direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D4.10D10, C20.48C24, C10.13C25, D10.7C24, D20.40C23, C10⋊22- (1+4), Dic5.8C24, Dic10.37C23, C4○D4⋊18D10, (C2×C10).4C24, (Q8×D5)⋊14C22, C4.63(C23×D5), C2.14(D5×C24), C5⋊D4.1C23, (C2×D4).254D10, C5⋊2(C2×2- (1+4)), C4○D20⋊26C22, (C2×Q8).212D10, (C4×D5).19C23, D4.29(C22×D5), (C5×D4).29C23, (C5×Q8).30C23, Q8.30(C22×D5), D4⋊2D5⋊13C22, (C2×C20).567C23, (C22×C4).292D10, C22.57(C23×D5), (C2×Dic10)⋊75C22, (C22×Dic10)⋊25C2, (C2×D20).298C22, (D4×C10).279C22, (Q8×C10).247C22, C23.215(C22×D5), (C22×C10).249C23, (C22×C20).303C22, (C2×Dic5).167C23, (C22×D5).260C23, (C22×Dic5).169C22, (C2×Q8×D5)⋊21C2, (C2×C4○D4)⋊14D5, (C10×C4○D4)⋊15C2, (C2×C4○D20)⋊38C2, (C2×D4⋊2D5)⋊30C2, (C5×C4○D4)⋊21C22, (C2×C4×D5).182C22, (C2×C4).253(C22×D5), (C2×C5⋊D4).152C22, SmallGroup(320,1620)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 2126 in 794 conjugacy classes, 447 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×10], C4 [×8], C4 [×12], C22, C22 [×6], C22 [×14], C5, C2×C4, C2×C4 [×15], C2×C4 [×54], D4 [×12], D4 [×28], Q8 [×4], Q8 [×36], C23 [×3], C23 [×2], D5 [×4], C10, C10 [×2], C10 [×6], C22×C4 [×3], C22×C4 [×12], C2×D4 [×3], C2×D4 [×7], C2×Q8, C2×Q8 [×49], C4○D4 [×8], C4○D4 [×72], Dic5 [×12], C20 [×8], D10 [×4], D10 [×4], C2×C10, C2×C10 [×6], C2×C10 [×6], C22×Q8 [×5], C2×C4○D4, C2×C4○D4 [×9], 2- (1+4) [×16], Dic10 [×36], C4×D5 [×24], D20 [×4], C2×Dic5 [×30], C5⋊D4 [×24], C2×C20, C2×C20 [×15], C5×D4 [×12], C5×Q8 [×4], C22×D5 [×2], C22×C10 [×3], C2×2- (1+4), C2×Dic10 [×33], C2×C4×D5 [×6], C2×D20, C4○D20 [×24], D4⋊2D5 [×48], Q8×D5 [×16], C22×Dic5 [×6], C2×C5⋊D4 [×6], C22×C20 [×3], D4×C10 [×3], Q8×C10, C5×C4○D4 [×8], C22×Dic10 [×3], C2×C4○D20 [×3], C2×D4⋊2D5 [×6], C2×Q8×D5 [×2], D4.10D10 [×16], C10×C4○D4, C2×D4.10D10
Quotients:
C1, C2 [×31], C22 [×155], C23 [×155], D5, C24 [×31], D10 [×15], 2- (1+4) [×2], C25, C22×D5 [×35], C2×2- (1+4), C23×D5 [×15], D4.10D10 [×2], D5×C24, C2×D4.10D10
Generators and relations
G = < a,b,c,d,e | a2=b4=c2=1, d10=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=ebe-1=b-1, dcd-1=b2c, ce=ec, ede-1=d9 >
►On 160 points
Generators in S160
(1 146)(2 147)(3 148)(4 149)(5 150)(6 151)(7 152)(8 153)(9 154)(10 155)(11 156)(12 157)(13 158)(14 159)(15 160)(16 141)(17 142)(18 143)(19 144)(20 145)(21 125)(22 126)(23 127)(24 128)(25 129)(26 130)(27 131)(28 132)(29 133)(30 134)(31 135)(32 136)(33 137)(34 138)(35 139)(36 140)(37 121)(38 122)(39 123)(40 124)(41 108)(42 109)(43 110)(44 111)(45 112)(46 113)(47 114)(48 115)(49 116)(50 117)(51 118)(52 119)(53 120)(54 101)(55 102)(56 103)(57 104)(58 105)(59 106)(60 107)(61 97)(62 98)(63 99)(64 100)(65 81)(66 82)(67 83)(68 84)(69 85)(70 86)(71 87)(72 88)(73 89)(74 90)(75 91)(76 92)(77 93)(78 94)(79 95)(80 96)
(1 89 11 99)(2 100 12 90)(3 91 13 81)(4 82 14 92)(5 93 15 83)(6 84 16 94)(7 95 17 85)(8 86 18 96)(9 97 19 87)(10 88 20 98)(21 107 31 117)(22 118 32 108)(23 109 33 119)(24 120 34 110)(25 111 35 101)(26 102 36 112)(27 113 37 103)(28 104 38 114)(29 115 39 105)(30 106 40 116)(41 126 51 136)(42 137 52 127)(43 128 53 138)(44 139 54 129)(45 130 55 140)(46 121 56 131)(47 132 57 122)(48 123 58 133)(49 134 59 124)(50 125 60 135)(61 144 71 154)(62 155 72 145)(63 146 73 156)(64 157 74 147)(65 148 75 158)(66 159 76 149)(67 150 77 160)(68 141 78 151)(69 152 79 142)(70 143 80 153)
(1 39)(2 30)(3 21)(4 32)(5 23)(6 34)(7 25)(8 36)(9 27)(10 38)(11 29)(12 40)(13 31)(14 22)(15 33)(16 24)(17 35)(18 26)(19 37)(20 28)(41 76)(42 67)(43 78)(44 69)(45 80)(46 71)(47 62)(48 73)(49 64)(50 75)(51 66)(52 77)(53 68)(54 79)(55 70)(56 61)(57 72)(58 63)(59 74)(60 65)(81 107)(82 118)(83 109)(84 120)(85 111)(86 102)(87 113)(88 104)(89 115)(90 106)(91 117)(92 108)(93 119)(94 110)(95 101)(96 112)(97 103)(98 114)(99 105)(100 116)(121 144)(122 155)(123 146)(124 157)(125 148)(126 159)(127 150)(128 141)(129 152)(130 143)(131 154)(132 145)(133 156)(134 147)(135 158)(136 149)(137 160)(138 151)(139 142)(140 153)
(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)
(1 155 11 145)(2 144 12 154)(3 153 13 143)(4 142 14 152)(5 151 15 141)(6 160 16 150)(7 149 17 159)(8 158 18 148)(9 147 19 157)(10 156 20 146)(21 140 31 130)(22 129 32 139)(23 138 33 128)(24 127 34 137)(25 136 35 126)(26 125 36 135)(27 134 37 124)(28 123 38 133)(29 132 39 122)(30 121 40 131)(41 111 51 101)(42 120 52 110)(43 109 53 119)(44 118 54 108)(45 107 55 117)(46 116 56 106)(47 105 57 115)(48 114 58 104)(49 103 59 113)(50 112 60 102)(61 90 71 100)(62 99 72 89)(63 88 73 98)(64 97 74 87)(65 86 75 96)(66 95 76 85)(67 84 77 94)(68 93 78 83)(69 82 79 92)(70 91 80 81)
G:=sub<Sym(160)| (1,146)(2,147)(3,148)(4,149)(5,150)(6,151)(7,152)(8,153)(9,154)(10,155)(11,156)(12,157)(13,158)(14,159)(15,160)(16,141)(17,142)(18,143)(19,144)(20,145)(21,125)(22,126)(23,127)(24,128)(25,129)(26,130)(27,131)(28,132)(29,133)(30,134)(31,135)(32,136)(33,137)(34,138)(35,139)(36,140)(37,121)(38,122)(39,123)(40,124)(41,108)(42,109)(43,110)(44,111)(45,112)(46,113)(47,114)(48,115)(49,116)(50,117)(51,118)(52,119)(53,120)(54,101)(55,102)(56,103)(57,104)(58,105)(59,106)(60,107)(61,97)(62,98)(63,99)(64,100)(65,81)(66,82)(67,83)(68,84)(69,85)(70,86)(71,87)(72,88)(73,89)(74,90)(75,91)(76,92)(77,93)(78,94)(79,95)(80,96), (1,89,11,99)(2,100,12,90)(3,91,13,81)(4,82,14,92)(5,93,15,83)(6,84,16,94)(7,95,17,85)(8,86,18,96)(9,97,19,87)(10,88,20,98)(21,107,31,117)(22,118,32,108)(23,109,33,119)(24,120,34,110)(25,111,35,101)(26,102,36,112)(27,113,37,103)(28,104,38,114)(29,115,39,105)(30,106,40,116)(41,126,51,136)(42,137,52,127)(43,128,53,138)(44,139,54,129)(45,130,55,140)(46,121,56,131)(47,132,57,122)(48,123,58,133)(49,134,59,124)(50,125,60,135)(61,144,71,154)(62,155,72,145)(63,146,73,156)(64,157,74,147)(65,148,75,158)(66,159,76,149)(67,150,77,160)(68,141,78,151)(69,152,79,142)(70,143,80,153), (1,39)(2,30)(3,21)(4,32)(5,23)(6,34)(7,25)(8,36)(9,27)(10,38)(11,29)(12,40)(13,31)(14,22)(15,33)(16,24)(17,35)(18,26)(19,37)(20,28)(41,76)(42,67)(43,78)(44,69)(45,80)(46,71)(47,62)(48,73)(49,64)(50,75)(51,66)(52,77)(53,68)(54,79)(55,70)(56,61)(57,72)(58,63)(59,74)(60,65)(81,107)(82,118)(83,109)(84,120)(85,111)(86,102)(87,113)(88,104)(89,115)(90,106)(91,117)(92,108)(93,119)(94,110)(95,101)(96,112)(97,103)(98,114)(99,105)(100,116)(121,144)(122,155)(123,146)(124,157)(125,148)(126,159)(127,150)(128,141)(129,152)(130,143)(131,154)(132,145)(133,156)(134,147)(135,158)(136,149)(137,160)(138,151)(139,142)(140,153), (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), (1,155,11,145)(2,144,12,154)(3,153,13,143)(4,142,14,152)(5,151,15,141)(6,160,16,150)(7,149,17,159)(8,158,18,148)(9,147,19,157)(10,156,20,146)(21,140,31,130)(22,129,32,139)(23,138,33,128)(24,127,34,137)(25,136,35,126)(26,125,36,135)(27,134,37,124)(28,123,38,133)(29,132,39,122)(30,121,40,131)(41,111,51,101)(42,120,52,110)(43,109,53,119)(44,118,54,108)(45,107,55,117)(46,116,56,106)(47,105,57,115)(48,114,58,104)(49,103,59,113)(50,112,60,102)(61,90,71,100)(62,99,72,89)(63,88,73,98)(64,97,74,87)(65,86,75,96)(66,95,76,85)(67,84,77,94)(68,93,78,83)(69,82,79,92)(70,91,80,81)>;
G:=Group( (1,146)(2,147)(3,148)(4,149)(5,150)(6,151)(7,152)(8,153)(9,154)(10,155)(11,156)(12,157)(13,158)(14,159)(15,160)(16,141)(17,142)(18,143)(19,144)(20,145)(21,125)(22,126)(23,127)(24,128)(25,129)(26,130)(27,131)(28,132)(29,133)(30,134)(31,135)(32,136)(33,137)(34,138)(35,139)(36,140)(37,121)(38,122)(39,123)(40,124)(41,108)(42,109)(43,110)(44,111)(45,112)(46,113)(47,114)(48,115)(49,116)(50,117)(51,118)(52,119)(53,120)(54,101)(55,102)(56,103)(57,104)(58,105)(59,106)(60,107)(61,97)(62,98)(63,99)(64,100)(65,81)(66,82)(67,83)(68,84)(69,85)(70,86)(71,87)(72,88)(73,89)(74,90)(75,91)(76,92)(77,93)(78,94)(79,95)(80,96), (1,89,11,99)(2,100,12,90)(3,91,13,81)(4,82,14,92)(5,93,15,83)(6,84,16,94)(7,95,17,85)(8,86,18,96)(9,97,19,87)(10,88,20,98)(21,107,31,117)(22,118,32,108)(23,109,33,119)(24,120,34,110)(25,111,35,101)(26,102,36,112)(27,113,37,103)(28,104,38,114)(29,115,39,105)(30,106,40,116)(41,126,51,136)(42,137,52,127)(43,128,53,138)(44,139,54,129)(45,130,55,140)(46,121,56,131)(47,132,57,122)(48,123,58,133)(49,134,59,124)(50,125,60,135)(61,144,71,154)(62,155,72,145)(63,146,73,156)(64,157,74,147)(65,148,75,158)(66,159,76,149)(67,150,77,160)(68,141,78,151)(69,152,79,142)(70,143,80,153), (1,39)(2,30)(3,21)(4,32)(5,23)(6,34)(7,25)(8,36)(9,27)(10,38)(11,29)(12,40)(13,31)(14,22)(15,33)(16,24)(17,35)(18,26)(19,37)(20,28)(41,76)(42,67)(43,78)(44,69)(45,80)(46,71)(47,62)(48,73)(49,64)(50,75)(51,66)(52,77)(53,68)(54,79)(55,70)(56,61)(57,72)(58,63)(59,74)(60,65)(81,107)(82,118)(83,109)(84,120)(85,111)(86,102)(87,113)(88,104)(89,115)(90,106)(91,117)(92,108)(93,119)(94,110)(95,101)(96,112)(97,103)(98,114)(99,105)(100,116)(121,144)(122,155)(123,146)(124,157)(125,148)(126,159)(127,150)(128,141)(129,152)(130,143)(131,154)(132,145)(133,156)(134,147)(135,158)(136,149)(137,160)(138,151)(139,142)(140,153), (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), (1,155,11,145)(2,144,12,154)(3,153,13,143)(4,142,14,152)(5,151,15,141)(6,160,16,150)(7,149,17,159)(8,158,18,148)(9,147,19,157)(10,156,20,146)(21,140,31,130)(22,129,32,139)(23,138,33,128)(24,127,34,137)(25,136,35,126)(26,125,36,135)(27,134,37,124)(28,123,38,133)(29,132,39,122)(30,121,40,131)(41,111,51,101)(42,120,52,110)(43,109,53,119)(44,118,54,108)(45,107,55,117)(46,116,56,106)(47,105,57,115)(48,114,58,104)(49,103,59,113)(50,112,60,102)(61,90,71,100)(62,99,72,89)(63,88,73,98)(64,97,74,87)(65,86,75,96)(66,95,76,85)(67,84,77,94)(68,93,78,83)(69,82,79,92)(70,91,80,81) );
G=PermutationGroup([(1,146),(2,147),(3,148),(4,149),(5,150),(6,151),(7,152),(8,153),(9,154),(10,155),(11,156),(12,157),(13,158),(14,159),(15,160),(16,141),(17,142),(18,143),(19,144),(20,145),(21,125),(22,126),(23,127),(24,128),(25,129),(26,130),(27,131),(28,132),(29,133),(30,134),(31,135),(32,136),(33,137),(34,138),(35,139),(36,140),(37,121),(38,122),(39,123),(40,124),(41,108),(42,109),(43,110),(44,111),(45,112),(46,113),(47,114),(48,115),(49,116),(50,117),(51,118),(52,119),(53,120),(54,101),(55,102),(56,103),(57,104),(58,105),(59,106),(60,107),(61,97),(62,98),(63,99),(64,100),(65,81),(66,82),(67,83),(68,84),(69,85),(70,86),(71,87),(72,88),(73,89),(74,90),(75,91),(76,92),(77,93),(78,94),(79,95),(80,96)], [(1,89,11,99),(2,100,12,90),(3,91,13,81),(4,82,14,92),(5,93,15,83),(6,84,16,94),(7,95,17,85),(8,86,18,96),(9,97,19,87),(10,88,20,98),(21,107,31,117),(22,118,32,108),(23,109,33,119),(24,120,34,110),(25,111,35,101),(26,102,36,112),(27,113,37,103),(28,104,38,114),(29,115,39,105),(30,106,40,116),(41,126,51,136),(42,137,52,127),(43,128,53,138),(44,139,54,129),(45,130,55,140),(46,121,56,131),(47,132,57,122),(48,123,58,133),(49,134,59,124),(50,125,60,135),(61,144,71,154),(62,155,72,145),(63,146,73,156),(64,157,74,147),(65,148,75,158),(66,159,76,149),(67,150,77,160),(68,141,78,151),(69,152,79,142),(70,143,80,153)], [(1,39),(2,30),(3,21),(4,32),(5,23),(6,34),(7,25),(8,36),(9,27),(10,38),(11,29),(12,40),(13,31),(14,22),(15,33),(16,24),(17,35),(18,26),(19,37),(20,28),(41,76),(42,67),(43,78),(44,69),(45,80),(46,71),(47,62),(48,73),(49,64),(50,75),(51,66),(52,77),(53,68),(54,79),(55,70),(56,61),(57,72),(58,63),(59,74),(60,65),(81,107),(82,118),(83,109),(84,120),(85,111),(86,102),(87,113),(88,104),(89,115),(90,106),(91,117),(92,108),(93,119),(94,110),(95,101),(96,112),(97,103),(98,114),(99,105),(100,116),(121,144),(122,155),(123,146),(124,157),(125,148),(126,159),(127,150),(128,141),(129,152),(130,143),(131,154),(132,145),(133,156),(134,147),(135,158),(136,149),(137,160),(138,151),(139,142),(140,153)], [(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)], [(1,155,11,145),(2,144,12,154),(3,153,13,143),(4,142,14,152),(5,151,15,141),(6,160,16,150),(7,149,17,159),(8,158,18,148),(9,147,19,157),(10,156,20,146),(21,140,31,130),(22,129,32,139),(23,138,33,128),(24,127,34,137),(25,136,35,126),(26,125,36,135),(27,134,37,124),(28,123,38,133),(29,132,39,122),(30,121,40,131),(41,111,51,101),(42,120,52,110),(43,109,53,119),(44,118,54,108),(45,107,55,117),(46,116,56,106),(47,105,57,115),(48,114,58,104),(49,103,59,113),(50,112,60,102),(61,90,71,100),(62,99,72,89),(63,88,73,98),(64,97,74,87),(65,86,75,96),(66,95,76,85),(67,84,77,94),(68,93,78,83),(69,82,79,92),(70,91,80,81)])
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 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 37 | 22 | 26 |
| 0 | 0 | 32 | 16 | 38 | 19 |
| 0 | 0 | 22 | 26 | 16 | 4 |
| 0 | 0 | 38 | 19 | 9 | 25 |
| 34 | 7 | 0 | 0 | 0 | 0 |
| 34 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 29 | 0 | 36 |
| 0 | 0 | 14 | 14 | 40 | 40 |
| 0 | 0 | 0 | 36 | 0 | 12 |
| 0 | 0 | 40 | 40 | 27 | 27 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 35 | 0 |
| 0 | 0 | 25 | 39 | 7 | 6 |
| 0 | 0 | 35 | 0 | 39 | 0 |
| 0 | 0 | 7 | 6 | 16 | 2 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,0,0,0,0,0,0,40,0,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,25,32,22,38,0,0,37,16,26,19,0,0,22,38,16,9,0,0,26,19,4,25],[34,34,0,0,0,0,7,1,0,0,0,0,0,0,0,14,0,40,0,0,29,14,36,40,0,0,0,40,0,27,0,0,36,40,12,27],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,2,25,35,7,0,0,0,39,0,6,0,0,35,7,39,16,0,0,0,6,0,2] >;
74 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 2K | 2L | 2M | 4A | ··· | 4H | 4I | ··· | 4T | 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 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 10 | 10 | 10 | 10 | 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 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | D10 | D10 | D10 | D10 | 2- (1+4) | D4.10D10 |
| kernel | C2×D4.10D10 | C22×Dic10 | C2×C4○D20 | C2×D4⋊2D5 | C2×Q8×D5 | D4.10D10 | C10×C4○D4 | C2×C4○D4 | C22×C4 | C2×D4 | C2×Q8 | C4○D4 | C10 | C2 |
| # reps | 1 | 3 | 3 | 6 | 2 | 16 | 1 | 2 | 6 | 6 | 2 | 16 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_2\times D_4._{10}D_{10} % in TeX
G:=Group("C2xD4.10D10"); // GroupNames label
G:=SmallGroup(320,1620);
// by ID
G=gap.SmallGroup(320,1620);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,297,136,1684,102,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^2=1,d^10=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=d*b*d^-1=e*b*e^-1=b^-1,d*c*d^-1=b^2*c,c*e=e*c,e*d*e^-1=d^9>;
// generators/relations
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