direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C4○D4×C2×C10, C10.23C25, C20.89C24, (C23×C4)⋊8C10, (C2×C20)⋊18C23, (C23×C20)⋊17C2, (C5×D4)⋊14C23, D4⋊3(C22×C10), C2.3(C24×C10), Q8⋊3(C22×C10), (C5×Q8)⋊13C23, (D4×C10)⋊70C22, (C22×D4)⋊13C10, C4.12(C23×C10), C24.34(C2×C10), (Q8×C10)⋊59C22, (C22×Q8)⋊11C10, (C2×C10).386C24, (C22×C20)⋊67C22, C22.1(C23×C10), C23.46(C22×C10), (C23×C10).94C22, (C22×C10).269C23, (D4×C2×C10)⋊28C2, (Q8×C2×C10)⋊23C2, (C2×D4)⋊19(C2×C10), (C2×C4)⋊5(C22×C10), (C2×Q8)⋊19(C2×C10), (C22×C4)⋊20(C2×C10), SmallGroup(320,1631)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1010 in 890 conjugacy classes, 770 normal (12 characteristic)
C1, C2, C2 [×6], C2 [×12], C4 [×16], C22 [×19], C22 [×36], C5, C2×C4 [×72], D4 [×48], Q8 [×16], C23, C23 [×18], C23 [×12], C10, C10 [×6], C10 [×12], C22×C4, C22×C4 [×39], C2×D4 [×36], C2×Q8 [×12], C4○D4 [×64], C24 [×3], C20 [×16], C2×C10 [×19], C2×C10 [×36], C23×C4 [×3], C22×D4 [×3], C22×Q8, C2×C4○D4 [×24], C2×C20 [×72], C5×D4 [×48], C5×Q8 [×16], C22×C10, C22×C10 [×18], C22×C10 [×12], C22×C4○D4, C22×C20, C22×C20 [×39], D4×C10 [×36], Q8×C10 [×12], C5×C4○D4 [×64], C23×C10 [×3], C23×C20 [×3], D4×C2×C10 [×3], Q8×C2×C10, C10×C4○D4 [×24], C4○D4×C2×C10
Quotients:
C1, C2 [×31], C22 [×155], C5, C23 [×155], C10 [×31], C4○D4 [×4], C24 [×31], C2×C10 [×155], C2×C4○D4 [×6], C25, C22×C10 [×155], C22×C4○D4, C5×C4○D4 [×4], C23×C10 [×31], C10×C4○D4 [×6], C24×C10, C4○D4×C2×C10
Generators and relations
G = < a,b,c,d,e | a2=b10=c4=e2=1, d2=c2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c2d >
►On 160 points
Generators in S160
(1 42)(2 43)(3 44)(4 45)(5 46)(6 47)(7 48)(8 49)(9 50)(10 41)(11 136)(12 137)(13 138)(14 139)(15 140)(16 131)(17 132)(18 133)(19 134)(20 135)(21 37)(22 38)(23 39)(24 40)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(51 85)(52 86)(53 87)(54 88)(55 89)(56 90)(57 81)(58 82)(59 83)(60 84)(61 77)(62 78)(63 79)(64 80)(65 71)(66 72)(67 73)(68 74)(69 75)(70 76)(91 125)(92 126)(93 127)(94 128)(95 129)(96 130)(97 121)(98 122)(99 123)(100 124)(101 117)(102 118)(103 119)(104 120)(105 111)(106 112)(107 113)(108 114)(109 115)(110 116)(141 157)(142 158)(143 159)(144 160)(145 151)(146 152)(147 153)(148 154)(149 155)(150 156)
(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 87 27 78)(2 88 28 79)(3 89 29 80)(4 90 30 71)(5 81 21 72)(6 82 22 73)(7 83 23 74)(8 84 24 75)(9 85 25 76)(10 86 26 77)(11 110 151 91)(12 101 152 92)(13 102 153 93)(14 103 154 94)(15 104 155 95)(16 105 156 96)(17 106 157 97)(18 107 158 98)(19 108 159 99)(20 109 160 100)(31 70 50 51)(32 61 41 52)(33 62 42 53)(34 63 43 54)(35 64 44 55)(36 65 45 56)(37 66 46 57)(38 67 47 58)(39 68 48 59)(40 69 49 60)(111 150 130 131)(112 141 121 132)(113 142 122 133)(114 143 123 134)(115 144 124 135)(116 145 125 136)(117 146 126 137)(118 147 127 138)(119 148 128 139)(120 149 129 140)
(1 127 27 118)(2 128 28 119)(3 129 29 120)(4 130 30 111)(5 121 21 112)(6 122 22 113)(7 123 23 114)(8 124 24 115)(9 125 25 116)(10 126 26 117)(11 70 151 51)(12 61 152 52)(13 62 153 53)(14 63 154 54)(15 64 155 55)(16 65 156 56)(17 66 157 57)(18 67 158 58)(19 68 159 59)(20 69 160 60)(31 110 50 91)(32 101 41 92)(33 102 42 93)(34 103 43 94)(35 104 44 95)(36 105 45 96)(37 106 46 97)(38 107 47 98)(39 108 48 99)(40 109 49 100)(71 150 90 131)(72 141 81 132)(73 142 82 133)(74 143 83 134)(75 144 84 135)(76 145 85 136)(77 146 86 137)(78 147 87 138)(79 148 88 139)(80 149 89 140)
(1 118)(2 119)(3 120)(4 111)(5 112)(6 113)(7 114)(8 115)(9 116)(10 117)(11 70)(12 61)(13 62)(14 63)(15 64)(16 65)(17 66)(18 67)(19 68)(20 69)(21 121)(22 122)(23 123)(24 124)(25 125)(26 126)(27 127)(28 128)(29 129)(30 130)(31 91)(32 92)(33 93)(34 94)(35 95)(36 96)(37 97)(38 98)(39 99)(40 100)(41 101)(42 102)(43 103)(44 104)(45 105)(46 106)(47 107)(48 108)(49 109)(50 110)(51 151)(52 152)(53 153)(54 154)(55 155)(56 156)(57 157)(58 158)(59 159)(60 160)(71 131)(72 132)(73 133)(74 134)(75 135)(76 136)(77 137)(78 138)(79 139)(80 140)(81 141)(82 142)(83 143)(84 144)(85 145)(86 146)(87 147)(88 148)(89 149)(90 150)
G:=sub<Sym(160)| (1,42)(2,43)(3,44)(4,45)(5,46)(6,47)(7,48)(8,49)(9,50)(10,41)(11,136)(12,137)(13,138)(14,139)(15,140)(16,131)(17,132)(18,133)(19,134)(20,135)(21,37)(22,38)(23,39)(24,40)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(51,85)(52,86)(53,87)(54,88)(55,89)(56,90)(57,81)(58,82)(59,83)(60,84)(61,77)(62,78)(63,79)(64,80)(65,71)(66,72)(67,73)(68,74)(69,75)(70,76)(91,125)(92,126)(93,127)(94,128)(95,129)(96,130)(97,121)(98,122)(99,123)(100,124)(101,117)(102,118)(103,119)(104,120)(105,111)(106,112)(107,113)(108,114)(109,115)(110,116)(141,157)(142,158)(143,159)(144,160)(145,151)(146,152)(147,153)(148,154)(149,155)(150,156), (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), (1,87,27,78)(2,88,28,79)(3,89,29,80)(4,90,30,71)(5,81,21,72)(6,82,22,73)(7,83,23,74)(8,84,24,75)(9,85,25,76)(10,86,26,77)(11,110,151,91)(12,101,152,92)(13,102,153,93)(14,103,154,94)(15,104,155,95)(16,105,156,96)(17,106,157,97)(18,107,158,98)(19,108,159,99)(20,109,160,100)(31,70,50,51)(32,61,41,52)(33,62,42,53)(34,63,43,54)(35,64,44,55)(36,65,45,56)(37,66,46,57)(38,67,47,58)(39,68,48,59)(40,69,49,60)(111,150,130,131)(112,141,121,132)(113,142,122,133)(114,143,123,134)(115,144,124,135)(116,145,125,136)(117,146,126,137)(118,147,127,138)(119,148,128,139)(120,149,129,140), (1,127,27,118)(2,128,28,119)(3,129,29,120)(4,130,30,111)(5,121,21,112)(6,122,22,113)(7,123,23,114)(8,124,24,115)(9,125,25,116)(10,126,26,117)(11,70,151,51)(12,61,152,52)(13,62,153,53)(14,63,154,54)(15,64,155,55)(16,65,156,56)(17,66,157,57)(18,67,158,58)(19,68,159,59)(20,69,160,60)(31,110,50,91)(32,101,41,92)(33,102,42,93)(34,103,43,94)(35,104,44,95)(36,105,45,96)(37,106,46,97)(38,107,47,98)(39,108,48,99)(40,109,49,100)(71,150,90,131)(72,141,81,132)(73,142,82,133)(74,143,83,134)(75,144,84,135)(76,145,85,136)(77,146,86,137)(78,147,87,138)(79,148,88,139)(80,149,89,140), (1,118)(2,119)(3,120)(4,111)(5,112)(6,113)(7,114)(8,115)(9,116)(10,117)(11,70)(12,61)(13,62)(14,63)(15,64)(16,65)(17,66)(18,67)(19,68)(20,69)(21,121)(22,122)(23,123)(24,124)(25,125)(26,126)(27,127)(28,128)(29,129)(30,130)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(37,97)(38,98)(39,99)(40,100)(41,101)(42,102)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,109)(50,110)(51,151)(52,152)(53,153)(54,154)(55,155)(56,156)(57,157)(58,158)(59,159)(60,160)(71,131)(72,132)(73,133)(74,134)(75,135)(76,136)(77,137)(78,138)(79,139)(80,140)(81,141)(82,142)(83,143)(84,144)(85,145)(86,146)(87,147)(88,148)(89,149)(90,150)>;
G:=Group( (1,42)(2,43)(3,44)(4,45)(5,46)(6,47)(7,48)(8,49)(9,50)(10,41)(11,136)(12,137)(13,138)(14,139)(15,140)(16,131)(17,132)(18,133)(19,134)(20,135)(21,37)(22,38)(23,39)(24,40)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(51,85)(52,86)(53,87)(54,88)(55,89)(56,90)(57,81)(58,82)(59,83)(60,84)(61,77)(62,78)(63,79)(64,80)(65,71)(66,72)(67,73)(68,74)(69,75)(70,76)(91,125)(92,126)(93,127)(94,128)(95,129)(96,130)(97,121)(98,122)(99,123)(100,124)(101,117)(102,118)(103,119)(104,120)(105,111)(106,112)(107,113)(108,114)(109,115)(110,116)(141,157)(142,158)(143,159)(144,160)(145,151)(146,152)(147,153)(148,154)(149,155)(150,156), (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), (1,87,27,78)(2,88,28,79)(3,89,29,80)(4,90,30,71)(5,81,21,72)(6,82,22,73)(7,83,23,74)(8,84,24,75)(9,85,25,76)(10,86,26,77)(11,110,151,91)(12,101,152,92)(13,102,153,93)(14,103,154,94)(15,104,155,95)(16,105,156,96)(17,106,157,97)(18,107,158,98)(19,108,159,99)(20,109,160,100)(31,70,50,51)(32,61,41,52)(33,62,42,53)(34,63,43,54)(35,64,44,55)(36,65,45,56)(37,66,46,57)(38,67,47,58)(39,68,48,59)(40,69,49,60)(111,150,130,131)(112,141,121,132)(113,142,122,133)(114,143,123,134)(115,144,124,135)(116,145,125,136)(117,146,126,137)(118,147,127,138)(119,148,128,139)(120,149,129,140), (1,127,27,118)(2,128,28,119)(3,129,29,120)(4,130,30,111)(5,121,21,112)(6,122,22,113)(7,123,23,114)(8,124,24,115)(9,125,25,116)(10,126,26,117)(11,70,151,51)(12,61,152,52)(13,62,153,53)(14,63,154,54)(15,64,155,55)(16,65,156,56)(17,66,157,57)(18,67,158,58)(19,68,159,59)(20,69,160,60)(31,110,50,91)(32,101,41,92)(33,102,42,93)(34,103,43,94)(35,104,44,95)(36,105,45,96)(37,106,46,97)(38,107,47,98)(39,108,48,99)(40,109,49,100)(71,150,90,131)(72,141,81,132)(73,142,82,133)(74,143,83,134)(75,144,84,135)(76,145,85,136)(77,146,86,137)(78,147,87,138)(79,148,88,139)(80,149,89,140), (1,118)(2,119)(3,120)(4,111)(5,112)(6,113)(7,114)(8,115)(9,116)(10,117)(11,70)(12,61)(13,62)(14,63)(15,64)(16,65)(17,66)(18,67)(19,68)(20,69)(21,121)(22,122)(23,123)(24,124)(25,125)(26,126)(27,127)(28,128)(29,129)(30,130)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(37,97)(38,98)(39,99)(40,100)(41,101)(42,102)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,109)(50,110)(51,151)(52,152)(53,153)(54,154)(55,155)(56,156)(57,157)(58,158)(59,159)(60,160)(71,131)(72,132)(73,133)(74,134)(75,135)(76,136)(77,137)(78,138)(79,139)(80,140)(81,141)(82,142)(83,143)(84,144)(85,145)(86,146)(87,147)(88,148)(89,149)(90,150) );
G=PermutationGroup([(1,42),(2,43),(3,44),(4,45),(5,46),(6,47),(7,48),(8,49),(9,50),(10,41),(11,136),(12,137),(13,138),(14,139),(15,140),(16,131),(17,132),(18,133),(19,134),(20,135),(21,37),(22,38),(23,39),(24,40),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(51,85),(52,86),(53,87),(54,88),(55,89),(56,90),(57,81),(58,82),(59,83),(60,84),(61,77),(62,78),(63,79),(64,80),(65,71),(66,72),(67,73),(68,74),(69,75),(70,76),(91,125),(92,126),(93,127),(94,128),(95,129),(96,130),(97,121),(98,122),(99,123),(100,124),(101,117),(102,118),(103,119),(104,120),(105,111),(106,112),(107,113),(108,114),(109,115),(110,116),(141,157),(142,158),(143,159),(144,160),(145,151),(146,152),(147,153),(148,154),(149,155),(150,156)], [(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)], [(1,87,27,78),(2,88,28,79),(3,89,29,80),(4,90,30,71),(5,81,21,72),(6,82,22,73),(7,83,23,74),(8,84,24,75),(9,85,25,76),(10,86,26,77),(11,110,151,91),(12,101,152,92),(13,102,153,93),(14,103,154,94),(15,104,155,95),(16,105,156,96),(17,106,157,97),(18,107,158,98),(19,108,159,99),(20,109,160,100),(31,70,50,51),(32,61,41,52),(33,62,42,53),(34,63,43,54),(35,64,44,55),(36,65,45,56),(37,66,46,57),(38,67,47,58),(39,68,48,59),(40,69,49,60),(111,150,130,131),(112,141,121,132),(113,142,122,133),(114,143,123,134),(115,144,124,135),(116,145,125,136),(117,146,126,137),(118,147,127,138),(119,148,128,139),(120,149,129,140)], [(1,127,27,118),(2,128,28,119),(3,129,29,120),(4,130,30,111),(5,121,21,112),(6,122,22,113),(7,123,23,114),(8,124,24,115),(9,125,25,116),(10,126,26,117),(11,70,151,51),(12,61,152,52),(13,62,153,53),(14,63,154,54),(15,64,155,55),(16,65,156,56),(17,66,157,57),(18,67,158,58),(19,68,159,59),(20,69,160,60),(31,110,50,91),(32,101,41,92),(33,102,42,93),(34,103,43,94),(35,104,44,95),(36,105,45,96),(37,106,46,97),(38,107,47,98),(39,108,48,99),(40,109,49,100),(71,150,90,131),(72,141,81,132),(73,142,82,133),(74,143,83,134),(75,144,84,135),(76,145,85,136),(77,146,86,137),(78,147,87,138),(79,148,88,139),(80,149,89,140)], [(1,118),(2,119),(3,120),(4,111),(5,112),(6,113),(7,114),(8,115),(9,116),(10,117),(11,70),(12,61),(13,62),(14,63),(15,64),(16,65),(17,66),(18,67),(19,68),(20,69),(21,121),(22,122),(23,123),(24,124),(25,125),(26,126),(27,127),(28,128),(29,129),(30,130),(31,91),(32,92),(33,93),(34,94),(35,95),(36,96),(37,97),(38,98),(39,99),(40,100),(41,101),(42,102),(43,103),(44,104),(45,105),(46,106),(47,107),(48,108),(49,109),(50,110),(51,151),(52,152),(53,153),(54,154),(55,155),(56,156),(57,157),(58,158),(59,159),(60,160),(71,131),(72,132),(73,133),(74,134),(75,135),(76,136),(77,137),(78,138),(79,139),(80,140),(81,141),(82,142),(83,143),(84,144),(85,145),(86,146),(87,147),(88,148),(89,149),(90,150)])
Matrix representation ►G ⊆ GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 37 | 0 |
| 0 | 0 | 0 | 37 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 40 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 39 |
| 0 | 0 | 1 | 40 |
| 1 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 39 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,37,0,0,0,0,37],[40,0,0,0,0,40,0,0,0,0,9,0,0,0,0,9],[40,0,0,0,0,1,0,0,0,0,1,1,0,0,39,40],[1,0,0,0,0,40,0,0,0,0,1,0,0,0,39,40] >;
200 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2S | 4A | ··· | 4H | 4I | ··· | 4T | 5A | 5B | 5C | 5D | 10A | ··· | 10AB | 10AC | ··· | 10BX | 20A | ··· | 20AF | 20AG | ··· | 20CB |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
200 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C4○D4 | C5×C4○D4 |
| kernel | C4○D4×C2×C10 | C23×C20 | D4×C2×C10 | Q8×C2×C10 | C10×C4○D4 | C22×C4○D4 | C23×C4 | C22×D4 | C22×Q8 | C2×C4○D4 | C2×C10 | C22 |
| # reps | 1 | 3 | 3 | 1 | 24 | 4 | 12 | 12 | 4 | 96 | 8 | 32 |
In GAP, Magma, Sage, TeX
C_4\circ D_4\times C_2\times C_{10} % in TeX
G:=Group("C4oD4xC2xC10"); // GroupNames label
G:=SmallGroup(320,1631);
// by ID
G=gap.SmallGroup(320,1631);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-5,-2,2269,856]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^10=c^4=e^2=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=c^2*d>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀