direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C4○D4×C2×C10, C10.23C25, C20.89C24, (C23×C4)⋊8C10, (C23×C20)⋊17C2, (C2×C20)⋊18C23, D4⋊3(C22×C10), (C5×D4)⋊14C23, 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
Generators and relations for C4○D4×C2×C10
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 >
Subgroups: 1010 in 890 conjugacy classes, 770 normal (12 characteristic)
C1, C2, C2, C2, C4, C22, C22, C5, C2×C4, D4, Q8, C23, C23, C23, C10, C10, C10, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, C20, C2×C10, C2×C10, C23×C4, C22×D4, C22×Q8, C2×C4○D4, C2×C20, C5×D4, C5×Q8, C22×C10, C22×C10, C22×C10, C22×C4○D4, C22×C20, C22×C20, D4×C10, Q8×C10, C5×C4○D4, C23×C10, C23×C20, D4×C2×C10, Q8×C2×C10, C10×C4○D4, C4○D4×C2×C10
Quotients: C1, C2, C22, C5, C23, C10, C4○D4, C24, C2×C10, C2×C4○D4, C25, C22×C10, C22×C4○D4, C5×C4○D4, C23×C10, C10×C4○D4, C24×C10, C4○D4×C2×C10
►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)(11 134)(12 135)(13 136)(14 137)(15 138)(16 139)(17 140)(18 131)(19 132)(20 133)(21 35)(22 36)(23 37)(24 38)(25 39)(26 40)(27 31)(28 32)(29 33)(30 34)(51 87)(52 88)(53 89)(54 90)(55 81)(56 82)(57 83)(58 84)(59 85)(60 86)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 71)(68 72)(69 73)(70 74)(91 127)(92 128)(93 129)(94 130)(95 121)(96 122)(97 123)(98 124)(99 125)(100 126)(101 115)(102 116)(103 117)(104 118)(105 119)(106 120)(107 111)(108 112)(109 113)(110 114)(141 155)(142 156)(143 157)(144 158)(145 159)(146 160)(147 151)(148 152)(149 153)(150 154)
(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 76)(2 88 28 77)(3 89 29 78)(4 90 30 79)(5 81 21 80)(6 82 22 71)(7 83 23 72)(8 84 24 73)(9 85 25 74)(10 86 26 75)(11 110 159 99)(12 101 160 100)(13 102 151 91)(14 103 152 92)(15 104 153 93)(16 105 154 94)(17 106 155 95)(18 107 156 96)(19 108 157 97)(20 109 158 98)(31 62 42 51)(32 63 43 52)(33 64 44 53)(34 65 45 54)(35 66 46 55)(36 67 47 56)(37 68 48 57)(38 69 49 58)(39 70 50 59)(40 61 41 60)(111 142 122 131)(112 143 123 132)(113 144 124 133)(114 145 125 134)(115 146 126 135)(116 147 127 136)(117 148 128 137)(118 149 129 138)(119 150 130 139)(120 141 121 140)
(1 127 27 116)(2 128 28 117)(3 129 29 118)(4 130 30 119)(5 121 21 120)(6 122 22 111)(7 123 23 112)(8 124 24 113)(9 125 25 114)(10 126 26 115)(11 70 159 59)(12 61 160 60)(13 62 151 51)(14 63 152 52)(15 64 153 53)(16 65 154 54)(17 66 155 55)(18 67 156 56)(19 68 157 57)(20 69 158 58)(31 102 42 91)(32 103 43 92)(33 104 44 93)(34 105 45 94)(35 106 46 95)(36 107 47 96)(37 108 48 97)(38 109 49 98)(39 110 50 99)(40 101 41 100)(71 142 82 131)(72 143 83 132)(73 144 84 133)(74 145 85 134)(75 146 86 135)(76 147 87 136)(77 148 88 137)(78 149 89 138)(79 150 90 139)(80 141 81 140)
(1 116)(2 117)(3 118)(4 119)(5 120)(6 111)(7 112)(8 113)(9 114)(10 115)(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)(11,134)(12,135)(13,136)(14,137)(15,138)(16,139)(17,140)(18,131)(19,132)(20,133)(21,35)(22,36)(23,37)(24,38)(25,39)(26,40)(27,31)(28,32)(29,33)(30,34)(51,87)(52,88)(53,89)(54,90)(55,81)(56,82)(57,83)(58,84)(59,85)(60,86)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,71)(68,72)(69,73)(70,74)(91,127)(92,128)(93,129)(94,130)(95,121)(96,122)(97,123)(98,124)(99,125)(100,126)(101,115)(102,116)(103,117)(104,118)(105,119)(106,120)(107,111)(108,112)(109,113)(110,114)(141,155)(142,156)(143,157)(144,158)(145,159)(146,160)(147,151)(148,152)(149,153)(150,154), (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), (1,87,27,76)(2,88,28,77)(3,89,29,78)(4,90,30,79)(5,81,21,80)(6,82,22,71)(7,83,23,72)(8,84,24,73)(9,85,25,74)(10,86,26,75)(11,110,159,99)(12,101,160,100)(13,102,151,91)(14,103,152,92)(15,104,153,93)(16,105,154,94)(17,106,155,95)(18,107,156,96)(19,108,157,97)(20,109,158,98)(31,62,42,51)(32,63,43,52)(33,64,44,53)(34,65,45,54)(35,66,46,55)(36,67,47,56)(37,68,48,57)(38,69,49,58)(39,70,50,59)(40,61,41,60)(111,142,122,131)(112,143,123,132)(113,144,124,133)(114,145,125,134)(115,146,126,135)(116,147,127,136)(117,148,128,137)(118,149,129,138)(119,150,130,139)(120,141,121,140), (1,127,27,116)(2,128,28,117)(3,129,29,118)(4,130,30,119)(5,121,21,120)(6,122,22,111)(7,123,23,112)(8,124,24,113)(9,125,25,114)(10,126,26,115)(11,70,159,59)(12,61,160,60)(13,62,151,51)(14,63,152,52)(15,64,153,53)(16,65,154,54)(17,66,155,55)(18,67,156,56)(19,68,157,57)(20,69,158,58)(31,102,42,91)(32,103,43,92)(33,104,44,93)(34,105,45,94)(35,106,46,95)(36,107,47,96)(37,108,48,97)(38,109,49,98)(39,110,50,99)(40,101,41,100)(71,142,82,131)(72,143,83,132)(73,144,84,133)(74,145,85,134)(75,146,86,135)(76,147,87,136)(77,148,88,137)(78,149,89,138)(79,150,90,139)(80,141,81,140), (1,116)(2,117)(3,118)(4,119)(5,120)(6,111)(7,112)(8,113)(9,114)(10,115)(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)(11,134)(12,135)(13,136)(14,137)(15,138)(16,139)(17,140)(18,131)(19,132)(20,133)(21,35)(22,36)(23,37)(24,38)(25,39)(26,40)(27,31)(28,32)(29,33)(30,34)(51,87)(52,88)(53,89)(54,90)(55,81)(56,82)(57,83)(58,84)(59,85)(60,86)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,71)(68,72)(69,73)(70,74)(91,127)(92,128)(93,129)(94,130)(95,121)(96,122)(97,123)(98,124)(99,125)(100,126)(101,115)(102,116)(103,117)(104,118)(105,119)(106,120)(107,111)(108,112)(109,113)(110,114)(141,155)(142,156)(143,157)(144,158)(145,159)(146,160)(147,151)(148,152)(149,153)(150,154), (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), (1,87,27,76)(2,88,28,77)(3,89,29,78)(4,90,30,79)(5,81,21,80)(6,82,22,71)(7,83,23,72)(8,84,24,73)(9,85,25,74)(10,86,26,75)(11,110,159,99)(12,101,160,100)(13,102,151,91)(14,103,152,92)(15,104,153,93)(16,105,154,94)(17,106,155,95)(18,107,156,96)(19,108,157,97)(20,109,158,98)(31,62,42,51)(32,63,43,52)(33,64,44,53)(34,65,45,54)(35,66,46,55)(36,67,47,56)(37,68,48,57)(38,69,49,58)(39,70,50,59)(40,61,41,60)(111,142,122,131)(112,143,123,132)(113,144,124,133)(114,145,125,134)(115,146,126,135)(116,147,127,136)(117,148,128,137)(118,149,129,138)(119,150,130,139)(120,141,121,140), (1,127,27,116)(2,128,28,117)(3,129,29,118)(4,130,30,119)(5,121,21,120)(6,122,22,111)(7,123,23,112)(8,124,24,113)(9,125,25,114)(10,126,26,115)(11,70,159,59)(12,61,160,60)(13,62,151,51)(14,63,152,52)(15,64,153,53)(16,65,154,54)(17,66,155,55)(18,67,156,56)(19,68,157,57)(20,69,158,58)(31,102,42,91)(32,103,43,92)(33,104,44,93)(34,105,45,94)(35,106,46,95)(36,107,47,96)(37,108,48,97)(38,109,49,98)(39,110,50,99)(40,101,41,100)(71,142,82,131)(72,143,83,132)(73,144,84,133)(74,145,85,134)(75,146,86,135)(76,147,87,136)(77,148,88,137)(78,149,89,138)(79,150,90,139)(80,141,81,140), (1,116)(2,117)(3,118)(4,119)(5,120)(6,111)(7,112)(8,113)(9,114)(10,115)(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),(11,134),(12,135),(13,136),(14,137),(15,138),(16,139),(17,140),(18,131),(19,132),(20,133),(21,35),(22,36),(23,37),(24,38),(25,39),(26,40),(27,31),(28,32),(29,33),(30,34),(51,87),(52,88),(53,89),(54,90),(55,81),(56,82),(57,83),(58,84),(59,85),(60,86),(61,75),(62,76),(63,77),(64,78),(65,79),(66,80),(67,71),(68,72),(69,73),(70,74),(91,127),(92,128),(93,129),(94,130),(95,121),(96,122),(97,123),(98,124),(99,125),(100,126),(101,115),(102,116),(103,117),(104,118),(105,119),(106,120),(107,111),(108,112),(109,113),(110,114),(141,155),(142,156),(143,157),(144,158),(145,159),(146,160),(147,151),(148,152),(149,153),(150,154)], [(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)], [(1,87,27,76),(2,88,28,77),(3,89,29,78),(4,90,30,79),(5,81,21,80),(6,82,22,71),(7,83,23,72),(8,84,24,73),(9,85,25,74),(10,86,26,75),(11,110,159,99),(12,101,160,100),(13,102,151,91),(14,103,152,92),(15,104,153,93),(16,105,154,94),(17,106,155,95),(18,107,156,96),(19,108,157,97),(20,109,158,98),(31,62,42,51),(32,63,43,52),(33,64,44,53),(34,65,45,54),(35,66,46,55),(36,67,47,56),(37,68,48,57),(38,69,49,58),(39,70,50,59),(40,61,41,60),(111,142,122,131),(112,143,123,132),(113,144,124,133),(114,145,125,134),(115,146,126,135),(116,147,127,136),(117,148,128,137),(118,149,129,138),(119,150,130,139),(120,141,121,140)], [(1,127,27,116),(2,128,28,117),(3,129,29,118),(4,130,30,119),(5,121,21,120),(6,122,22,111),(7,123,23,112),(8,124,24,113),(9,125,25,114),(10,126,26,115),(11,70,159,59),(12,61,160,60),(13,62,151,51),(14,63,152,52),(15,64,153,53),(16,65,154,54),(17,66,155,55),(18,67,156,56),(19,68,157,57),(20,69,158,58),(31,102,42,91),(32,103,43,92),(33,104,44,93),(34,105,45,94),(35,106,46,95),(36,107,47,96),(37,108,48,97),(38,109,49,98),(39,110,50,99),(40,101,41,100),(71,142,82,131),(72,143,83,132),(73,144,84,133),(74,145,85,134),(75,146,86,135),(76,147,87,136),(77,148,88,137),(78,149,89,138),(79,150,90,139),(80,141,81,140)], [(1,116),(2,117),(3,118),(4,119),(5,120),(6,111),(7,112),(8,113),(9,114),(10,115),(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)]])
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 |
Matrix representation of C4○D4×C2×C10 ►in 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] >;
C4○D4×C2×C10 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