?
G = C10.1442+ (1+4) order 320 = 26·5
53rd non-split extension by C10 of 2+ (1+4) acting via 2+ (1+4)/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.1442+ (1+4), C10.1062- (1+4), C4○D4⋊5Dic5, D4⋊8(C2×Dic5), Q8⋊7(C2×Dic5), (D4×Dic5)⋊41C2, (Q8×Dic5)⋊28C2, (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×C10).239C23, (C22×C20).295C22, C5⋊6(C23.33C23), (C2×Dic5).302C23, (C4×Dic5).183C22, 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
Subgroups: 734 in 294 conjugacy classes, 191 normal (18 characteristic)
C1, C2 [×3], C2 [×6], C4 [×8], C4 [×8], C22, C22 [×6], C22 [×6], C5, C2×C4, C2×C4 [×15], C2×C4 [×14], D4 [×12], Q8 [×4], C23 [×3], C10 [×3], C10 [×6], C42 [×6], C22⋊C4 [×6], C4⋊C4 [×10], C22×C4 [×3], C22×C4 [×6], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5 [×8], C20 [×8], C2×C10, C2×C10 [×6], C2×C10 [×6], C2×C4⋊C4 [×3], C42⋊C2 [×3], C4×D4 [×6], C4×Q8 [×2], C2×C4○D4, C2×Dic5 [×8], C2×Dic5 [×6], C2×C20, C2×C20 [×15], C5×D4 [×12], C5×Q8 [×4], C22×C10 [×3], C23.33C23, C4×Dic5 [×6], C4⋊Dic5, C4⋊Dic5 [×9], C23.D5 [×6], C22×Dic5 [×6], C22×C20 [×3], D4×C10 [×3], Q8×C10, C5×C4○D4 [×8], C2×C4⋊Dic5 [×3], C23.21D10 [×3], D4×Dic5 [×6], Q8×Dic5 [×2], C10×C4○D4, C10.1442+ (1+4)
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D5, C22×C4 [×14], C24, Dic5 [×8], D10 [×7], C23×C4, 2+ (1+4), 2- (1+4), C2×Dic5 [×28], C22×D5 [×7], C23.33C23, C22×Dic5 [×14], C23×D5, D4⋊8D10, D4.10D10, C23×Dic5, C10.1442+ (1+4)
Generators and relations
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 >
►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 80 30 81)(2 71 21 82)(3 72 22 83)(4 73 23 84)(5 74 24 85)(6 75 25 86)(7 76 26 87)(8 77 27 88)(9 78 28 89)(10 79 29 90)(11 106 153 93)(12 107 154 94)(13 108 155 95)(14 109 156 96)(15 110 157 97)(16 101 158 98)(17 102 159 99)(18 103 160 100)(19 104 151 91)(20 105 152 92)(31 61 41 51)(32 62 42 52)(33 63 43 53)(34 64 44 54)(35 65 45 55)(36 66 46 56)(37 67 47 57)(38 68 48 58)(39 69 49 59)(40 70 50 60)(111 131 124 144)(112 132 125 145)(113 133 126 146)(114 134 127 147)(115 135 128 148)(116 136 129 149)(117 137 130 150)(118 138 121 141)(119 139 122 142)(120 140 123 143)
(1 75)(2 76)(3 77)(4 78)(5 79)(6 80)(7 71)(8 72)(9 73)(10 74)(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 66)(32 67)(33 68)(34 69)(35 70)(36 61)(37 62)(38 63)(39 64)(40 65)(41 56)(42 57)(43 58)(44 59)(45 60)(46 51)(47 52)(48 53)(49 54)(50 55)(91 156)(92 157)(93 158)(94 159)(95 160)(96 151)(97 152)(98 153)(99 154)(100 155)(111 149)(112 150)(113 141)(114 142)(115 143)(116 144)(117 145)(118 146)(119 147)(120 148)(121 133)(122 134)(123 135)(124 136)(125 137)(126 138)(127 139)(128 140)(129 131)(130 132)
(1 50 30 40)(2 41 21 31)(3 42 22 32)(4 43 23 33)(5 44 24 34)(6 45 25 35)(7 46 26 36)(8 47 27 37)(9 48 28 38)(10 49 29 39)(11 133 153 146)(12 134 154 147)(13 135 155 148)(14 136 156 149)(15 137 157 150)(16 138 158 141)(17 139 159 142)(18 140 160 143)(19 131 151 144)(20 132 152 145)(51 82 61 71)(52 83 62 72)(53 84 63 73)(54 85 64 74)(55 86 65 75)(56 87 66 76)(57 88 67 77)(58 89 68 78)(59 90 69 79)(60 81 70 80)(91 111 104 124)(92 112 105 125)(93 113 106 126)(94 114 107 127)(95 115 108 128)(96 116 109 129)(97 117 110 130)(98 118 101 121)(99 119 102 122)(100 120 103 123)
(1 132 6 137)(2 131 7 136)(3 140 8 135)(4 139 9 134)(5 138 10 133)(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)(31 151 36 156)(32 160 37 155)(33 159 38 154)(34 158 39 153)(35 157 40 152)(51 104 56 109)(52 103 57 108)(53 102 58 107)(54 101 59 106)(55 110 60 105)(61 91 66 96)(62 100 67 95)(63 99 68 94)(64 98 69 93)(65 97 70 92)(71 124 76 129)(72 123 77 128)(73 122 78 127)(74 121 79 126)(75 130 80 125)(81 112 86 117)(82 111 87 116)(83 120 88 115)(84 119 89 114)(85 118 90 113)
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), (1,80,30,81)(2,71,21,82)(3,72,22,83)(4,73,23,84)(5,74,24,85)(6,75,25,86)(7,76,26,87)(8,77,27,88)(9,78,28,89)(10,79,29,90)(11,106,153,93)(12,107,154,94)(13,108,155,95)(14,109,156,96)(15,110,157,97)(16,101,158,98)(17,102,159,99)(18,103,160,100)(19,104,151,91)(20,105,152,92)(31,61,41,51)(32,62,42,52)(33,63,43,53)(34,64,44,54)(35,65,45,55)(36,66,46,56)(37,67,47,57)(38,68,48,58)(39,69,49,59)(40,70,50,60)(111,131,124,144)(112,132,125,145)(113,133,126,146)(114,134,127,147)(115,135,128,148)(116,136,129,149)(117,137,130,150)(118,138,121,141)(119,139,122,142)(120,140,123,143), (1,75)(2,76)(3,77)(4,78)(5,79)(6,80)(7,71)(8,72)(9,73)(10,74)(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)(31,66)(32,67)(33,68)(34,69)(35,70)(36,61)(37,62)(38,63)(39,64)(40,65)(41,56)(42,57)(43,58)(44,59)(45,60)(46,51)(47,52)(48,53)(49,54)(50,55)(91,156)(92,157)(93,158)(94,159)(95,160)(96,151)(97,152)(98,153)(99,154)(100,155)(111,149)(112,150)(113,141)(114,142)(115,143)(116,144)(117,145)(118,146)(119,147)(120,148)(121,133)(122,134)(123,135)(124,136)(125,137)(126,138)(127,139)(128,140)(129,131)(130,132), (1,50,30,40)(2,41,21,31)(3,42,22,32)(4,43,23,33)(5,44,24,34)(6,45,25,35)(7,46,26,36)(8,47,27,37)(9,48,28,38)(10,49,29,39)(11,133,153,146)(12,134,154,147)(13,135,155,148)(14,136,156,149)(15,137,157,150)(16,138,158,141)(17,139,159,142)(18,140,160,143)(19,131,151,144)(20,132,152,145)(51,82,61,71)(52,83,62,72)(53,84,63,73)(54,85,64,74)(55,86,65,75)(56,87,66,76)(57,88,67,77)(58,89,68,78)(59,90,69,79)(60,81,70,80)(91,111,104,124)(92,112,105,125)(93,113,106,126)(94,114,107,127)(95,115,108,128)(96,116,109,129)(97,117,110,130)(98,118,101,121)(99,119,102,122)(100,120,103,123), (1,132,6,137)(2,131,7,136)(3,140,8,135)(4,139,9,134)(5,138,10,133)(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)(31,151,36,156)(32,160,37,155)(33,159,38,154)(34,158,39,153)(35,157,40,152)(51,104,56,109)(52,103,57,108)(53,102,58,107)(54,101,59,106)(55,110,60,105)(61,91,66,96)(62,100,67,95)(63,99,68,94)(64,98,69,93)(65,97,70,92)(71,124,76,129)(72,123,77,128)(73,122,78,127)(74,121,79,126)(75,130,80,125)(81,112,86,117)(82,111,87,116)(83,120,88,115)(84,119,89,114)(85,118,90,113)>;
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), (1,80,30,81)(2,71,21,82)(3,72,22,83)(4,73,23,84)(5,74,24,85)(6,75,25,86)(7,76,26,87)(8,77,27,88)(9,78,28,89)(10,79,29,90)(11,106,153,93)(12,107,154,94)(13,108,155,95)(14,109,156,96)(15,110,157,97)(16,101,158,98)(17,102,159,99)(18,103,160,100)(19,104,151,91)(20,105,152,92)(31,61,41,51)(32,62,42,52)(33,63,43,53)(34,64,44,54)(35,65,45,55)(36,66,46,56)(37,67,47,57)(38,68,48,58)(39,69,49,59)(40,70,50,60)(111,131,124,144)(112,132,125,145)(113,133,126,146)(114,134,127,147)(115,135,128,148)(116,136,129,149)(117,137,130,150)(118,138,121,141)(119,139,122,142)(120,140,123,143), (1,75)(2,76)(3,77)(4,78)(5,79)(6,80)(7,71)(8,72)(9,73)(10,74)(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)(31,66)(32,67)(33,68)(34,69)(35,70)(36,61)(37,62)(38,63)(39,64)(40,65)(41,56)(42,57)(43,58)(44,59)(45,60)(46,51)(47,52)(48,53)(49,54)(50,55)(91,156)(92,157)(93,158)(94,159)(95,160)(96,151)(97,152)(98,153)(99,154)(100,155)(111,149)(112,150)(113,141)(114,142)(115,143)(116,144)(117,145)(118,146)(119,147)(120,148)(121,133)(122,134)(123,135)(124,136)(125,137)(126,138)(127,139)(128,140)(129,131)(130,132), (1,50,30,40)(2,41,21,31)(3,42,22,32)(4,43,23,33)(5,44,24,34)(6,45,25,35)(7,46,26,36)(8,47,27,37)(9,48,28,38)(10,49,29,39)(11,133,153,146)(12,134,154,147)(13,135,155,148)(14,136,156,149)(15,137,157,150)(16,138,158,141)(17,139,159,142)(18,140,160,143)(19,131,151,144)(20,132,152,145)(51,82,61,71)(52,83,62,72)(53,84,63,73)(54,85,64,74)(55,86,65,75)(56,87,66,76)(57,88,67,77)(58,89,68,78)(59,90,69,79)(60,81,70,80)(91,111,104,124)(92,112,105,125)(93,113,106,126)(94,114,107,127)(95,115,108,128)(96,116,109,129)(97,117,110,130)(98,118,101,121)(99,119,102,122)(100,120,103,123), (1,132,6,137)(2,131,7,136)(3,140,8,135)(4,139,9,134)(5,138,10,133)(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)(31,151,36,156)(32,160,37,155)(33,159,38,154)(34,158,39,153)(35,157,40,152)(51,104,56,109)(52,103,57,108)(53,102,58,107)(54,101,59,106)(55,110,60,105)(61,91,66,96)(62,100,67,95)(63,99,68,94)(64,98,69,93)(65,97,70,92)(71,124,76,129)(72,123,77,128)(73,122,78,127)(74,121,79,126)(75,130,80,125)(81,112,86,117)(82,111,87,116)(83,120,88,115)(84,119,89,114)(85,118,90,113) );
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)], [(1,80,30,81),(2,71,21,82),(3,72,22,83),(4,73,23,84),(5,74,24,85),(6,75,25,86),(7,76,26,87),(8,77,27,88),(9,78,28,89),(10,79,29,90),(11,106,153,93),(12,107,154,94),(13,108,155,95),(14,109,156,96),(15,110,157,97),(16,101,158,98),(17,102,159,99),(18,103,160,100),(19,104,151,91),(20,105,152,92),(31,61,41,51),(32,62,42,52),(33,63,43,53),(34,64,44,54),(35,65,45,55),(36,66,46,56),(37,67,47,57),(38,68,48,58),(39,69,49,59),(40,70,50,60),(111,131,124,144),(112,132,125,145),(113,133,126,146),(114,134,127,147),(115,135,128,148),(116,136,129,149),(117,137,130,150),(118,138,121,141),(119,139,122,142),(120,140,123,143)], [(1,75),(2,76),(3,77),(4,78),(5,79),(6,80),(7,71),(8,72),(9,73),(10,74),(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),(31,66),(32,67),(33,68),(34,69),(35,70),(36,61),(37,62),(38,63),(39,64),(40,65),(41,56),(42,57),(43,58),(44,59),(45,60),(46,51),(47,52),(48,53),(49,54),(50,55),(91,156),(92,157),(93,158),(94,159),(95,160),(96,151),(97,152),(98,153),(99,154),(100,155),(111,149),(112,150),(113,141),(114,142),(115,143),(116,144),(117,145),(118,146),(119,147),(120,148),(121,133),(122,134),(123,135),(124,136),(125,137),(126,138),(127,139),(128,140),(129,131),(130,132)], [(1,50,30,40),(2,41,21,31),(3,42,22,32),(4,43,23,33),(5,44,24,34),(6,45,25,35),(7,46,26,36),(8,47,27,37),(9,48,28,38),(10,49,29,39),(11,133,153,146),(12,134,154,147),(13,135,155,148),(14,136,156,149),(15,137,157,150),(16,138,158,141),(17,139,159,142),(18,140,160,143),(19,131,151,144),(20,132,152,145),(51,82,61,71),(52,83,62,72),(53,84,63,73),(54,85,64,74),(55,86,65,75),(56,87,66,76),(57,88,67,77),(58,89,68,78),(59,90,69,79),(60,81,70,80),(91,111,104,124),(92,112,105,125),(93,113,106,126),(94,114,107,127),(95,115,108,128),(96,116,109,129),(97,117,110,130),(98,118,101,121),(99,119,102,122),(100,120,103,123)], [(1,132,6,137),(2,131,7,136),(3,140,8,135),(4,139,9,134),(5,138,10,133),(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),(31,151,36,156),(32,160,37,155),(33,159,38,154),(34,158,39,153),(35,157,40,152),(51,104,56,109),(52,103,57,108),(53,102,58,107),(54,101,59,106),(55,110,60,105),(61,91,66,96),(62,100,67,95),(63,99,68,94),(64,98,69,93),(65,97,70,92),(71,124,76,129),(72,123,77,128),(73,122,78,127),(74,121,79,126),(75,130,80,125),(81,112,86,117),(82,111,87,116),(83,120,88,115),(84,119,89,114),(85,118,90,113)])
Matrix representation ►G ⊆ 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] >;
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 | C10.1442+ (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 |
In GAP, Magma, Sage, TeX
C_{10}._{144}2_+^{(1+4)} % in TeX
G:=Group("C10.144ES+(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
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