direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Q8.D10, Q16⋊11D10, D40⋊17C22, C40.34C23, C20.12C24, D20.7C23, C4.48(D4×D5), (C2×D40)⋊20C2, C10⋊4(C4○D8), (C10×Q16)⋊8C2, (C2×Q16)⋊13D5, (C4×D5).70D4, C20.87(C2×D4), D10.23(C2×D4), (C2×C8).247D10, (C8×D5)⋊15C22, Q8⋊D5⋊10C22, (C5×Q16)⋊9C22, C4.12(C23×D5), C8.40(C22×D5), (C5×Q8).6C23, Q8.6(C22×D5), (C2×C40).99C22, C5⋊2C8.23C23, (C2×Q8).154D10, Q8⋊2D5⋊7C22, (C4×D5).64C23, (C22×D5).94D4, C22.144(D4×D5), (C2×C20).529C23, (C2×Dic5).285D4, Dic5.125(C2×D4), C10.113(C22×D4), (C2×D20).186C22, (Q8×C10).151C22, (D5×C2×C8)⋊6C2, C5⋊4(C2×C4○D8), C2.86(C2×D4×D5), (C2×Q8⋊D5)⋊28C2, (C2×Q8⋊2D5)⋊16C2, (C2×C10).402(C2×D4), (C2×C4×D5).330C22, (C2×C4).617(C22×D5), (C2×C5⋊2C8).294C22, SmallGroup(320,1437)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1054 in 266 conjugacy classes, 103 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×6], C22, C22 [×12], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×15], D4 [×14], Q8 [×4], Q8 [×2], C23 [×3], D5 [×6], C10, C10 [×2], C2×C8, C2×C8 [×5], D8 [×4], SD16 [×8], Q16 [×4], C22×C4 [×3], C2×D4 [×4], C2×Q8 [×2], C4○D4 [×12], Dic5 [×2], C20 [×2], C20 [×4], D10 [×2], D10 [×10], C2×C10, C22×C8, C2×D8, C2×SD16 [×2], C2×Q16, C4○D8 [×8], C2×C4○D4 [×2], C5⋊2C8 [×2], C40 [×2], C4×D5 [×4], C4×D5 [×8], D20 [×4], D20 [×10], C2×Dic5, C2×C20, C2×C20 [×2], C5×Q8 [×4], C5×Q8 [×2], C22×D5, C22×D5 [×2], C2×C4○D8, C8×D5 [×4], D40 [×4], C2×C5⋊2C8, Q8⋊D5 [×8], C2×C40, C5×Q16 [×4], C2×C4×D5, C2×C4×D5 [×2], C2×D20 [×2], C2×D20 [×2], Q8⋊2D5 [×8], Q8⋊2D5 [×4], Q8×C10 [×2], D5×C2×C8, C2×D40, Q8.D10 [×8], C2×Q8⋊D5 [×2], C10×Q16, C2×Q8⋊2D5 [×2], C2×Q8.D10
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C4○D8 [×2], C22×D4, C22×D5 [×7], C2×C4○D8, D4×D5 [×2], C23×D5, Q8.D10 [×2], C2×D4×D5, C2×Q8.D10
Generators and relations
G = < a,b,c,d,e | a2=b4=e2=1, c2=d10=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=dbd-1=ebe=b-1, dcd-1=b-1c, ece=bc, ede=b2d9 >
►On 160 points
Generators in S160
(1 69)(2 70)(3 71)(4 72)(5 73)(6 74)(7 75)(8 76)(9 77)(10 78)(11 79)(12 80)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 157)(22 158)(23 159)(24 160)(25 141)(26 142)(27 143)(28 144)(29 145)(30 146)(31 147)(32 148)(33 149)(34 150)(35 151)(36 152)(37 153)(38 154)(39 155)(40 156)(41 110)(42 111)(43 112)(44 113)(45 114)(46 115)(47 116)(48 117)(49 118)(50 119)(51 120)(52 101)(53 102)(54 103)(55 104)(56 105)(57 106)(58 107)(59 108)(60 109)(81 124)(82 125)(83 126)(84 127)(85 128)(86 129)(87 130)(88 131)(89 132)(90 133)(91 134)(92 135)(93 136)(94 137)(95 138)(96 139)(97 140)(98 121)(99 122)(100 123)
(1 42 11 52)(2 53 12 43)(3 44 13 54)(4 55 14 45)(5 46 15 56)(6 57 16 47)(7 48 17 58)(8 59 18 49)(9 50 19 60)(10 41 20 51)(21 98 31 88)(22 89 32 99)(23 100 33 90)(24 91 34 81)(25 82 35 92)(26 93 36 83)(27 84 37 94)(28 95 38 85)(29 86 39 96)(30 97 40 87)(61 103 71 113)(62 114 72 104)(63 105 73 115)(64 116 74 106)(65 107 75 117)(66 118 76 108)(67 109 77 119)(68 120 78 110)(69 111 79 101)(70 102 80 112)(121 147 131 157)(122 158 132 148)(123 149 133 159)(124 160 134 150)(125 151 135 141)(126 142 136 152)(127 153 137 143)(128 144 138 154)(129 155 139 145)(130 146 140 156)
(1 30 11 40)(2 98 12 88)(3 32 13 22)(4 100 14 90)(5 34 15 24)(6 82 16 92)(7 36 17 26)(8 84 18 94)(9 38 19 28)(10 86 20 96)(21 43 31 53)(23 45 33 55)(25 47 35 57)(27 49 37 59)(29 51 39 41)(42 87 52 97)(44 89 54 99)(46 91 56 81)(48 93 58 83)(50 95 60 85)(61 158 71 148)(62 133 72 123)(63 160 73 150)(64 135 74 125)(65 142 75 152)(66 137 76 127)(67 144 77 154)(68 139 78 129)(69 146 79 156)(70 121 80 131)(101 140 111 130)(102 157 112 147)(103 122 113 132)(104 159 114 149)(105 124 115 134)(106 141 116 151)(107 126 117 136)(108 143 118 153)(109 128 119 138)(110 145 120 155)
(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)
(1 139)(2 138)(3 137)(4 136)(5 135)(6 134)(7 133)(8 132)(9 131)(10 130)(11 129)(12 128)(13 127)(14 126)(15 125)(16 124)(17 123)(18 122)(19 121)(20 140)(21 109)(22 108)(23 107)(24 106)(25 105)(26 104)(27 103)(28 102)(29 101)(30 120)(31 119)(32 118)(33 117)(34 116)(35 115)(36 114)(37 113)(38 112)(39 111)(40 110)(41 156)(42 155)(43 154)(44 153)(45 152)(46 151)(47 150)(48 149)(49 148)(50 147)(51 146)(52 145)(53 144)(54 143)(55 142)(56 141)(57 160)(58 159)(59 158)(60 157)(61 84)(62 83)(63 82)(64 81)(65 100)(66 99)(67 98)(68 97)(69 96)(70 95)(71 94)(72 93)(73 92)(74 91)(75 90)(76 89)(77 88)(78 87)(79 86)(80 85)
G:=sub<Sym(160)| (1,69)(2,70)(3,71)(4,72)(5,73)(6,74)(7,75)(8,76)(9,77)(10,78)(11,79)(12,80)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,157)(22,158)(23,159)(24,160)(25,141)(26,142)(27,143)(28,144)(29,145)(30,146)(31,147)(32,148)(33,149)(34,150)(35,151)(36,152)(37,153)(38,154)(39,155)(40,156)(41,110)(42,111)(43,112)(44,113)(45,114)(46,115)(47,116)(48,117)(49,118)(50,119)(51,120)(52,101)(53,102)(54,103)(55,104)(56,105)(57,106)(58,107)(59,108)(60,109)(81,124)(82,125)(83,126)(84,127)(85,128)(86,129)(87,130)(88,131)(89,132)(90,133)(91,134)(92,135)(93,136)(94,137)(95,138)(96,139)(97,140)(98,121)(99,122)(100,123), (1,42,11,52)(2,53,12,43)(3,44,13,54)(4,55,14,45)(5,46,15,56)(6,57,16,47)(7,48,17,58)(8,59,18,49)(9,50,19,60)(10,41,20,51)(21,98,31,88)(22,89,32,99)(23,100,33,90)(24,91,34,81)(25,82,35,92)(26,93,36,83)(27,84,37,94)(28,95,38,85)(29,86,39,96)(30,97,40,87)(61,103,71,113)(62,114,72,104)(63,105,73,115)(64,116,74,106)(65,107,75,117)(66,118,76,108)(67,109,77,119)(68,120,78,110)(69,111,79,101)(70,102,80,112)(121,147,131,157)(122,158,132,148)(123,149,133,159)(124,160,134,150)(125,151,135,141)(126,142,136,152)(127,153,137,143)(128,144,138,154)(129,155,139,145)(130,146,140,156), (1,30,11,40)(2,98,12,88)(3,32,13,22)(4,100,14,90)(5,34,15,24)(6,82,16,92)(7,36,17,26)(8,84,18,94)(9,38,19,28)(10,86,20,96)(21,43,31,53)(23,45,33,55)(25,47,35,57)(27,49,37,59)(29,51,39,41)(42,87,52,97)(44,89,54,99)(46,91,56,81)(48,93,58,83)(50,95,60,85)(61,158,71,148)(62,133,72,123)(63,160,73,150)(64,135,74,125)(65,142,75,152)(66,137,76,127)(67,144,77,154)(68,139,78,129)(69,146,79,156)(70,121,80,131)(101,140,111,130)(102,157,112,147)(103,122,113,132)(104,159,114,149)(105,124,115,134)(106,141,116,151)(107,126,117,136)(108,143,118,153)(109,128,119,138)(110,145,120,155), (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), (1,139)(2,138)(3,137)(4,136)(5,135)(6,134)(7,133)(8,132)(9,131)(10,130)(11,129)(12,128)(13,127)(14,126)(15,125)(16,124)(17,123)(18,122)(19,121)(20,140)(21,109)(22,108)(23,107)(24,106)(25,105)(26,104)(27,103)(28,102)(29,101)(30,120)(31,119)(32,118)(33,117)(34,116)(35,115)(36,114)(37,113)(38,112)(39,111)(40,110)(41,156)(42,155)(43,154)(44,153)(45,152)(46,151)(47,150)(48,149)(49,148)(50,147)(51,146)(52,145)(53,144)(54,143)(55,142)(56,141)(57,160)(58,159)(59,158)(60,157)(61,84)(62,83)(63,82)(64,81)(65,100)(66,99)(67,98)(68,97)(69,96)(70,95)(71,94)(72,93)(73,92)(74,91)(75,90)(76,89)(77,88)(78,87)(79,86)(80,85)>;
G:=Group( (1,69)(2,70)(3,71)(4,72)(5,73)(6,74)(7,75)(8,76)(9,77)(10,78)(11,79)(12,80)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,157)(22,158)(23,159)(24,160)(25,141)(26,142)(27,143)(28,144)(29,145)(30,146)(31,147)(32,148)(33,149)(34,150)(35,151)(36,152)(37,153)(38,154)(39,155)(40,156)(41,110)(42,111)(43,112)(44,113)(45,114)(46,115)(47,116)(48,117)(49,118)(50,119)(51,120)(52,101)(53,102)(54,103)(55,104)(56,105)(57,106)(58,107)(59,108)(60,109)(81,124)(82,125)(83,126)(84,127)(85,128)(86,129)(87,130)(88,131)(89,132)(90,133)(91,134)(92,135)(93,136)(94,137)(95,138)(96,139)(97,140)(98,121)(99,122)(100,123), (1,42,11,52)(2,53,12,43)(3,44,13,54)(4,55,14,45)(5,46,15,56)(6,57,16,47)(7,48,17,58)(8,59,18,49)(9,50,19,60)(10,41,20,51)(21,98,31,88)(22,89,32,99)(23,100,33,90)(24,91,34,81)(25,82,35,92)(26,93,36,83)(27,84,37,94)(28,95,38,85)(29,86,39,96)(30,97,40,87)(61,103,71,113)(62,114,72,104)(63,105,73,115)(64,116,74,106)(65,107,75,117)(66,118,76,108)(67,109,77,119)(68,120,78,110)(69,111,79,101)(70,102,80,112)(121,147,131,157)(122,158,132,148)(123,149,133,159)(124,160,134,150)(125,151,135,141)(126,142,136,152)(127,153,137,143)(128,144,138,154)(129,155,139,145)(130,146,140,156), (1,30,11,40)(2,98,12,88)(3,32,13,22)(4,100,14,90)(5,34,15,24)(6,82,16,92)(7,36,17,26)(8,84,18,94)(9,38,19,28)(10,86,20,96)(21,43,31,53)(23,45,33,55)(25,47,35,57)(27,49,37,59)(29,51,39,41)(42,87,52,97)(44,89,54,99)(46,91,56,81)(48,93,58,83)(50,95,60,85)(61,158,71,148)(62,133,72,123)(63,160,73,150)(64,135,74,125)(65,142,75,152)(66,137,76,127)(67,144,77,154)(68,139,78,129)(69,146,79,156)(70,121,80,131)(101,140,111,130)(102,157,112,147)(103,122,113,132)(104,159,114,149)(105,124,115,134)(106,141,116,151)(107,126,117,136)(108,143,118,153)(109,128,119,138)(110,145,120,155), (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), (1,139)(2,138)(3,137)(4,136)(5,135)(6,134)(7,133)(8,132)(9,131)(10,130)(11,129)(12,128)(13,127)(14,126)(15,125)(16,124)(17,123)(18,122)(19,121)(20,140)(21,109)(22,108)(23,107)(24,106)(25,105)(26,104)(27,103)(28,102)(29,101)(30,120)(31,119)(32,118)(33,117)(34,116)(35,115)(36,114)(37,113)(38,112)(39,111)(40,110)(41,156)(42,155)(43,154)(44,153)(45,152)(46,151)(47,150)(48,149)(49,148)(50,147)(51,146)(52,145)(53,144)(54,143)(55,142)(56,141)(57,160)(58,159)(59,158)(60,157)(61,84)(62,83)(63,82)(64,81)(65,100)(66,99)(67,98)(68,97)(69,96)(70,95)(71,94)(72,93)(73,92)(74,91)(75,90)(76,89)(77,88)(78,87)(79,86)(80,85) );
G=PermutationGroup([(1,69),(2,70),(3,71),(4,72),(5,73),(6,74),(7,75),(8,76),(9,77),(10,78),(11,79),(12,80),(13,61),(14,62),(15,63),(16,64),(17,65),(18,66),(19,67),(20,68),(21,157),(22,158),(23,159),(24,160),(25,141),(26,142),(27,143),(28,144),(29,145),(30,146),(31,147),(32,148),(33,149),(34,150),(35,151),(36,152),(37,153),(38,154),(39,155),(40,156),(41,110),(42,111),(43,112),(44,113),(45,114),(46,115),(47,116),(48,117),(49,118),(50,119),(51,120),(52,101),(53,102),(54,103),(55,104),(56,105),(57,106),(58,107),(59,108),(60,109),(81,124),(82,125),(83,126),(84,127),(85,128),(86,129),(87,130),(88,131),(89,132),(90,133),(91,134),(92,135),(93,136),(94,137),(95,138),(96,139),(97,140),(98,121),(99,122),(100,123)], [(1,42,11,52),(2,53,12,43),(3,44,13,54),(4,55,14,45),(5,46,15,56),(6,57,16,47),(7,48,17,58),(8,59,18,49),(9,50,19,60),(10,41,20,51),(21,98,31,88),(22,89,32,99),(23,100,33,90),(24,91,34,81),(25,82,35,92),(26,93,36,83),(27,84,37,94),(28,95,38,85),(29,86,39,96),(30,97,40,87),(61,103,71,113),(62,114,72,104),(63,105,73,115),(64,116,74,106),(65,107,75,117),(66,118,76,108),(67,109,77,119),(68,120,78,110),(69,111,79,101),(70,102,80,112),(121,147,131,157),(122,158,132,148),(123,149,133,159),(124,160,134,150),(125,151,135,141),(126,142,136,152),(127,153,137,143),(128,144,138,154),(129,155,139,145),(130,146,140,156)], [(1,30,11,40),(2,98,12,88),(3,32,13,22),(4,100,14,90),(5,34,15,24),(6,82,16,92),(7,36,17,26),(8,84,18,94),(9,38,19,28),(10,86,20,96),(21,43,31,53),(23,45,33,55),(25,47,35,57),(27,49,37,59),(29,51,39,41),(42,87,52,97),(44,89,54,99),(46,91,56,81),(48,93,58,83),(50,95,60,85),(61,158,71,148),(62,133,72,123),(63,160,73,150),(64,135,74,125),(65,142,75,152),(66,137,76,127),(67,144,77,154),(68,139,78,129),(69,146,79,156),(70,121,80,131),(101,140,111,130),(102,157,112,147),(103,122,113,132),(104,159,114,149),(105,124,115,134),(106,141,116,151),(107,126,117,136),(108,143,118,153),(109,128,119,138),(110,145,120,155)], [(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)], [(1,139),(2,138),(3,137),(4,136),(5,135),(6,134),(7,133),(8,132),(9,131),(10,130),(11,129),(12,128),(13,127),(14,126),(15,125),(16,124),(17,123),(18,122),(19,121),(20,140),(21,109),(22,108),(23,107),(24,106),(25,105),(26,104),(27,103),(28,102),(29,101),(30,120),(31,119),(32,118),(33,117),(34,116),(35,115),(36,114),(37,113),(38,112),(39,111),(40,110),(41,156),(42,155),(43,154),(44,153),(45,152),(46,151),(47,150),(48,149),(49,148),(50,147),(51,146),(52,145),(53,144),(54,143),(55,142),(56,141),(57,160),(58,159),(59,158),(60,157),(61,84),(62,83),(63,82),(64,81),(65,100),(66,99),(67,98),(68,97),(69,96),(70,95),(71,94),(72,93),(73,92),(74,91),(75,90),(76,89),(77,88),(78,87),(79,86),(80,85)])
Matrix representation ►G ⊆ GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 0 | 1 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 9 |
| 6 | 35 | 0 | 0 |
| 6 | 1 | 0 | 0 |
| 0 | 0 | 15 | 15 |
| 0 | 0 | 15 | 26 |
| 0 | 40 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 12 | 29 |
| 0 | 0 | 29 | 29 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,40,0],[40,0,0,0,0,40,0,0,0,0,32,0,0,0,0,9],[6,6,0,0,35,1,0,0,0,0,15,15,0,0,15,26],[0,40,0,0,40,0,0,0,0,0,12,29,0,0,29,29] >;
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10F | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | D10 | D10 | D10 | C4○D8 | D4×D5 | D4×D5 | Q8.D10 |
| kernel | C2×Q8.D10 | D5×C2×C8 | C2×D40 | Q8.D10 | C2×Q8⋊D5 | C10×Q16 | C2×Q8⋊2D5 | C4×D5 | C2×Dic5 | C22×D5 | C2×Q16 | C2×C8 | Q16 | C2×Q8 | C10 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 8 | 2 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 8 | 4 | 8 | 2 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_2\times Q_8.D_{10} % in TeX
G:=Group("C2xQ8.D10"); // GroupNames label
G:=SmallGroup(320,1437);
// by ID
G=gap.SmallGroup(320,1437);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,184,1123,185,136,438,235,102,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=e^2=1,c^2=d^10=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=d*b*d^-1=e*b*e=b^-1,d*c*d^-1=b^-1*c,e*c*e=b*c,e*d*e=b^2*d^9>;
// generators/relations
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