metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C20).289D4, (C22×D5).4Q8, C22.50(Q8×D5), C2.7(C20⋊D4), (C2×Dic5).69D4, C22.248(D4×D5), (C22×C4).44D10, C10.16(C4⋊1D4), C5⋊3(C23.4Q8), C2.8(D10⋊3Q8), C10.50(C22⋊Q8), C2.22(D10⋊Q8), (C22×C20).31C22, (C23×D5).21C22, C23.381(C22×D5), C10.10C42⋊43C2, C22.109(C4○D20), (C22×C10).356C23, C22.52(Q8⋊2D5), C2.21(D10.13D4), C10.54(C22.D4), (C22×Dic5).61C22, C2.14(C23.23D10), (C2×C4⋊C4)⋊10D5, (C10×C4⋊C4)⋊26C2, (C2×C10).85(C2×Q8), (C2×C10).453(C2×D4), (C2×C4).42(C5⋊D4), (C2×C10.D4)⋊14C2, C22.141(C2×C5⋊D4), (C2×D10⋊C4).16C2, (C2×C10).191(C4○D4), SmallGroup(320,619)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C20).289D4
G = < a,b,c,d | a2=b20=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=ab-1, dbd=ab9, dcd=ab10c-1 >
Subgroups: 774 in 186 conjugacy classes, 61 normal (25 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×9], C22 [×3], C22 [×4], C22 [×10], C5, C2×C4 [×2], C2×C4 [×19], C23, C23 [×8], D5 [×2], C10 [×3], C10 [×4], C22⋊C4 [×6], C4⋊C4 [×6], C22×C4, C22×C4 [×2], C22×C4 [×3], C24, Dic5 [×5], C20 [×4], D10 [×10], C2×C10 [×3], C2×C10 [×4], C2.C42, C2×C22⋊C4 [×3], C2×C4⋊C4, C2×C4⋊C4 [×2], C2×Dic5 [×4], C2×Dic5 [×7], C2×C20 [×2], C2×C20 [×8], C22×D5 [×2], C22×D5 [×6], C22×C10, C23.4Q8, C10.D4 [×4], D10⋊C4 [×6], C5×C4⋊C4 [×2], C22×Dic5, C22×Dic5 [×2], C22×C20, C22×C20 [×2], C23×D5, C10.10C42, C2×C10.D4 [×2], C2×D10⋊C4, C2×D10⋊C4 [×2], C10×C4⋊C4, (C2×C20).289D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], Q8 [×2], C23, D5, C2×D4 [×3], C2×Q8, C4○D4 [×3], D10 [×3], C22⋊Q8 [×3], C22.D4 [×3], C4⋊1D4, C5⋊D4 [×2], C22×D5, C23.4Q8, C4○D20 [×2], D4×D5 [×2], Q8×D5, Q8⋊2D5, C2×C5⋊D4, D10.13D4 [×2], D10⋊Q8 [×2], C23.23D10, C20⋊D4, D10⋊3Q8, (C2×C20).289D4
►On 160 points
Generators in S160
(1 98)(2 99)(3 100)(4 81)(5 82)(6 83)(7 84)(8 85)(9 86)(10 87)(11 88)(12 89)(13 90)(14 91)(15 92)(16 93)(17 94)(18 95)(19 96)(20 97)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(31 41)(32 42)(33 43)(34 44)(35 45)(36 46)(37 47)(38 48)(39 49)(40 50)(61 139)(62 140)(63 121)(64 122)(65 123)(66 124)(67 125)(68 126)(69 127)(70 128)(71 129)(72 130)(73 131)(74 132)(75 133)(76 134)(77 135)(78 136)(79 137)(80 138)(101 160)(102 141)(103 142)(104 143)(105 144)(106 145)(107 146)(108 147)(109 148)(110 149)(111 150)(112 151)(113 152)(114 153)(115 154)(116 155)(117 156)(118 157)(119 158)(120 159)
(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)
(1 121 29 117)(2 62 30 155)(3 139 31 115)(4 80 32 153)(5 137 33 113)(6 78 34 151)(7 135 35 111)(8 76 36 149)(9 133 37 109)(10 74 38 147)(11 131 39 107)(12 72 40 145)(13 129 21 105)(14 70 22 143)(15 127 23 103)(16 68 24 141)(17 125 25 101)(18 66 26 159)(19 123 27 119)(20 64 28 157)(41 154 100 61)(42 114 81 138)(43 152 82 79)(44 112 83 136)(45 150 84 77)(46 110 85 134)(47 148 86 75)(48 108 87 132)(49 146 88 73)(50 106 89 130)(51 144 90 71)(52 104 91 128)(53 142 92 69)(54 102 93 126)(55 160 94 67)(56 120 95 124)(57 158 96 65)(58 118 97 122)(59 156 98 63)(60 116 99 140)
(2 87)(3 19)(4 85)(5 17)(6 83)(7 15)(8 81)(9 13)(10 99)(12 97)(14 95)(16 93)(18 91)(20 89)(21 37)(22 56)(23 35)(24 54)(25 33)(26 52)(27 31)(28 50)(30 48)(32 46)(34 44)(36 42)(38 60)(40 58)(41 57)(43 55)(45 53)(47 51)(61 109)(62 157)(63 107)(64 155)(65 105)(66 153)(67 103)(68 151)(69 101)(70 149)(71 119)(72 147)(73 117)(74 145)(75 115)(76 143)(77 113)(78 141)(79 111)(80 159)(82 94)(84 92)(86 90)(96 100)(102 136)(104 134)(106 132)(108 130)(110 128)(112 126)(114 124)(116 122)(118 140)(120 138)(121 146)(123 144)(125 142)(127 160)(129 158)(131 156)(133 154)(135 152)(137 150)(139 148)
G:=sub<Sym(160)| (1,98)(2,99)(3,100)(4,81)(5,82)(6,83)(7,84)(8,85)(9,86)(10,87)(11,88)(12,89)(13,90)(14,91)(15,92)(16,93)(17,94)(18,95)(19,96)(20,97)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(31,41)(32,42)(33,43)(34,44)(35,45)(36,46)(37,47)(38,48)(39,49)(40,50)(61,139)(62,140)(63,121)(64,122)(65,123)(66,124)(67,125)(68,126)(69,127)(70,128)(71,129)(72,130)(73,131)(74,132)(75,133)(76,134)(77,135)(78,136)(79,137)(80,138)(101,160)(102,141)(103,142)(104,143)(105,144)(106,145)(107,146)(108,147)(109,148)(110,149)(111,150)(112,151)(113,152)(114,153)(115,154)(116,155)(117,156)(118,157)(119,158)(120,159), (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), (1,121,29,117)(2,62,30,155)(3,139,31,115)(4,80,32,153)(5,137,33,113)(6,78,34,151)(7,135,35,111)(8,76,36,149)(9,133,37,109)(10,74,38,147)(11,131,39,107)(12,72,40,145)(13,129,21,105)(14,70,22,143)(15,127,23,103)(16,68,24,141)(17,125,25,101)(18,66,26,159)(19,123,27,119)(20,64,28,157)(41,154,100,61)(42,114,81,138)(43,152,82,79)(44,112,83,136)(45,150,84,77)(46,110,85,134)(47,148,86,75)(48,108,87,132)(49,146,88,73)(50,106,89,130)(51,144,90,71)(52,104,91,128)(53,142,92,69)(54,102,93,126)(55,160,94,67)(56,120,95,124)(57,158,96,65)(58,118,97,122)(59,156,98,63)(60,116,99,140), (2,87)(3,19)(4,85)(5,17)(6,83)(7,15)(8,81)(9,13)(10,99)(12,97)(14,95)(16,93)(18,91)(20,89)(21,37)(22,56)(23,35)(24,54)(25,33)(26,52)(27,31)(28,50)(30,48)(32,46)(34,44)(36,42)(38,60)(40,58)(41,57)(43,55)(45,53)(47,51)(61,109)(62,157)(63,107)(64,155)(65,105)(66,153)(67,103)(68,151)(69,101)(70,149)(71,119)(72,147)(73,117)(74,145)(75,115)(76,143)(77,113)(78,141)(79,111)(80,159)(82,94)(84,92)(86,90)(96,100)(102,136)(104,134)(106,132)(108,130)(110,128)(112,126)(114,124)(116,122)(118,140)(120,138)(121,146)(123,144)(125,142)(127,160)(129,158)(131,156)(133,154)(135,152)(137,150)(139,148)>;
G:=Group( (1,98)(2,99)(3,100)(4,81)(5,82)(6,83)(7,84)(8,85)(9,86)(10,87)(11,88)(12,89)(13,90)(14,91)(15,92)(16,93)(17,94)(18,95)(19,96)(20,97)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(31,41)(32,42)(33,43)(34,44)(35,45)(36,46)(37,47)(38,48)(39,49)(40,50)(61,139)(62,140)(63,121)(64,122)(65,123)(66,124)(67,125)(68,126)(69,127)(70,128)(71,129)(72,130)(73,131)(74,132)(75,133)(76,134)(77,135)(78,136)(79,137)(80,138)(101,160)(102,141)(103,142)(104,143)(105,144)(106,145)(107,146)(108,147)(109,148)(110,149)(111,150)(112,151)(113,152)(114,153)(115,154)(116,155)(117,156)(118,157)(119,158)(120,159), (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), (1,121,29,117)(2,62,30,155)(3,139,31,115)(4,80,32,153)(5,137,33,113)(6,78,34,151)(7,135,35,111)(8,76,36,149)(9,133,37,109)(10,74,38,147)(11,131,39,107)(12,72,40,145)(13,129,21,105)(14,70,22,143)(15,127,23,103)(16,68,24,141)(17,125,25,101)(18,66,26,159)(19,123,27,119)(20,64,28,157)(41,154,100,61)(42,114,81,138)(43,152,82,79)(44,112,83,136)(45,150,84,77)(46,110,85,134)(47,148,86,75)(48,108,87,132)(49,146,88,73)(50,106,89,130)(51,144,90,71)(52,104,91,128)(53,142,92,69)(54,102,93,126)(55,160,94,67)(56,120,95,124)(57,158,96,65)(58,118,97,122)(59,156,98,63)(60,116,99,140), (2,87)(3,19)(4,85)(5,17)(6,83)(7,15)(8,81)(9,13)(10,99)(12,97)(14,95)(16,93)(18,91)(20,89)(21,37)(22,56)(23,35)(24,54)(25,33)(26,52)(27,31)(28,50)(30,48)(32,46)(34,44)(36,42)(38,60)(40,58)(41,57)(43,55)(45,53)(47,51)(61,109)(62,157)(63,107)(64,155)(65,105)(66,153)(67,103)(68,151)(69,101)(70,149)(71,119)(72,147)(73,117)(74,145)(75,115)(76,143)(77,113)(78,141)(79,111)(80,159)(82,94)(84,92)(86,90)(96,100)(102,136)(104,134)(106,132)(108,130)(110,128)(112,126)(114,124)(116,122)(118,140)(120,138)(121,146)(123,144)(125,142)(127,160)(129,158)(131,156)(133,154)(135,152)(137,150)(139,148) );
G=PermutationGroup([(1,98),(2,99),(3,100),(4,81),(5,82),(6,83),(7,84),(8,85),(9,86),(10,87),(11,88),(12,89),(13,90),(14,91),(15,92),(16,93),(17,94),(18,95),(19,96),(20,97),(21,51),(22,52),(23,53),(24,54),(25,55),(26,56),(27,57),(28,58),(29,59),(30,60),(31,41),(32,42),(33,43),(34,44),(35,45),(36,46),(37,47),(38,48),(39,49),(40,50),(61,139),(62,140),(63,121),(64,122),(65,123),(66,124),(67,125),(68,126),(69,127),(70,128),(71,129),(72,130),(73,131),(74,132),(75,133),(76,134),(77,135),(78,136),(79,137),(80,138),(101,160),(102,141),(103,142),(104,143),(105,144),(106,145),(107,146),(108,147),(109,148),(110,149),(111,150),(112,151),(113,152),(114,153),(115,154),(116,155),(117,156),(118,157),(119,158),(120,159)], [(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)], [(1,121,29,117),(2,62,30,155),(3,139,31,115),(4,80,32,153),(5,137,33,113),(6,78,34,151),(7,135,35,111),(8,76,36,149),(9,133,37,109),(10,74,38,147),(11,131,39,107),(12,72,40,145),(13,129,21,105),(14,70,22,143),(15,127,23,103),(16,68,24,141),(17,125,25,101),(18,66,26,159),(19,123,27,119),(20,64,28,157),(41,154,100,61),(42,114,81,138),(43,152,82,79),(44,112,83,136),(45,150,84,77),(46,110,85,134),(47,148,86,75),(48,108,87,132),(49,146,88,73),(50,106,89,130),(51,144,90,71),(52,104,91,128),(53,142,92,69),(54,102,93,126),(55,160,94,67),(56,120,95,124),(57,158,96,65),(58,118,97,122),(59,156,98,63),(60,116,99,140)], [(2,87),(3,19),(4,85),(5,17),(6,83),(7,15),(8,81),(9,13),(10,99),(12,97),(14,95),(16,93),(18,91),(20,89),(21,37),(22,56),(23,35),(24,54),(25,33),(26,52),(27,31),(28,50),(30,48),(32,46),(34,44),(36,42),(38,60),(40,58),(41,57),(43,55),(45,53),(47,51),(61,109),(62,157),(63,107),(64,155),(65,105),(66,153),(67,103),(68,151),(69,101),(70,149),(71,119),(72,147),(73,117),(74,145),(75,115),(76,143),(77,113),(78,141),(79,111),(80,159),(82,94),(84,92),(86,90),(96,100),(102,136),(104,134),(106,132),(108,130),(110,128),(112,126),(114,124),(116,122),(118,140),(120,138),(121,146),(123,144),(125,142),(127,160),(129,158),(131,156),(133,154),(135,152),(137,150),(139,148)])
62 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | ··· | 4F | 4G | ··· | 4L | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20X |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 20 | 20 | 4 | ··· | 4 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | D4 | D4 | Q8 | D5 | C4○D4 | D10 | C5⋊D4 | C4○D20 | D4×D5 | Q8×D5 | Q8⋊2D5 |
| kernel | (C2×C20).289D4 | C10.10C42 | C2×C10.D4 | C2×D10⋊C4 | C10×C4⋊C4 | C2×Dic5 | C2×C20 | C22×D5 | C2×C4⋊C4 | C2×C10 | C22×C4 | C2×C4 | C22 | C22 | C22 | C22 |
| # reps | 1 | 1 | 2 | 3 | 1 | 4 | 2 | 2 | 2 | 6 | 6 | 8 | 16 | 4 | 2 | 2 |
Matrix representation of (C2×C20).289D4 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 21 | 11 | 0 | 0 | 0 | 0 |
| 12 | 20 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 8 |
| 0 | 0 | 0 | 0 | 31 | 32 |
| 40 | 16 | 0 | 0 | 0 | 0 |
| 5 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 20 | 0 | 0 |
| 0 | 0 | 35 | 23 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 10 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 6 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 8 | 40 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[21,12,0,0,0,0,11,20,0,0,0,0,0,0,0,1,0,0,0,0,40,6,0,0,0,0,0,0,9,31,0,0,0,0,8,32],[40,5,0,0,0,0,16,1,0,0,0,0,0,0,18,35,0,0,0,0,20,23,0,0,0,0,0,0,1,0,0,0,0,0,10,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,6,40,0,0,0,0,0,0,1,8,0,0,0,0,0,40] >;
(C2×C20).289D4 in GAP, Magma, Sage, TeX
(C_2\times C_{20})._{289}D_4 % in TeX
G:=Group("(C2xC20).289D4"); // GroupNames label
G:=SmallGroup(320,619);
// by ID
G=gap.SmallGroup(320,619);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,232,254,387,268,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^20=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a*b^-1,d*b*d=a*b^9,d*c*d=a*b^10*c^-1>;
// generators/relations