direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C20.17D4, C24.37D10, C20.251(C2×D4), (C2×C20).209D4, (C2×D4).229D10, C10⋊3(C4.4D4), (C22×D4).11D5, (C2×C20).540C23, (C2×C10).292C24, (C4×Dic5)⋊67C22, (C22×C4).378D10, C10.140(C22×D4), C23.D5⋊58C22, (C22×Dic10)⋊20C2, (C2×Dic10)⋊67C22, (D4×C10).269C22, (C23×C10).74C22, C23.134(C22×D5), C22.306(C23×D5), C22.78(D4⋊2D5), (C22×C10).228C23, (C22×C20).273C22, (C2×Dic5).292C23, (C22×Dic5).254C22, (D4×C2×C10).8C2, C5⋊4(C2×C4.4D4), (C2×C4×Dic5)⋊11C2, C4.23(C2×C5⋊D4), C10.104(C2×C4○D4), (C2×C10).579(C2×D4), C2.68(C2×D4⋊2D5), (C2×C23.D5)⋊25C2, C2.13(C22×C5⋊D4), (C2×C4).153(C5⋊D4), (C2×C4).623(C22×D5), C22.109(C2×C5⋊D4), (C2×C10).176(C4○D4), SmallGroup(320,1469)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 958 in 330 conjugacy classes, 127 normal (15 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×4], C4 [×8], C22, C22 [×6], C22 [×20], C5, C2×C4 [×6], C2×C4 [×16], D4 [×8], Q8 [×8], C23, C23 [×4], C23 [×12], C10, C10 [×6], C10 [×4], C42 [×4], C22⋊C4 [×16], C22×C4, C22×C4 [×4], C2×D4 [×4], C2×D4 [×4], C2×Q8 [×8], C24 [×2], Dic5 [×8], C20 [×4], C2×C10, C2×C10 [×6], C2×C10 [×20], C2×C42, C2×C22⋊C4 [×4], C4.4D4 [×8], C22×D4, C22×Q8, Dic10 [×8], C2×Dic5 [×8], C2×Dic5 [×8], C2×C20 [×6], C5×D4 [×8], C22×C10, C22×C10 [×4], C22×C10 [×12], C2×C4.4D4, C4×Dic5 [×4], C23.D5 [×16], C2×Dic10 [×4], C2×Dic10 [×4], C22×Dic5 [×4], C22×C20, D4×C10 [×4], D4×C10 [×4], C23×C10 [×2], C2×C4×Dic5, C20.17D4 [×8], C2×C23.D5 [×4], C22×Dic10, D4×C2×C10, C2×C20.17D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×4], C24, D10 [×7], C4.4D4 [×4], C22×D4, C2×C4○D4 [×2], C5⋊D4 [×4], C22×D5 [×7], C2×C4.4D4, D4⋊2D5 [×4], C2×C5⋊D4 [×6], C23×D5, C20.17D4 [×4], C2×D4⋊2D5 [×2], C22×C5⋊D4, C2×C20.17D4
Generators and relations
G = < a,b,c,d | a2=b20=c4=1, d2=b10, ab=ba, ac=ca, ad=da, cbc-1=b9, dbd-1=b-1, dcd-1=b10c-1 >
►On 160 points
Generators in S160
(1 113)(2 114)(3 115)(4 116)(5 117)(6 118)(7 119)(8 120)(9 101)(10 102)(11 103)(12 104)(13 105)(14 106)(15 107)(16 108)(17 109)(18 110)(19 111)(20 112)(21 138)(22 139)(23 140)(24 121)(25 122)(26 123)(27 124)(28 125)(29 126)(30 127)(31 128)(32 129)(33 130)(34 131)(35 132)(36 133)(37 134)(38 135)(39 136)(40 137)(41 153)(42 154)(43 155)(44 156)(45 157)(46 158)(47 159)(48 160)(49 141)(50 142)(51 143)(52 144)(53 145)(54 146)(55 147)(56 148)(57 149)(58 150)(59 151)(60 152)(61 85)(62 86)(63 87)(64 88)(65 89)(66 90)(67 91)(68 92)(69 93)(70 94)(71 95)(72 96)(73 97)(74 98)(75 99)(76 100)(77 81)(78 82)(79 83)(80 84)
(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 38 77 152)(2 27 78 141)(3 36 79 150)(4 25 80 159)(5 34 61 148)(6 23 62 157)(7 32 63 146)(8 21 64 155)(9 30 65 144)(10 39 66 153)(11 28 67 142)(12 37 68 151)(13 26 69 160)(14 35 70 149)(15 24 71 158)(16 33 72 147)(17 22 73 156)(18 31 74 145)(19 40 75 154)(20 29 76 143)(41 102 136 90)(42 111 137 99)(43 120 138 88)(44 109 139 97)(45 118 140 86)(46 107 121 95)(47 116 122 84)(48 105 123 93)(49 114 124 82)(50 103 125 91)(51 112 126 100)(52 101 127 89)(53 110 128 98)(54 119 129 87)(55 108 130 96)(56 117 131 85)(57 106 132 94)(58 115 133 83)(59 104 134 92)(60 113 135 81)
(1 135 11 125)(2 134 12 124)(3 133 13 123)(4 132 14 122)(5 131 15 121)(6 130 16 140)(7 129 17 139)(8 128 18 138)(9 127 19 137)(10 126 20 136)(21 120 31 110)(22 119 32 109)(23 118 33 108)(24 117 34 107)(25 116 35 106)(26 115 36 105)(27 114 37 104)(28 113 38 103)(29 112 39 102)(30 111 40 101)(41 66 51 76)(42 65 52 75)(43 64 53 74)(44 63 54 73)(45 62 55 72)(46 61 56 71)(47 80 57 70)(48 79 58 69)(49 78 59 68)(50 77 60 67)(81 152 91 142)(82 151 92 141)(83 150 93 160)(84 149 94 159)(85 148 95 158)(86 147 96 157)(87 146 97 156)(88 145 98 155)(89 144 99 154)(90 143 100 153)
G:=sub<Sym(160)| (1,113)(2,114)(3,115)(4,116)(5,117)(6,118)(7,119)(8,120)(9,101)(10,102)(11,103)(12,104)(13,105)(14,106)(15,107)(16,108)(17,109)(18,110)(19,111)(20,112)(21,138)(22,139)(23,140)(24,121)(25,122)(26,123)(27,124)(28,125)(29,126)(30,127)(31,128)(32,129)(33,130)(34,131)(35,132)(36,133)(37,134)(38,135)(39,136)(40,137)(41,153)(42,154)(43,155)(44,156)(45,157)(46,158)(47,159)(48,160)(49,141)(50,142)(51,143)(52,144)(53,145)(54,146)(55,147)(56,148)(57,149)(58,150)(59,151)(60,152)(61,85)(62,86)(63,87)(64,88)(65,89)(66,90)(67,91)(68,92)(69,93)(70,94)(71,95)(72,96)(73,97)(74,98)(75,99)(76,100)(77,81)(78,82)(79,83)(80,84), (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), (1,38,77,152)(2,27,78,141)(3,36,79,150)(4,25,80,159)(5,34,61,148)(6,23,62,157)(7,32,63,146)(8,21,64,155)(9,30,65,144)(10,39,66,153)(11,28,67,142)(12,37,68,151)(13,26,69,160)(14,35,70,149)(15,24,71,158)(16,33,72,147)(17,22,73,156)(18,31,74,145)(19,40,75,154)(20,29,76,143)(41,102,136,90)(42,111,137,99)(43,120,138,88)(44,109,139,97)(45,118,140,86)(46,107,121,95)(47,116,122,84)(48,105,123,93)(49,114,124,82)(50,103,125,91)(51,112,126,100)(52,101,127,89)(53,110,128,98)(54,119,129,87)(55,108,130,96)(56,117,131,85)(57,106,132,94)(58,115,133,83)(59,104,134,92)(60,113,135,81), (1,135,11,125)(2,134,12,124)(3,133,13,123)(4,132,14,122)(5,131,15,121)(6,130,16,140)(7,129,17,139)(8,128,18,138)(9,127,19,137)(10,126,20,136)(21,120,31,110)(22,119,32,109)(23,118,33,108)(24,117,34,107)(25,116,35,106)(26,115,36,105)(27,114,37,104)(28,113,38,103)(29,112,39,102)(30,111,40,101)(41,66,51,76)(42,65,52,75)(43,64,53,74)(44,63,54,73)(45,62,55,72)(46,61,56,71)(47,80,57,70)(48,79,58,69)(49,78,59,68)(50,77,60,67)(81,152,91,142)(82,151,92,141)(83,150,93,160)(84,149,94,159)(85,148,95,158)(86,147,96,157)(87,146,97,156)(88,145,98,155)(89,144,99,154)(90,143,100,153)>;
G:=Group( (1,113)(2,114)(3,115)(4,116)(5,117)(6,118)(7,119)(8,120)(9,101)(10,102)(11,103)(12,104)(13,105)(14,106)(15,107)(16,108)(17,109)(18,110)(19,111)(20,112)(21,138)(22,139)(23,140)(24,121)(25,122)(26,123)(27,124)(28,125)(29,126)(30,127)(31,128)(32,129)(33,130)(34,131)(35,132)(36,133)(37,134)(38,135)(39,136)(40,137)(41,153)(42,154)(43,155)(44,156)(45,157)(46,158)(47,159)(48,160)(49,141)(50,142)(51,143)(52,144)(53,145)(54,146)(55,147)(56,148)(57,149)(58,150)(59,151)(60,152)(61,85)(62,86)(63,87)(64,88)(65,89)(66,90)(67,91)(68,92)(69,93)(70,94)(71,95)(72,96)(73,97)(74,98)(75,99)(76,100)(77,81)(78,82)(79,83)(80,84), (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), (1,38,77,152)(2,27,78,141)(3,36,79,150)(4,25,80,159)(5,34,61,148)(6,23,62,157)(7,32,63,146)(8,21,64,155)(9,30,65,144)(10,39,66,153)(11,28,67,142)(12,37,68,151)(13,26,69,160)(14,35,70,149)(15,24,71,158)(16,33,72,147)(17,22,73,156)(18,31,74,145)(19,40,75,154)(20,29,76,143)(41,102,136,90)(42,111,137,99)(43,120,138,88)(44,109,139,97)(45,118,140,86)(46,107,121,95)(47,116,122,84)(48,105,123,93)(49,114,124,82)(50,103,125,91)(51,112,126,100)(52,101,127,89)(53,110,128,98)(54,119,129,87)(55,108,130,96)(56,117,131,85)(57,106,132,94)(58,115,133,83)(59,104,134,92)(60,113,135,81), (1,135,11,125)(2,134,12,124)(3,133,13,123)(4,132,14,122)(5,131,15,121)(6,130,16,140)(7,129,17,139)(8,128,18,138)(9,127,19,137)(10,126,20,136)(21,120,31,110)(22,119,32,109)(23,118,33,108)(24,117,34,107)(25,116,35,106)(26,115,36,105)(27,114,37,104)(28,113,38,103)(29,112,39,102)(30,111,40,101)(41,66,51,76)(42,65,52,75)(43,64,53,74)(44,63,54,73)(45,62,55,72)(46,61,56,71)(47,80,57,70)(48,79,58,69)(49,78,59,68)(50,77,60,67)(81,152,91,142)(82,151,92,141)(83,150,93,160)(84,149,94,159)(85,148,95,158)(86,147,96,157)(87,146,97,156)(88,145,98,155)(89,144,99,154)(90,143,100,153) );
G=PermutationGroup([(1,113),(2,114),(3,115),(4,116),(5,117),(6,118),(7,119),(8,120),(9,101),(10,102),(11,103),(12,104),(13,105),(14,106),(15,107),(16,108),(17,109),(18,110),(19,111),(20,112),(21,138),(22,139),(23,140),(24,121),(25,122),(26,123),(27,124),(28,125),(29,126),(30,127),(31,128),(32,129),(33,130),(34,131),(35,132),(36,133),(37,134),(38,135),(39,136),(40,137),(41,153),(42,154),(43,155),(44,156),(45,157),(46,158),(47,159),(48,160),(49,141),(50,142),(51,143),(52,144),(53,145),(54,146),(55,147),(56,148),(57,149),(58,150),(59,151),(60,152),(61,85),(62,86),(63,87),(64,88),(65,89),(66,90),(67,91),(68,92),(69,93),(70,94),(71,95),(72,96),(73,97),(74,98),(75,99),(76,100),(77,81),(78,82),(79,83),(80,84)], [(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)], [(1,38,77,152),(2,27,78,141),(3,36,79,150),(4,25,80,159),(5,34,61,148),(6,23,62,157),(7,32,63,146),(8,21,64,155),(9,30,65,144),(10,39,66,153),(11,28,67,142),(12,37,68,151),(13,26,69,160),(14,35,70,149),(15,24,71,158),(16,33,72,147),(17,22,73,156),(18,31,74,145),(19,40,75,154),(20,29,76,143),(41,102,136,90),(42,111,137,99),(43,120,138,88),(44,109,139,97),(45,118,140,86),(46,107,121,95),(47,116,122,84),(48,105,123,93),(49,114,124,82),(50,103,125,91),(51,112,126,100),(52,101,127,89),(53,110,128,98),(54,119,129,87),(55,108,130,96),(56,117,131,85),(57,106,132,94),(58,115,133,83),(59,104,134,92),(60,113,135,81)], [(1,135,11,125),(2,134,12,124),(3,133,13,123),(4,132,14,122),(5,131,15,121),(6,130,16,140),(7,129,17,139),(8,128,18,138),(9,127,19,137),(10,126,20,136),(21,120,31,110),(22,119,32,109),(23,118,33,108),(24,117,34,107),(25,116,35,106),(26,115,36,105),(27,114,37,104),(28,113,38,103),(29,112,39,102),(30,111,40,101),(41,66,51,76),(42,65,52,75),(43,64,53,74),(44,63,54,73),(45,62,55,72),(46,61,56,71),(47,80,57,70),(48,79,58,69),(49,78,59,68),(50,77,60,67),(81,152,91,142),(82,151,92,141),(83,150,93,160),(84,149,94,159),(85,148,95,158),(86,147,96,157),(87,146,97,156),(88,145,98,155),(89,144,99,154),(90,143,100,153)])
Matrix representation ►G ⊆ GL5(𝔽41)
| 40 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 23 | 0 |
| 0 | 0 | 0 | 18 | 25 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 23 | 39 |
| 0 | 0 | 0 | 19 | 18 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 23 | 39 |
| 0 | 0 | 0 | 18 | 18 |
G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,1,0,0,0,0,0,0,23,18,0,0,0,0,25],[1,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,23,19,0,0,0,39,18],[1,0,0,0,0,0,9,0,0,0,0,0,32,0,0,0,0,0,23,18,0,0,0,39,18] >;
68 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 4M | 4N | 4O | 4P | 5A | 5B | 10A | ··· | 10N | 10O | ··· | 10AD | 20A | ··· | 20H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | D10 | C5⋊D4 | D4⋊2D5 |
| kernel | C2×C20.17D4 | C2×C4×Dic5 | C20.17D4 | C2×C23.D5 | C22×Dic10 | D4×C2×C10 | C2×C20 | C22×D4 | C2×C10 | C22×C4 | C2×D4 | C24 | C2×C4 | C22 |
| # reps | 1 | 1 | 8 | 4 | 1 | 1 | 4 | 2 | 8 | 2 | 8 | 4 | 16 | 8 |
In GAP, Magma, Sage, TeX
C_2\times C_{20}._{17}D_4 % in TeX
G:=Group("C2xC20.17D4"); // GroupNames label
G:=SmallGroup(320,1469);
// by ID
G=gap.SmallGroup(320,1469);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,100,1571,185,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^20=c^4=1,d^2=b^10,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^9,d*b*d^-1=b^-1,d*c*d^-1=b^10*c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀