?
G = C10.342+ (1+4) order 320 = 26·5
34th non-split extension by C10 of 2+ (1+4) acting via 2+ (1+4)/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.342+ (1+4), C4⋊D4⋊8D5, C4⋊C4.90D10, (D4×Dic5)⋊17C2, Dic5⋊4D4⋊7C2, (C2×D4).153D10, (C2×C20).36C23, C22⋊C4.48D10, Dic5⋊D4⋊29C2, (C2×C10).145C24, (C22×C4).220D10, D10.12D4⋊16C2, C2.36(D4⋊6D10), Dic5.38(C4○D4), Dic5.Q8⋊11C2, (D4×C10).119C22, C23.11D10⋊5C2, C22.1(D4⋊2D5), C23.D10⋊15C2, C4⋊Dic5.206C22, (C22×C10).16C23, (C2×Dic5).66C23, (C22×D5).63C23, C23.179(C22×D5), C22.166(C23×D5), C23.D5.22C22, D10⋊C4.13C22, C23.18D10⋊20C2, (C22×C20).378C22, C5⋊6(C22.47C24), (C4×Dic5).100C22, C10.D4.16C22, (C22×Dic5).106C22, (C5×C4⋊D4)⋊9C2, (C4×C5⋊D4)⋊53C2, C2.36(D5×C4○D4), C4⋊C4⋊D5⋊12C2, C10.81(C2×C4○D4), C2.33(C2×D4⋊2D5), (C2×C4×D5).259C22, (C2×C10).21(C4○D4), (C2×C10.D4)⋊40C2, (C5×C4⋊C4).141C22, (C2×C4).293(C22×D5), (C2×C5⋊D4).26C22, (C5×C22⋊C4).10C22, SmallGroup(320,1273)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 790 in 238 conjugacy classes, 97 normal (91 characteristic)
C1, C2 [×3], C2 [×5], C4 [×12], C22, C22 [×2], C22 [×11], C5, C2×C4 [×4], C2×C4 [×15], D4 [×10], C23 [×3], C23, D5, C10 [×3], C10 [×4], C42 [×3], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×9], C22×C4, C22×C4 [×5], C2×D4 [×3], C2×D4 [×3], Dic5 [×2], Dic5 [×6], C20 [×4], D10 [×3], C2×C10, C2×C10 [×2], C2×C10 [×8], C2×C4⋊C4, C42⋊C2, C4×D4 [×4], C4⋊D4, C4⋊D4 [×3], C22.D4 [×2], C42.C2, C42⋊2C2 [×2], C4×D5, C2×Dic5 [×7], C2×Dic5 [×6], C5⋊D4 [×5], C2×C20 [×4], C2×C20, C5×D4 [×5], C22×D5, C22×C10 [×3], C22.47C24, C4×Dic5 [×3], C10.D4 [×7], C4⋊Dic5 [×2], D10⋊C4 [×3], C23.D5 [×5], C5×C22⋊C4 [×2], C5×C4⋊C4, C2×C4×D5, C22×Dic5 [×4], C2×C5⋊D4 [×3], C22×C20, D4×C10 [×3], C23.11D10, C23.D10, Dic5⋊4D4, D10.12D4, Dic5.Q8, C4⋊C4⋊D5, C2×C10.D4, C4×C5⋊D4, D4×Dic5 [×2], C23.18D10, Dic5⋊D4 [×3], C5×C4⋊D4, C10.342+ (1+4)
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×4], C24, D10 [×7], C2×C4○D4 [×2], 2+ (1+4), C22×D5 [×7], C22.47C24, D4⋊2D5 [×2], C23×D5, C2×D4⋊2D5, D4⋊6D10, D5×C4○D4, C10.342+ (1+4)
Generators and relations
G = < a,b,c,d,e | a10=b4=e2=1, c2=a5, d2=b2, bab-1=cac-1=a-1, ad=da, ae=ea, cbc-1=b-1, dbd-1=ebe=a5b, cd=dc, ce=ec, ede=a5b2d >
►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)
(1 110 25 93)(2 109 26 92)(3 108 27 91)(4 107 28 100)(5 106 29 99)(6 105 30 98)(7 104 21 97)(8 103 22 96)(9 102 23 95)(10 101 24 94)(11 85 158 78)(12 84 159 77)(13 83 160 76)(14 82 151 75)(15 81 152 74)(16 90 153 73)(17 89 154 72)(18 88 155 71)(19 87 156 80)(20 86 157 79)(31 122 48 115)(32 121 49 114)(33 130 50 113)(34 129 41 112)(35 128 42 111)(36 127 43 120)(37 126 44 119)(38 125 45 118)(39 124 46 117)(40 123 47 116)(51 147 68 140)(52 146 69 139)(53 145 70 138)(54 144 61 137)(55 143 62 136)(56 142 63 135)(57 141 64 134)(58 150 65 133)(59 149 66 132)(60 148 67 131)
(1 65 6 70)(2 64 7 69)(3 63 8 68)(4 62 9 67)(5 61 10 66)(11 125 16 130)(12 124 17 129)(13 123 18 128)(14 122 19 127)(15 121 20 126)(21 52 26 57)(22 51 27 56)(23 60 28 55)(24 59 29 54)(25 58 30 53)(31 87 36 82)(32 86 37 81)(33 85 38 90)(34 84 39 89)(35 83 40 88)(41 77 46 72)(42 76 47 71)(43 75 48 80)(44 74 49 79)(45 73 50 78)(91 135 96 140)(92 134 97 139)(93 133 98 138)(94 132 99 137)(95 131 100 136)(101 149 106 144)(102 148 107 143)(103 147 108 142)(104 146 109 141)(105 145 110 150)(111 160 116 155)(112 159 117 154)(113 158 118 153)(114 157 119 152)(115 156 120 151)
(1 118 25 125)(2 119 26 126)(3 120 27 127)(4 111 28 128)(5 112 29 129)(6 113 30 130)(7 114 21 121)(8 115 22 122)(9 116 23 123)(10 117 24 124)(11 70 158 53)(12 61 159 54)(13 62 160 55)(14 63 151 56)(15 64 152 57)(16 65 153 58)(17 66 154 59)(18 67 155 60)(19 68 156 51)(20 69 157 52)(31 91 48 108)(32 92 49 109)(33 93 50 110)(34 94 41 101)(35 95 42 102)(36 96 43 103)(37 97 44 104)(38 98 45 105)(39 99 46 106)(40 100 47 107)(71 143 88 136)(72 144 89 137)(73 145 90 138)(74 146 81 139)(75 147 82 140)(76 148 83 131)(77 149 84 132)(78 150 85 133)(79 141 86 134)(80 142 87 135)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 154)(22 155)(23 156)(24 157)(25 158)(26 159)(27 160)(28 151)(29 152)(30 153)(31 148)(32 149)(33 150)(34 141)(35 142)(36 143)(37 144)(38 145)(39 146)(40 147)(41 134)(42 135)(43 136)(44 137)(45 138)(46 139)(47 140)(48 131)(49 132)(50 133)(51 111)(52 112)(53 113)(54 114)(55 115)(56 116)(57 117)(58 118)(59 119)(60 120)(61 121)(62 122)(63 123)(64 124)(65 125)(66 126)(67 127)(68 128)(69 129)(70 130)(71 91)(72 92)(73 93)(74 94)(75 95)(76 96)(77 97)(78 98)(79 99)(80 100)(81 101)(82 102)(83 103)(84 104)(85 105)(86 106)(87 107)(88 108)(89 109)(90 110)
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), (1,110,25,93)(2,109,26,92)(3,108,27,91)(4,107,28,100)(5,106,29,99)(6,105,30,98)(7,104,21,97)(8,103,22,96)(9,102,23,95)(10,101,24,94)(11,85,158,78)(12,84,159,77)(13,83,160,76)(14,82,151,75)(15,81,152,74)(16,90,153,73)(17,89,154,72)(18,88,155,71)(19,87,156,80)(20,86,157,79)(31,122,48,115)(32,121,49,114)(33,130,50,113)(34,129,41,112)(35,128,42,111)(36,127,43,120)(37,126,44,119)(38,125,45,118)(39,124,46,117)(40,123,47,116)(51,147,68,140)(52,146,69,139)(53,145,70,138)(54,144,61,137)(55,143,62,136)(56,142,63,135)(57,141,64,134)(58,150,65,133)(59,149,66,132)(60,148,67,131), (1,65,6,70)(2,64,7,69)(3,63,8,68)(4,62,9,67)(5,61,10,66)(11,125,16,130)(12,124,17,129)(13,123,18,128)(14,122,19,127)(15,121,20,126)(21,52,26,57)(22,51,27,56)(23,60,28,55)(24,59,29,54)(25,58,30,53)(31,87,36,82)(32,86,37,81)(33,85,38,90)(34,84,39,89)(35,83,40,88)(41,77,46,72)(42,76,47,71)(43,75,48,80)(44,74,49,79)(45,73,50,78)(91,135,96,140)(92,134,97,139)(93,133,98,138)(94,132,99,137)(95,131,100,136)(101,149,106,144)(102,148,107,143)(103,147,108,142)(104,146,109,141)(105,145,110,150)(111,160,116,155)(112,159,117,154)(113,158,118,153)(114,157,119,152)(115,156,120,151), (1,118,25,125)(2,119,26,126)(3,120,27,127)(4,111,28,128)(5,112,29,129)(6,113,30,130)(7,114,21,121)(8,115,22,122)(9,116,23,123)(10,117,24,124)(11,70,158,53)(12,61,159,54)(13,62,160,55)(14,63,151,56)(15,64,152,57)(16,65,153,58)(17,66,154,59)(18,67,155,60)(19,68,156,51)(20,69,157,52)(31,91,48,108)(32,92,49,109)(33,93,50,110)(34,94,41,101)(35,95,42,102)(36,96,43,103)(37,97,44,104)(38,98,45,105)(39,99,46,106)(40,100,47,107)(71,143,88,136)(72,144,89,137)(73,145,90,138)(74,146,81,139)(75,147,82,140)(76,148,83,131)(77,149,84,132)(78,150,85,133)(79,141,86,134)(80,142,87,135), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,154)(22,155)(23,156)(24,157)(25,158)(26,159)(27,160)(28,151)(29,152)(30,153)(31,148)(32,149)(33,150)(34,141)(35,142)(36,143)(37,144)(38,145)(39,146)(40,147)(41,134)(42,135)(43,136)(44,137)(45,138)(46,139)(47,140)(48,131)(49,132)(50,133)(51,111)(52,112)(53,113)(54,114)(55,115)(56,116)(57,117)(58,118)(59,119)(60,120)(61,121)(62,122)(63,123)(64,124)(65,125)(66,126)(67,127)(68,128)(69,129)(70,130)(71,91)(72,92)(73,93)(74,94)(75,95)(76,96)(77,97)(78,98)(79,99)(80,100)(81,101)(82,102)(83,103)(84,104)(85,105)(86,106)(87,107)(88,108)(89,109)(90,110)>;
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), (1,110,25,93)(2,109,26,92)(3,108,27,91)(4,107,28,100)(5,106,29,99)(6,105,30,98)(7,104,21,97)(8,103,22,96)(9,102,23,95)(10,101,24,94)(11,85,158,78)(12,84,159,77)(13,83,160,76)(14,82,151,75)(15,81,152,74)(16,90,153,73)(17,89,154,72)(18,88,155,71)(19,87,156,80)(20,86,157,79)(31,122,48,115)(32,121,49,114)(33,130,50,113)(34,129,41,112)(35,128,42,111)(36,127,43,120)(37,126,44,119)(38,125,45,118)(39,124,46,117)(40,123,47,116)(51,147,68,140)(52,146,69,139)(53,145,70,138)(54,144,61,137)(55,143,62,136)(56,142,63,135)(57,141,64,134)(58,150,65,133)(59,149,66,132)(60,148,67,131), (1,65,6,70)(2,64,7,69)(3,63,8,68)(4,62,9,67)(5,61,10,66)(11,125,16,130)(12,124,17,129)(13,123,18,128)(14,122,19,127)(15,121,20,126)(21,52,26,57)(22,51,27,56)(23,60,28,55)(24,59,29,54)(25,58,30,53)(31,87,36,82)(32,86,37,81)(33,85,38,90)(34,84,39,89)(35,83,40,88)(41,77,46,72)(42,76,47,71)(43,75,48,80)(44,74,49,79)(45,73,50,78)(91,135,96,140)(92,134,97,139)(93,133,98,138)(94,132,99,137)(95,131,100,136)(101,149,106,144)(102,148,107,143)(103,147,108,142)(104,146,109,141)(105,145,110,150)(111,160,116,155)(112,159,117,154)(113,158,118,153)(114,157,119,152)(115,156,120,151), (1,118,25,125)(2,119,26,126)(3,120,27,127)(4,111,28,128)(5,112,29,129)(6,113,30,130)(7,114,21,121)(8,115,22,122)(9,116,23,123)(10,117,24,124)(11,70,158,53)(12,61,159,54)(13,62,160,55)(14,63,151,56)(15,64,152,57)(16,65,153,58)(17,66,154,59)(18,67,155,60)(19,68,156,51)(20,69,157,52)(31,91,48,108)(32,92,49,109)(33,93,50,110)(34,94,41,101)(35,95,42,102)(36,96,43,103)(37,97,44,104)(38,98,45,105)(39,99,46,106)(40,100,47,107)(71,143,88,136)(72,144,89,137)(73,145,90,138)(74,146,81,139)(75,147,82,140)(76,148,83,131)(77,149,84,132)(78,150,85,133)(79,141,86,134)(80,142,87,135), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,154)(22,155)(23,156)(24,157)(25,158)(26,159)(27,160)(28,151)(29,152)(30,153)(31,148)(32,149)(33,150)(34,141)(35,142)(36,143)(37,144)(38,145)(39,146)(40,147)(41,134)(42,135)(43,136)(44,137)(45,138)(46,139)(47,140)(48,131)(49,132)(50,133)(51,111)(52,112)(53,113)(54,114)(55,115)(56,116)(57,117)(58,118)(59,119)(60,120)(61,121)(62,122)(63,123)(64,124)(65,125)(66,126)(67,127)(68,128)(69,129)(70,130)(71,91)(72,92)(73,93)(74,94)(75,95)(76,96)(77,97)(78,98)(79,99)(80,100)(81,101)(82,102)(83,103)(84,104)(85,105)(86,106)(87,107)(88,108)(89,109)(90,110) );
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)], [(1,110,25,93),(2,109,26,92),(3,108,27,91),(4,107,28,100),(5,106,29,99),(6,105,30,98),(7,104,21,97),(8,103,22,96),(9,102,23,95),(10,101,24,94),(11,85,158,78),(12,84,159,77),(13,83,160,76),(14,82,151,75),(15,81,152,74),(16,90,153,73),(17,89,154,72),(18,88,155,71),(19,87,156,80),(20,86,157,79),(31,122,48,115),(32,121,49,114),(33,130,50,113),(34,129,41,112),(35,128,42,111),(36,127,43,120),(37,126,44,119),(38,125,45,118),(39,124,46,117),(40,123,47,116),(51,147,68,140),(52,146,69,139),(53,145,70,138),(54,144,61,137),(55,143,62,136),(56,142,63,135),(57,141,64,134),(58,150,65,133),(59,149,66,132),(60,148,67,131)], [(1,65,6,70),(2,64,7,69),(3,63,8,68),(4,62,9,67),(5,61,10,66),(11,125,16,130),(12,124,17,129),(13,123,18,128),(14,122,19,127),(15,121,20,126),(21,52,26,57),(22,51,27,56),(23,60,28,55),(24,59,29,54),(25,58,30,53),(31,87,36,82),(32,86,37,81),(33,85,38,90),(34,84,39,89),(35,83,40,88),(41,77,46,72),(42,76,47,71),(43,75,48,80),(44,74,49,79),(45,73,50,78),(91,135,96,140),(92,134,97,139),(93,133,98,138),(94,132,99,137),(95,131,100,136),(101,149,106,144),(102,148,107,143),(103,147,108,142),(104,146,109,141),(105,145,110,150),(111,160,116,155),(112,159,117,154),(113,158,118,153),(114,157,119,152),(115,156,120,151)], [(1,118,25,125),(2,119,26,126),(3,120,27,127),(4,111,28,128),(5,112,29,129),(6,113,30,130),(7,114,21,121),(8,115,22,122),(9,116,23,123),(10,117,24,124),(11,70,158,53),(12,61,159,54),(13,62,160,55),(14,63,151,56),(15,64,152,57),(16,65,153,58),(17,66,154,59),(18,67,155,60),(19,68,156,51),(20,69,157,52),(31,91,48,108),(32,92,49,109),(33,93,50,110),(34,94,41,101),(35,95,42,102),(36,96,43,103),(37,97,44,104),(38,98,45,105),(39,99,46,106),(40,100,47,107),(71,143,88,136),(72,144,89,137),(73,145,90,138),(74,146,81,139),(75,147,82,140),(76,148,83,131),(77,149,84,132),(78,150,85,133),(79,141,86,134),(80,142,87,135)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(21,154),(22,155),(23,156),(24,157),(25,158),(26,159),(27,160),(28,151),(29,152),(30,153),(31,148),(32,149),(33,150),(34,141),(35,142),(36,143),(37,144),(38,145),(39,146),(40,147),(41,134),(42,135),(43,136),(44,137),(45,138),(46,139),(47,140),(48,131),(49,132),(50,133),(51,111),(52,112),(53,113),(54,114),(55,115),(56,116),(57,117),(58,118),(59,119),(60,120),(61,121),(62,122),(63,123),(64,124),(65,125),(66,126),(67,127),(68,128),(69,129),(70,130),(71,91),(72,92),(73,93),(74,94),(75,95),(76,96),(77,97),(78,98),(79,99),(80,100),(81,101),(82,102),(83,103),(84,104),(85,105),(86,106),(87,107),(88,108),(89,109),(90,110)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 35 |
| 0 | 0 | 0 | 0 | 6 | 35 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 35 | 6 |
| 0 | 0 | 0 | 0 | 1 | 6 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 35 | 6 |
| 0 | 0 | 0 | 0 | 1 | 6 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,6,0,0,0,0,35,35],[0,9,0,0,0,0,9,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,35,1,0,0,0,0,6,6],[9,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,35,1,0,0,0,0,6,6],[32,0,0,0,0,0,0,9,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,32,0,0,0,0,9,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | ··· | 4M | 4N | 4O | 4P | 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 | 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 | 2 | 2 | 4 | 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 | 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 | C2 | D5 | C4○D4 | C4○D4 | D10 | D10 | D10 | D10 | 2+ (1+4) | D4⋊2D5 | D4⋊6D10 | D5×C4○D4 |
| kernel | C10.342+ (1+4) | C23.11D10 | C23.D10 | Dic5⋊4D4 | D10.12D4 | Dic5.Q8 | C4⋊C4⋊D5 | C2×C10.D4 | C4×C5⋊D4 | D4×Dic5 | C23.18D10 | Dic5⋊D4 | C5×C4⋊D4 | C4⋊D4 | Dic5 | C2×C10 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C10 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 3 | 1 | 2 | 4 | 4 | 4 | 2 | 2 | 6 | 1 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_{10}._{34}2_+^{(1+4)} % in TeX
G:=Group("C10.34ES+(2,2)"); // GroupNames label
G:=SmallGroup(320,1273);
// by ID
G=gap.SmallGroup(320,1273);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,758,100,794,297,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^4=e^2=1,c^2=a^5,d^2=b^2,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,d*b*d^-1=e*b*e=a^5*b,c*d=d*c,c*e=e*c,e*d*e=a^5*b^2*d>;
// generators/relations
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