4, C2×C4⋊Dic5, C23.21D10, D4×Dic5, Q8×Dic5, C10×C4○D4, C10.1062- 1+4
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, C24, Dic5, D10, C23×C4, 2+ 1+4, 2- 1+4, C2×Dic5, C22×D5, C23.33C23, C22×Dic5, C23×D5, D4⋊8D10, D4.10D10, C23×Dic5, C10.1062- 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 76 30 81)(2 77 21 82)(3 78 22 83)(4 79 23 84)(5 80 24 85)(6 71 25 86)(7 72 26 87)(8 73 27 88)(9 74 28 89)(10 75 29 90)(11 106 155 95)(12 107 156 96)(13 108 157 97)(14 109 158 98)(15 110 159 99)(16 101 160 100)(17 102 151 91)(18 103 152 92)(19 104 153 93)(20 105 154 94)(31 65 45 51)(32 66 46 52)(33 67 47 53)(34 68 48 54)(35 69 49 55)(36 70 50 56)(37 61 41 57)(38 62 42 58)(39 63 43 59)(40 64 44 60)(111 131 122 142)(112 132 123 143)(113 133 124 144)(114 134 125 145)(115 135 126 146)(116 136 127 147)(117 137 128 148)(118 138 129 149)(119 139 130 150)(120 140 121 141)
(1 71)(2 72)(3 73)(4 74)(5 75)(6 76)(7 77)(8 78)(9 79)(10 80)(11 101)(12 102)(13 103)(14 104)(15 105)(16 106)(17 107)(18 108)(19 109)(20 110)(21 87)(22 88)(23 89)(24 90)(25 81)(26 82)(27 83)(28 84)(29 85)(30 86)(31 70)(32 61)(33 62)(34 63)(35 64)(36 65)(37 66)(38 67)(39 68)(40 69)(41 52)(42 53)(43 54)(44 55)(45 56)(46 57)(47 58)(48 59)(49 60)(50 51)(91 156)(92 157)(93 158)(94 159)(95 160)(96 151)(97 152)(98 153)(99 154)(100 155)(111 147)(112 148)(113 149)(114 150)(115 141)(116 142)(117 143)(118 144)(119 145)(120 146)(121 135)(122 136)(123 137)(124 138)(125 139)(126 140)(127 131)(128 132)(129 133)(130 134)
(1 50 30 36)(2 41 21 37)(3 42 22 38)(4 43 23 39)(5 44 24 40)(6 45 25 31)(7 46 26 32)(8 47 27 33)(9 48 28 34)(10 49 29 35)(11 135 155 146)(12 136 156 147)(13 137 157 148)(14 138 158 149)(15 139 159 150)(16 140 160 141)(17 131 151 142)(18 132 152 143)(19 133 153 144)(20 134 154 145)(51 86 65 71)(52 87 66 72)(53 88 67 73)(54 89 68 74)(55 90 69 75)(56 81 70 76)(57 82 61 77)(58 83 62 78)(59 84 63 79)(60 85 64 80)(91 111 102 122)(92 112 103 123)(93 113 104 124)(94 114 105 125)(95 115 106 126)(96 116 107 127)(97 117 108 128)(98 118 109 129)(99 119 110 130)(100 120 101 121)
(1 134 6 139)(2 133 7 138)(3 132 8 137)(4 131 9 136)(5 140 10 135)(11 44 16 49)(12 43 17 48)(13 42 18 47)(14 41 19 46)(15 50 20 45)(21 144 26 149)(22 143 27 148)(23 142 28 147)(24 141 29 146)(25 150 30 145)(31 159 36 154)(32 158 37 153)(33 157 38 152)(34 156 39 151)(35 155 40 160)(51 110 56 105)(52 109 57 104)(53 108 58 103)(54 107 59 102)(55 106 60 101)(61 93 66 98)(62 92 67 97)(63 91 68 96)(64 100 69 95)(65 99 70 94)(71 130 76 125)(72 129 77 124)(73 128 78 123)(74 127 79 122)(75 126 80 121)(81 114 86 119)(82 113 87 118)(83 112 88 117)(84 111 89 116)(85 120 90 115)
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), (1,76,30,81)(2,77,21,82)(3,78,22,83)(4,79,23,84)(5,80,24,85)(6,71,25,86)(7,72,26,87)(8,73,27,88)(9,74,28,89)(10,75,29,90)(11,106,155,95)(12,107,156,96)(13,108,157,97)(14,109,158,98)(15,110,159,99)(16,101,160,100)(17,102,151,91)(18,103,152,92)(19,104,153,93)(20,105,154,94)(31,65,45,51)(32,66,46,52)(33,67,47,53)(34,68,48,54)(35,69,49,55)(36,70,50,56)(37,61,41,57)(38,62,42,58)(39,63,43,59)(40,64,44,60)(111,131,122,142)(112,132,123,143)(113,133,124,144)(114,134,125,145)(115,135,126,146)(116,136,127,147)(117,137,128,148)(118,138,129,149)(119,139,130,150)(120,140,121,141), (1,71)(2,72)(3,73)(4,74)(5,75)(6,76)(7,77)(8,78)(9,79)(10,80)(11,101)(12,102)(13,103)(14,104)(15,105)(16,106)(17,107)(18,108)(19,109)(20,110)(21,87)(22,88)(23,89)(24,90)(25,81)(26,82)(27,83)(28,84)(29,85)(30,86)(31,70)(32,61)(33,62)(34,63)(35,64)(36,65)(37,66)(38,67)(39,68)(40,69)(41,52)(42,53)(43,54)(44,55)(45,56)(46,57)(47,58)(48,59)(49,60)(50,51)(91,156)(92,157)(93,158)(94,159)(95,160)(96,151)(97,152)(98,153)(99,154)(100,155)(111,147)(112,148)(113,149)(114,150)(115,141)(116,142)(117,143)(118,144)(119,145)(120,146)(121,135)(122,136)(123,137)(124,138)(125,139)(126,140)(127,131)(128,132)(129,133)(130,134), (1,50,30,36)(2,41,21,37)(3,42,22,38)(4,43,23,39)(5,44,24,40)(6,45,25,31)(7,46,26,32)(8,47,27,33)(9,48,28,34)(10,49,29,35)(11,135,155,146)(12,136,156,147)(13,137,157,148)(14,138,158,149)(15,139,159,150)(16,140,160,141)(17,131,151,142)(18,132,152,143)(19,133,153,144)(20,134,154,145)(51,86,65,71)(52,87,66,72)(53,88,67,73)(54,89,68,74)(55,90,69,75)(56,81,70,76)(57,82,61,77)(58,83,62,78)(59,84,63,79)(60,85,64,80)(91,111,102,122)(92,112,103,123)(93,113,104,124)(94,114,105,125)(95,115,106,126)(96,116,107,127)(97,117,108,128)(98,118,109,129)(99,119,110,130)(100,120,101,121), (1,134,6,139)(2,133,7,138)(3,132,8,137)(4,131,9,136)(5,140,10,135)(11,44,16,49)(12,43,17,48)(13,42,18,47)(14,41,19,46)(15,50,20,45)(21,144,26,149)(22,143,27,148)(23,142,28,147)(24,141,29,146)(25,150,30,145)(31,159,36,154)(32,158,37,153)(33,157,38,152)(34,156,39,151)(35,155,40,160)(51,110,56,105)(52,109,57,104)(53,108,58,103)(54,107,59,102)(55,106,60,101)(61,93,66,98)(62,92,67,97)(63,91,68,96)(64,100,69,95)(65,99,70,94)(71,130,76,125)(72,129,77,124)(73,128,78,123)(74,127,79,122)(75,126,80,121)(81,114,86,119)(82,113,87,118)(83,112,88,117)(84,111,89,116)(85,120,90,115)>;
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), (1,76,30,81)(2,77,21,82)(3,78,22,83)(4,79,23,84)(5,80,24,85)(6,71,25,86)(7,72,26,87)(8,73,27,88)(9,74,28,89)(10,75,29,90)(11,106,155,95)(12,107,156,96)(13,108,157,97)(14,109,158,98)(15,110,159,99)(16,101,160,100)(17,102,151,91)(18,103,152,92)(19,104,153,93)(20,105,154,94)(31,65,45,51)(32,66,46,52)(33,67,47,53)(34,68,48,54)(35,69,49,55)(36,70,50,56)(37,61,41,57)(38,62,42,58)(39,63,43,59)(40,64,44,60)(111,131,122,142)(112,132,123,143)(113,133,124,144)(114,134,125,145)(115,135,126,146)(116,136,127,147)(117,137,128,148)(118,138,129,149)(119,139,130,150)(120,140,121,141), (1,71)(2,72)(3,73)(4,74)(5,75)(6,76)(7,77)(8,78)(9,79)(10,80)(11,101)(12,102)(13,103)(14,104)(15,105)(16,106)(17,107)(18,108)(19,109)(20,110)(21,87)(22,88)(23,89)(24,90)(25,81)(26,82)(27,83)(28,84)(29,85)(30,86)(31,70)(32,61)(33,62)(34,63)(35,64)(36,65)(37,66)(38,67)(39,68)(40,69)(41,52)(42,53)(43,54)(44,55)(45,56)(46,57)(47,58)(48,59)(49,60)(50,51)(91,156)(92,157)(93,158)(94,159)(95,160)(96,151)(97,152)(98,153)(99,154)(100,155)(111,147)(112,148)(113,149)(114,150)(115,141)(116,142)(117,143)(118,144)(119,145)(120,146)(121,135)(122,136)(123,137)(124,138)(125,139)(126,140)(127,131)(128,132)(129,133)(130,134), (1,50,30,36)(2,41,21,37)(3,42,22,38)(4,43,23,39)(5,44,24,40)(6,45,25,31)(7,46,26,32)(8,47,27,33)(9,48,28,34)(10,49,29,35)(11,135,155,146)(12,136,156,147)(13,137,157,148)(14,138,158,149)(15,139,159,150)(16,140,160,141)(17,131,151,142)(18,132,152,143)(19,133,153,144)(20,134,154,145)(51,86,65,71)(52,87,66,72)(53,88,67,73)(54,89,68,74)(55,90,69,75)(56,81,70,76)(57,82,61,77)(58,83,62,78)(59,84,63,79)(60,85,64,80)(91,111,102,122)(92,112,103,123)(93,113,104,124)(94,114,105,125)(95,115,106,126)(96,116,107,127)(97,117,108,128)(98,118,109,129)(99,119,110,130)(100,120,101,121), (1,134,6,139)(2,133,7,138)(3,132,8,137)(4,131,9,136)(5,140,10,135)(11,44,16,49)(12,43,17,48)(13,42,18,47)(14,41,19,46)(15,50,20,45)(21,144,26,149)(22,143,27,148)(23,142,28,147)(24,141,29,146)(25,150,30,145)(31,159,36,154)(32,158,37,153)(33,157,38,152)(34,156,39,151)(35,155,40,160)(51,110,56,105)(52,109,57,104)(53,108,58,103)(54,107,59,102)(55,106,60,101)(61,93,66,98)(62,92,67,97)(63,91,68,96)(64,100,69,95)(65,99,70,94)(71,130,76,125)(72,129,77,124)(73,128,78,123)(74,127,79,122)(75,126,80,121)(81,114,86,119)(82,113,87,118)(83,112,88,117)(84,111,89,116)(85,120,90,115) );
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)], [(1,76,30,81),(2,77,21,82),(3,78,22,83),(4,79,23,84),(5,80,24,85),(6,71,25,86),(7,72,26,87),(8,73,27,88),(9,74,28,89),(10,75,29,90),(11,106,155,95),(12,107,156,96),(13,108,157,97),(14,109,158,98),(15,110,159,99),(16,101,160,100),(17,102,151,91),(18,103,152,92),(19,104,153,93),(20,105,154,94),(31,65,45,51),(32,66,46,52),(33,67,47,53),(34,68,48,54),(35,69,49,55),(36,70,50,56),(37,61,41,57),(38,62,42,58),(39,63,43,59),(40,64,44,60),(111,131,122,142),(112,132,123,143),(113,133,124,144),(114,134,125,145),(115,135,126,146),(116,136,127,147),(117,137,128,148),(118,138,129,149),(119,139,130,150),(120,140,121,141)], [(1,71),(2,72),(3,73),(4,74),(5,75),(6,76),(7,77),(8,78),(9,79),(10,80),(11,101),(12,102),(13,103),(14,104),(15,105),(16,106),(17,107),(18,108),(19,109),(20,110),(21,87),(22,88),(23,89),(24,90),(25,81),(26,82),(27,83),(28,84),(29,85),(30,86),(31,70),(32,61),(33,62),(34,63),(35,64),(36,65),(37,66),(38,67),(39,68),(40,69),(41,52),(42,53),(43,54),(44,55),(45,56),(46,57),(47,58),(48,59),(49,60),(50,51),(91,156),(92,157),(93,158),(94,159),(95,160),(96,151),(97,152),(98,153),(99,154),(100,155),(111,147),(112,148),(113,149),(114,150),(115,141),(116,142),(117,143),(118,144),(119,145),(120,146),(121,135),(122,136),(123,137),(124,138),(125,139),(126,140),(127,131),(128,132),(129,133),(130,134)], [(1,50,30,36),(2,41,21,37),(3,42,22,38),(4,43,23,39),(5,44,24,40),(6,45,25,31),(7,46,26,32),(8,47,27,33),(9,48,28,34),(10,49,29,35),(11,135,155,146),(12,136,156,147),(13,137,157,148),(14,138,158,149),(15,139,159,150),(16,140,160,141),(17,131,151,142),(18,132,152,143),(19,133,153,144),(20,134,154,145),(51,86,65,71),(52,87,66,72),(53,88,67,73),(54,89,68,74),(55,90,69,75),(56,81,70,76),(57,82,61,77),(58,83,62,78),(59,84,63,79),(60,85,64,80),(91,111,102,122),(92,112,103,123),(93,113,104,124),(94,114,105,125),(95,115,106,126),(96,116,107,127),(97,117,108,128),(98,118,109,129),(99,119,110,130),(100,120,101,121)], [(1,134,6,139),(2,133,7,138),(3,132,8,137),(4,131,9,136),(5,140,10,135),(11,44,16,49),(12,43,17,48),(13,42,18,47),(14,41,19,46),(15,50,20,45),(21,144,26,149),(22,143,27,148),(23,142,28,147),(24,141,29,146),(25,150,30,145),(31,159,36,154),(32,158,37,153),(33,157,38,152),(34,156,39,151),(35,155,40,160),(51,110,56,105),(52,109,57,104),(53,108,58,103),(54,107,59,102),(55,106,60,101),(61,93,66,98),(62,92,67,97),(63,91,68,96),(64,100,69,95),(65,99,70,94),(71,130,76,125),(72,129,77,124),(73,128,78,123),(74,127,79,122),(75,126,80,121),(81,114,86,119),(82,113,87,118),(83,112,88,117),(84,111,89,116),(85,120,90,115)]])
74 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 4A | ··· | 4H | 4I | ··· | 4X | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10R | 20A | ··· | 20H | 20I | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
74 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D5 | D10 | D10 | D10 | Dic5 | 2+ 1+4 | 2- 1+4 | D4⋊8D10 | D4.10D10 |
| kernel | C10.1062- 1+4 | C2×C4⋊Dic5 | C23.21D10 | D4×Dic5 | Q8×Dic5 | C10×C4○D4 | C5×C4○D4 | C2×C4○D4 | C22×C4 | C2×D4 | C2×Q8 | C4○D4 | C10 | C10 | C2 | C2 |
| # reps | 1 | 3 | 3 | 6 | 2 | 1 | 16 | 2 | 6 | 6 | 2 | 16 | 1 | 1 | 4 | 4 |
Matrix representation of C10.1062- 1+4 ►in GL6(𝔽41)
| 0 | 34 | 0 | 0 | 0 | 0 |
| 6 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 7 | 0 | 0 |
| 0 | 0 | 34 | 7 | 0 | 0 |
| 0 | 0 | 6 | 13 | 6 | 7 |
| 0 | 0 | 8 | 22 | 35 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 5 | 18 | 21 |
| 0 | 0 | 11 | 38 | 0 | 21 |
| 0 | 0 | 12 | 27 | 2 | 34 |
| 0 | 0 | 32 | 34 | 14 | 33 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 36 | 23 | 20 |
| 0 | 0 | 30 | 3 | 0 | 20 |
| 0 | 0 | 38 | 5 | 39 | 7 |
| 0 | 0 | 9 | 3 | 27 | 8 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 32 | 0 | 0 |
| 0 | 0 | 9 | 30 | 0 | 0 |
| 0 | 0 | 4 | 36 | 2 | 32 |
| 0 | 0 | 19 | 1 | 37 | 39 |
| 23 | 1 | 0 | 0 | 0 | 0 |
| 3 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 14 | 0 | 0 |
| 0 | 0 | 30 | 27 | 0 | 0 |
| 0 | 0 | 11 | 26 | 25 | 14 |
| 0 | 0 | 22 | 33 | 14 | 16 |
G:=sub<GL(6,GF(41))| [0,6,0,0,0,0,34,35,0,0,0,0,0,0,40,34,6,8,0,0,7,7,13,22,0,0,0,0,6,35,0,0,0,0,7,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,11,12,32,0,0,5,38,27,34,0,0,18,0,2,14,0,0,21,21,34,33],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,30,38,9,0,0,36,3,5,3,0,0,23,0,39,27,0,0,20,20,7,8],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,11,9,4,19,0,0,32,30,36,1,0,0,0,0,2,37,0,0,0,0,32,39],[23,3,0,0,0,0,1,18,0,0,0,0,0,0,14,30,11,22,0,0,14,27,26,33,0,0,0,0,25,14,0,0,0,0,14,16] >;
C10.1062- 1+4 in GAP, Magma, Sage, TeX
C_{10}._{106}2_-^{1+4} % in TeX
G:=Group("C10.106ES-(2,2)"); // GroupNames label
G:=SmallGroup(320,1499);
// by ID
G=gap.SmallGroup(320,1499);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,387,1123,136,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^4=c^2=1,d^2=b^2,e^2=a^5,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a^-1,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b^2*d>;
// generators/relations