?
G = C10.1082- (1+4) order 320 = 26·5
63rd non-split extension by C10 of 2- (1+4) acting via 2- (1+4)/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.1082- (1+4), C10.1472+ (1+4), (C2×C20)⋊18D4, C20⋊2D4⋊43C2, C20⋊7D4⋊49C2, C20.431(C2×D4), D10⋊3Q8⋊45C2, (C2×D4).238D10, (C2×Q8).195D10, Dic5⋊D4⋊45C2, C20.48D4⋊49C2, (C2×C20).654C23, (C2×C10).319C24, (C22×C4).289D10, C10.169(C22×D4), C2.71(D4⋊8D10), (C2×D20).239C22, (D4×C10).315C22, C4⋊Dic5.393C22, (Q8×C10).245C22, C22.328(C23×D5), C23.140(C22×D5), C23.D5.77C22, D10⋊C4.79C22, (C22×C20).321C22, (C22×C10).245C23, C5⋊5(C22.31C24), (C2×Dic5).165C23, C10.D4.93C22, (C22×D5).140C23, C2.71(D4.10D10), (C2×Dic10).267C22, (C22×Dic5).168C22, (C2×C4○D4)⋊11D5, (C2×C4)⋊8(C5⋊D4), (C10×C4○D4)⋊11C2, (C2×C4○D20)⋊33C2, (C2×C4⋊Dic5)⋊48C2, (C2×C10).84(C2×D4), C4.101(C2×C5⋊D4), C22.2(C2×C5⋊D4), (C2×C4×D5).179C22, C2.42(C22×C5⋊D4), (C2×C4).641(C22×D5), (C2×C5⋊D4).82C22, SmallGroup(320,1505)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1022 in 294 conjugacy classes, 111 normal (29 characteristic)
C1, C2 [×3], C2 [×6], C4 [×4], C4 [×8], C22, C22 [×2], C22 [×14], C5, C2×C4 [×2], C2×C4 [×6], C2×C4 [×16], D4 [×16], Q8 [×4], C23, C23 [×2], C23 [×2], D5 [×2], C10 [×3], C10 [×4], C22⋊C4 [×8], C4⋊C4 [×8], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×D4, C2×D4 [×2], C2×D4 [×7], C2×Q8, C2×Q8, C4○D4 [×8], Dic5 [×6], C20 [×4], C20 [×2], D10 [×6], C2×C10, C2×C10 [×2], C2×C10 [×8], C2×C4⋊C4, C4⋊D4 [×8], C22⋊Q8 [×4], C2×C4○D4, C2×C4○D4, Dic10 [×2], C4×D5 [×4], D20 [×2], C2×Dic5 [×6], C2×Dic5 [×2], C5⋊D4 [×8], C2×C20 [×2], C2×C20 [×6], C2×C20 [×4], C5×D4 [×6], C5×Q8 [×2], C22×D5 [×2], C22×C10, C22×C10 [×2], C22.31C24, C10.D4 [×4], C4⋊Dic5 [×4], D10⋊C4 [×4], C23.D5 [×4], C2×Dic10, C2×C4×D5 [×2], C2×D20, C4○D20 [×4], C22×Dic5 [×2], C2×C5⋊D4 [×6], C22×C20, C22×C20 [×2], D4×C10, D4×C10 [×2], Q8×C10, C5×C4○D4 [×4], C20.48D4 [×2], C2×C4⋊Dic5, C20⋊7D4 [×2], C20⋊2D4 [×2], Dic5⋊D4 [×4], D10⋊3Q8 [×2], C2×C4○D20, C10×C4○D4, C10.1082- (1+4)
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, 2+ (1+4), 2- (1+4), C5⋊D4 [×4], C22×D5 [×7], C22.31C24, C2×C5⋊D4 [×6], C23×D5, D4⋊8D10, D4.10D10, C22×C5⋊D4, C10.1082- (1+4)
Generators and relations
G = < a,b,c,d,e | a10=b4=c2=1, d2=b2, e2=a5b2, bab-1=dad-1=a-1, ac=ca, ae=ea, cbc=b-1, bd=db, ebe-1=a5b, dcd-1=a5c, 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 38 24 50)(2 37 25 49)(3 36 26 48)(4 35 27 47)(5 34 28 46)(6 33 29 45)(7 32 30 44)(8 31 21 43)(9 40 22 42)(10 39 23 41)(11 133 153 145)(12 132 154 144)(13 131 155 143)(14 140 156 142)(15 139 157 141)(16 138 158 150)(17 137 159 149)(18 136 160 148)(19 135 151 147)(20 134 152 146)(51 82 63 80)(52 81 64 79)(53 90 65 78)(54 89 66 77)(55 88 67 76)(56 87 68 75)(57 86 69 74)(58 85 70 73)(59 84 61 72)(60 83 62 71)(91 115 103 127)(92 114 104 126)(93 113 105 125)(94 112 106 124)(95 111 107 123)(96 120 108 122)(97 119 109 121)(98 118 110 130)(99 117 101 129)(100 116 102 128)
(1 45)(2 46)(3 47)(4 48)(5 49)(6 50)(7 41)(8 42)(9 43)(10 44)(11 145)(12 146)(13 147)(14 148)(15 149)(16 150)(17 141)(18 142)(19 143)(20 144)(21 40)(22 31)(23 32)(24 33)(25 34)(26 35)(27 36)(28 37)(29 38)(30 39)(51 88)(52 89)(53 90)(54 81)(55 82)(56 83)(57 84)(58 85)(59 86)(60 87)(61 74)(62 75)(63 76)(64 77)(65 78)(66 79)(67 80)(68 71)(69 72)(70 73)(91 128)(92 129)(93 130)(94 121)(95 122)(96 123)(97 124)(98 125)(99 126)(100 127)(101 114)(102 115)(103 116)(104 117)(105 118)(106 119)(107 120)(108 111)(109 112)(110 113)(131 151)(132 152)(133 153)(134 154)(135 155)(136 156)(137 157)(138 158)(139 159)(140 160)
(1 78 24 90)(2 77 25 89)(3 76 26 88)(4 75 27 87)(5 74 28 86)(6 73 29 85)(7 72 30 84)(8 71 21 83)(9 80 22 82)(10 79 23 81)(11 93 153 105)(12 92 154 104)(13 91 155 103)(14 100 156 102)(15 99 157 101)(16 98 158 110)(17 97 159 109)(18 96 160 108)(19 95 151 107)(20 94 152 106)(31 60 43 62)(32 59 44 61)(33 58 45 70)(34 57 46 69)(35 56 47 68)(36 55 48 67)(37 54 49 66)(38 53 50 65)(39 52 41 64)(40 51 42 63)(111 147 123 135)(112 146 124 134)(113 145 125 133)(114 144 126 132)(115 143 127 131)(116 142 128 140)(117 141 129 139)(118 150 130 138)(119 149 121 137)(120 148 122 136)
(1 158 29 11)(2 159 30 12)(3 160 21 13)(4 151 22 14)(5 152 23 15)(6 153 24 16)(7 154 25 17)(8 155 26 18)(9 156 27 19)(10 157 28 20)(31 148 48 131)(32 149 49 132)(33 150 50 133)(34 141 41 134)(35 142 42 135)(36 143 43 136)(37 144 44 137)(38 145 45 138)(39 146 46 139)(40 147 47 140)(51 111 68 128)(52 112 69 129)(53 113 70 130)(54 114 61 121)(55 115 62 122)(56 116 63 123)(57 117 64 124)(58 118 65 125)(59 119 66 126)(60 120 67 127)(71 91 88 108)(72 92 89 109)(73 93 90 110)(74 94 81 101)(75 95 82 102)(76 96 83 103)(77 97 84 104)(78 98 85 105)(79 99 86 106)(80 100 87 107)
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), (1,38,24,50)(2,37,25,49)(3,36,26,48)(4,35,27,47)(5,34,28,46)(6,33,29,45)(7,32,30,44)(8,31,21,43)(9,40,22,42)(10,39,23,41)(11,133,153,145)(12,132,154,144)(13,131,155,143)(14,140,156,142)(15,139,157,141)(16,138,158,150)(17,137,159,149)(18,136,160,148)(19,135,151,147)(20,134,152,146)(51,82,63,80)(52,81,64,79)(53,90,65,78)(54,89,66,77)(55,88,67,76)(56,87,68,75)(57,86,69,74)(58,85,70,73)(59,84,61,72)(60,83,62,71)(91,115,103,127)(92,114,104,126)(93,113,105,125)(94,112,106,124)(95,111,107,123)(96,120,108,122)(97,119,109,121)(98,118,110,130)(99,117,101,129)(100,116,102,128), (1,45)(2,46)(3,47)(4,48)(5,49)(6,50)(7,41)(8,42)(9,43)(10,44)(11,145)(12,146)(13,147)(14,148)(15,149)(16,150)(17,141)(18,142)(19,143)(20,144)(21,40)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36)(28,37)(29,38)(30,39)(51,88)(52,89)(53,90)(54,81)(55,82)(56,83)(57,84)(58,85)(59,86)(60,87)(61,74)(62,75)(63,76)(64,77)(65,78)(66,79)(67,80)(68,71)(69,72)(70,73)(91,128)(92,129)(93,130)(94,121)(95,122)(96,123)(97,124)(98,125)(99,126)(100,127)(101,114)(102,115)(103,116)(104,117)(105,118)(106,119)(107,120)(108,111)(109,112)(110,113)(131,151)(132,152)(133,153)(134,154)(135,155)(136,156)(137,157)(138,158)(139,159)(140,160), (1,78,24,90)(2,77,25,89)(3,76,26,88)(4,75,27,87)(5,74,28,86)(6,73,29,85)(7,72,30,84)(8,71,21,83)(9,80,22,82)(10,79,23,81)(11,93,153,105)(12,92,154,104)(13,91,155,103)(14,100,156,102)(15,99,157,101)(16,98,158,110)(17,97,159,109)(18,96,160,108)(19,95,151,107)(20,94,152,106)(31,60,43,62)(32,59,44,61)(33,58,45,70)(34,57,46,69)(35,56,47,68)(36,55,48,67)(37,54,49,66)(38,53,50,65)(39,52,41,64)(40,51,42,63)(111,147,123,135)(112,146,124,134)(113,145,125,133)(114,144,126,132)(115,143,127,131)(116,142,128,140)(117,141,129,139)(118,150,130,138)(119,149,121,137)(120,148,122,136), (1,158,29,11)(2,159,30,12)(3,160,21,13)(4,151,22,14)(5,152,23,15)(6,153,24,16)(7,154,25,17)(8,155,26,18)(9,156,27,19)(10,157,28,20)(31,148,48,131)(32,149,49,132)(33,150,50,133)(34,141,41,134)(35,142,42,135)(36,143,43,136)(37,144,44,137)(38,145,45,138)(39,146,46,139)(40,147,47,140)(51,111,68,128)(52,112,69,129)(53,113,70,130)(54,114,61,121)(55,115,62,122)(56,116,63,123)(57,117,64,124)(58,118,65,125)(59,119,66,126)(60,120,67,127)(71,91,88,108)(72,92,89,109)(73,93,90,110)(74,94,81,101)(75,95,82,102)(76,96,83,103)(77,97,84,104)(78,98,85,105)(79,99,86,106)(80,100,87,107)>;
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), (1,38,24,50)(2,37,25,49)(3,36,26,48)(4,35,27,47)(5,34,28,46)(6,33,29,45)(7,32,30,44)(8,31,21,43)(9,40,22,42)(10,39,23,41)(11,133,153,145)(12,132,154,144)(13,131,155,143)(14,140,156,142)(15,139,157,141)(16,138,158,150)(17,137,159,149)(18,136,160,148)(19,135,151,147)(20,134,152,146)(51,82,63,80)(52,81,64,79)(53,90,65,78)(54,89,66,77)(55,88,67,76)(56,87,68,75)(57,86,69,74)(58,85,70,73)(59,84,61,72)(60,83,62,71)(91,115,103,127)(92,114,104,126)(93,113,105,125)(94,112,106,124)(95,111,107,123)(96,120,108,122)(97,119,109,121)(98,118,110,130)(99,117,101,129)(100,116,102,128), (1,45)(2,46)(3,47)(4,48)(5,49)(6,50)(7,41)(8,42)(9,43)(10,44)(11,145)(12,146)(13,147)(14,148)(15,149)(16,150)(17,141)(18,142)(19,143)(20,144)(21,40)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36)(28,37)(29,38)(30,39)(51,88)(52,89)(53,90)(54,81)(55,82)(56,83)(57,84)(58,85)(59,86)(60,87)(61,74)(62,75)(63,76)(64,77)(65,78)(66,79)(67,80)(68,71)(69,72)(70,73)(91,128)(92,129)(93,130)(94,121)(95,122)(96,123)(97,124)(98,125)(99,126)(100,127)(101,114)(102,115)(103,116)(104,117)(105,118)(106,119)(107,120)(108,111)(109,112)(110,113)(131,151)(132,152)(133,153)(134,154)(135,155)(136,156)(137,157)(138,158)(139,159)(140,160), (1,78,24,90)(2,77,25,89)(3,76,26,88)(4,75,27,87)(5,74,28,86)(6,73,29,85)(7,72,30,84)(8,71,21,83)(9,80,22,82)(10,79,23,81)(11,93,153,105)(12,92,154,104)(13,91,155,103)(14,100,156,102)(15,99,157,101)(16,98,158,110)(17,97,159,109)(18,96,160,108)(19,95,151,107)(20,94,152,106)(31,60,43,62)(32,59,44,61)(33,58,45,70)(34,57,46,69)(35,56,47,68)(36,55,48,67)(37,54,49,66)(38,53,50,65)(39,52,41,64)(40,51,42,63)(111,147,123,135)(112,146,124,134)(113,145,125,133)(114,144,126,132)(115,143,127,131)(116,142,128,140)(117,141,129,139)(118,150,130,138)(119,149,121,137)(120,148,122,136), (1,158,29,11)(2,159,30,12)(3,160,21,13)(4,151,22,14)(5,152,23,15)(6,153,24,16)(7,154,25,17)(8,155,26,18)(9,156,27,19)(10,157,28,20)(31,148,48,131)(32,149,49,132)(33,150,50,133)(34,141,41,134)(35,142,42,135)(36,143,43,136)(37,144,44,137)(38,145,45,138)(39,146,46,139)(40,147,47,140)(51,111,68,128)(52,112,69,129)(53,113,70,130)(54,114,61,121)(55,115,62,122)(56,116,63,123)(57,117,64,124)(58,118,65,125)(59,119,66,126)(60,120,67,127)(71,91,88,108)(72,92,89,109)(73,93,90,110)(74,94,81,101)(75,95,82,102)(76,96,83,103)(77,97,84,104)(78,98,85,105)(79,99,86,106)(80,100,87,107) );
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)], [(1,38,24,50),(2,37,25,49),(3,36,26,48),(4,35,27,47),(5,34,28,46),(6,33,29,45),(7,32,30,44),(8,31,21,43),(9,40,22,42),(10,39,23,41),(11,133,153,145),(12,132,154,144),(13,131,155,143),(14,140,156,142),(15,139,157,141),(16,138,158,150),(17,137,159,149),(18,136,160,148),(19,135,151,147),(20,134,152,146),(51,82,63,80),(52,81,64,79),(53,90,65,78),(54,89,66,77),(55,88,67,76),(56,87,68,75),(57,86,69,74),(58,85,70,73),(59,84,61,72),(60,83,62,71),(91,115,103,127),(92,114,104,126),(93,113,105,125),(94,112,106,124),(95,111,107,123),(96,120,108,122),(97,119,109,121),(98,118,110,130),(99,117,101,129),(100,116,102,128)], [(1,45),(2,46),(3,47),(4,48),(5,49),(6,50),(7,41),(8,42),(9,43),(10,44),(11,145),(12,146),(13,147),(14,148),(15,149),(16,150),(17,141),(18,142),(19,143),(20,144),(21,40),(22,31),(23,32),(24,33),(25,34),(26,35),(27,36),(28,37),(29,38),(30,39),(51,88),(52,89),(53,90),(54,81),(55,82),(56,83),(57,84),(58,85),(59,86),(60,87),(61,74),(62,75),(63,76),(64,77),(65,78),(66,79),(67,80),(68,71),(69,72),(70,73),(91,128),(92,129),(93,130),(94,121),(95,122),(96,123),(97,124),(98,125),(99,126),(100,127),(101,114),(102,115),(103,116),(104,117),(105,118),(106,119),(107,120),(108,111),(109,112),(110,113),(131,151),(132,152),(133,153),(134,154),(135,155),(136,156),(137,157),(138,158),(139,159),(140,160)], [(1,78,24,90),(2,77,25,89),(3,76,26,88),(4,75,27,87),(5,74,28,86),(6,73,29,85),(7,72,30,84),(8,71,21,83),(9,80,22,82),(10,79,23,81),(11,93,153,105),(12,92,154,104),(13,91,155,103),(14,100,156,102),(15,99,157,101),(16,98,158,110),(17,97,159,109),(18,96,160,108),(19,95,151,107),(20,94,152,106),(31,60,43,62),(32,59,44,61),(33,58,45,70),(34,57,46,69),(35,56,47,68),(36,55,48,67),(37,54,49,66),(38,53,50,65),(39,52,41,64),(40,51,42,63),(111,147,123,135),(112,146,124,134),(113,145,125,133),(114,144,126,132),(115,143,127,131),(116,142,128,140),(117,141,129,139),(118,150,130,138),(119,149,121,137),(120,148,122,136)], [(1,158,29,11),(2,159,30,12),(3,160,21,13),(4,151,22,14),(5,152,23,15),(6,153,24,16),(7,154,25,17),(8,155,26,18),(9,156,27,19),(10,157,28,20),(31,148,48,131),(32,149,49,132),(33,150,50,133),(34,141,41,134),(35,142,42,135),(36,143,43,136),(37,144,44,137),(38,145,45,138),(39,146,46,139),(40,147,47,140),(51,111,68,128),(52,112,69,129),(53,113,70,130),(54,114,61,121),(55,115,62,122),(56,116,63,123),(57,117,64,124),(58,118,65,125),(59,119,66,126),(60,120,67,127),(71,91,88,108),(72,92,89,109),(73,93,90,110),(74,94,81,101),(75,95,82,102),(76,96,83,103),(77,97,84,104),(78,98,85,105),(79,99,86,106),(80,100,87,107)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 35 | 0 | 0 |
| 0 | 0 | 6 | 35 | 0 | 0 |
| 0 | 0 | 6 | 0 | 34 | 35 |
| 0 | 0 | 40 | 35 | 7 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 35 | 7 | 12 |
| 0 | 0 | 35 | 35 | 6 | 5 |
| 0 | 0 | 12 | 6 | 7 | 6 |
| 0 | 0 | 4 | 11 | 40 | 5 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 35 | 40 |
| 0 | 0 | 40 | 40 | 2 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 40 | 0 | 35 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 2 | 15 | 0 |
| 0 | 0 | 25 | 15 | 18 | 26 |
| 0 | 0 | 39 | 0 | 29 | 2 |
| 0 | 0 | 14 | 2 | 4 | 12 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 13 | 3 | 26 |
| 0 | 0 | 2 | 2 | 0 | 26 |
| 0 | 0 | 13 | 0 | 30 | 28 |
| 0 | 0 | 2 | 13 | 25 | 37 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,6,6,40,0,0,35,35,0,35,0,0,0,0,34,7,0,0,0,0,35,0],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,35,35,12,4,0,0,35,35,6,11,0,0,7,6,7,40,0,0,12,5,6,5],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,40,0,0,0,40,0,0,0,0,35,2,1,35,0,0,40,1,0,0],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,26,25,39,14,0,0,2,15,0,2,0,0,15,18,29,4,0,0,0,26,2,12],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,13,2,13,2,0,0,13,2,0,13,0,0,3,0,30,25,0,0,26,26,28,37] >;
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4L | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10R | 20A | ··· | 20H | 20I | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | D10 | D10 | D10 | C5⋊D4 | 2+ (1+4) | 2- (1+4) | D4⋊8D10 | D4.10D10 |
| kernel | C10.1082- (1+4) | C20.48D4 | C2×C4⋊Dic5 | C20⋊7D4 | C20⋊2D4 | Dic5⋊D4 | D10⋊3Q8 | C2×C4○D20 | C10×C4○D4 | C2×C20 | C2×C4○D4 | C22×C4 | C2×D4 | C2×Q8 | C2×C4 | C10 | C10 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 1 | 1 | 4 | 2 | 6 | 6 | 2 | 16 | 1 | 1 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_{10}._{108}2_-^{(1+4)} % in TeX
G:=Group("C10.108ES-(2,2)"); // GroupNames label
G:=SmallGroup(320,1505);
// by ID
G=gap.SmallGroup(320,1505);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,675,570,297,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^4=c^2=1,d^2=b^2,e^2=a^5*b^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,e*b*e^-1=a^5*b,d*c*d^-1=a^5*c,c*e=e*c,e*d*e^-1=b^2*d>;
// generators/relations
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