metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4.89(C2×D20), (C2×D20).26C4, (C2×C20).171D4, (C2×C8).189D10, (C2×C4).152D20, C20.444(C2×D4), (C2×M4(2))⋊9D5, D10⋊1C8⋊40C2, C23.29(C4×D5), C10.58(C8○D4), C20.73(C22⋊C4), (C10×M4(2))⋊17C2, (C2×C20).869C23, (C2×C40).319C22, (C2×Dic10).27C4, (C22×C4).350D10, C4.12(D10⋊C4), C2.19(D20.2C4), C22.2(D10⋊C4), (C22×C20).186C22, (C2×C4).84(C4×D5), (C22×C5⋊2C8)⋊6C2, (C2×C5⋊D4).21C4, C4.135(C2×C5⋊D4), C22.148(C2×C4×D5), (C2×C20).279(C2×C4), C5⋊6((C22×C8)⋊C2), (C2×C4○D20).11C2, C10.97(C2×C22⋊C4), (C2×C4×D5).237C22, C2.28(C2×D10⋊C4), (C2×C4).141(C5⋊D4), (C2×Dic5).37(C2×C4), (C22×D5).31(C2×C4), (C2×C4).811(C22×D5), (C2×C10).84(C22⋊C4), (C2×C10).240(C22×C4), (C22×C10).137(C2×C4), (C2×C5⋊2C8).333C22, SmallGroup(320,756)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4.89(C2×D20)
G = < a,b,c,d | a4=b2=1, c20=a2, d2=a, ab=ba, ac=ca, ad=da, cbc-1=a2b, bd=db, dcd-1=a-1c19 >
Subgroups: 574 in 158 conjugacy classes, 63 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×8], C5, C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×6], D4 [×6], Q8 [×2], C23, C23 [×2], D5 [×2], C10, C10 [×2], C10 [×2], C2×C8 [×2], C2×C8 [×4], M4(2) [×2], C22×C4, C22×C4 [×2], C2×D4 [×3], C2×Q8, C4○D4 [×4], Dic5 [×2], C20 [×2], C20 [×2], D10 [×6], C2×C10, C2×C10 [×2], C2×C10 [×2], C22⋊C8 [×4], C22×C8, C2×M4(2), C2×C4○D4, C5⋊2C8 [×2], C40 [×2], Dic10 [×2], C4×D5 [×4], D20 [×2], C2×Dic5 [×2], C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×4], C22×D5 [×2], C22×C10, (C22×C8)⋊C2, C2×C5⋊2C8 [×2], C2×C5⋊2C8 [×2], C2×C40 [×2], C5×M4(2) [×2], C2×Dic10, C2×C4×D5 [×2], C2×D20, C4○D20 [×4], C2×C5⋊D4 [×2], C22×C20, D10⋊1C8 [×4], C22×C5⋊2C8, C10×M4(2), C2×C4○D20, C4.89(C2×D20)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], D10 [×3], C2×C22⋊C4, C8○D4 [×2], C4×D5 [×2], D20 [×2], C5⋊D4 [×2], C22×D5, (C22×C8)⋊C2, D10⋊C4 [×4], C2×C4×D5, C2×D20, C2×C5⋊D4, D20.2C4 [×2], C2×D10⋊C4, C4.89(C2×D20)
►On 160 points
Generators in S160
(1 117 21 97)(2 118 22 98)(3 119 23 99)(4 120 24 100)(5 81 25 101)(6 82 26 102)(7 83 27 103)(8 84 28 104)(9 85 29 105)(10 86 30 106)(11 87 31 107)(12 88 32 108)(13 89 33 109)(14 90 34 110)(15 91 35 111)(16 92 36 112)(17 93 37 113)(18 94 38 114)(19 95 39 115)(20 96 40 116)(41 142 61 122)(42 143 62 123)(43 144 63 124)(44 145 64 125)(45 146 65 126)(46 147 66 127)(47 148 67 128)(48 149 68 129)(49 150 69 130)(50 151 70 131)(51 152 71 132)(52 153 72 133)(53 154 73 134)(54 155 74 135)(55 156 75 136)(56 157 76 137)(57 158 77 138)(58 159 78 139)(59 160 79 140)(60 121 80 141)
(1 130)(2 151)(3 132)(4 153)(5 134)(6 155)(7 136)(8 157)(9 138)(10 159)(11 140)(12 121)(13 142)(14 123)(15 144)(16 125)(17 146)(18 127)(19 148)(20 129)(21 150)(22 131)(23 152)(24 133)(25 154)(26 135)(27 156)(28 137)(29 158)(30 139)(31 160)(32 141)(33 122)(34 143)(35 124)(36 145)(37 126)(38 147)(39 128)(40 149)(41 109)(42 90)(43 111)(44 92)(45 113)(46 94)(47 115)(48 96)(49 117)(50 98)(51 119)(52 100)(53 81)(54 102)(55 83)(56 104)(57 85)(58 106)(59 87)(60 108)(61 89)(62 110)(63 91)(64 112)(65 93)(66 114)(67 95)(68 116)(69 97)(70 118)(71 99)(72 120)(73 101)(74 82)(75 103)(76 84)(77 105)(78 86)(79 107)(80 88)
(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)
(1 116 117 20 21 96 97 40)(2 19 118 95 22 39 98 115)(3 94 119 38 23 114 99 18)(4 37 120 113 24 17 100 93)(5 112 81 16 25 92 101 36)(6 15 82 91 26 35 102 111)(7 90 83 34 27 110 103 14)(8 33 84 109 28 13 104 89)(9 108 85 12 29 88 105 32)(10 11 86 87 30 31 106 107)(41 137 142 56 61 157 122 76)(42 55 143 156 62 75 123 136)(43 155 144 74 63 135 124 54)(44 73 145 134 64 53 125 154)(45 133 146 52 65 153 126 72)(46 51 147 152 66 71 127 132)(47 151 148 70 67 131 128 50)(48 69 149 130 68 49 129 150)(57 121 158 80 77 141 138 60)(58 79 159 140 78 59 139 160)
G:=sub<Sym(160)| (1,117,21,97)(2,118,22,98)(3,119,23,99)(4,120,24,100)(5,81,25,101)(6,82,26,102)(7,83,27,103)(8,84,28,104)(9,85,29,105)(10,86,30,106)(11,87,31,107)(12,88,32,108)(13,89,33,109)(14,90,34,110)(15,91,35,111)(16,92,36,112)(17,93,37,113)(18,94,38,114)(19,95,39,115)(20,96,40,116)(41,142,61,122)(42,143,62,123)(43,144,63,124)(44,145,64,125)(45,146,65,126)(46,147,66,127)(47,148,67,128)(48,149,68,129)(49,150,69,130)(50,151,70,131)(51,152,71,132)(52,153,72,133)(53,154,73,134)(54,155,74,135)(55,156,75,136)(56,157,76,137)(57,158,77,138)(58,159,78,139)(59,160,79,140)(60,121,80,141), (1,130)(2,151)(3,132)(4,153)(5,134)(6,155)(7,136)(8,157)(9,138)(10,159)(11,140)(12,121)(13,142)(14,123)(15,144)(16,125)(17,146)(18,127)(19,148)(20,129)(21,150)(22,131)(23,152)(24,133)(25,154)(26,135)(27,156)(28,137)(29,158)(30,139)(31,160)(32,141)(33,122)(34,143)(35,124)(36,145)(37,126)(38,147)(39,128)(40,149)(41,109)(42,90)(43,111)(44,92)(45,113)(46,94)(47,115)(48,96)(49,117)(50,98)(51,119)(52,100)(53,81)(54,102)(55,83)(56,104)(57,85)(58,106)(59,87)(60,108)(61,89)(62,110)(63,91)(64,112)(65,93)(66,114)(67,95)(68,116)(69,97)(70,118)(71,99)(72,120)(73,101)(74,82)(75,103)(76,84)(77,105)(78,86)(79,107)(80,88), (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), (1,116,117,20,21,96,97,40)(2,19,118,95,22,39,98,115)(3,94,119,38,23,114,99,18)(4,37,120,113,24,17,100,93)(5,112,81,16,25,92,101,36)(6,15,82,91,26,35,102,111)(7,90,83,34,27,110,103,14)(8,33,84,109,28,13,104,89)(9,108,85,12,29,88,105,32)(10,11,86,87,30,31,106,107)(41,137,142,56,61,157,122,76)(42,55,143,156,62,75,123,136)(43,155,144,74,63,135,124,54)(44,73,145,134,64,53,125,154)(45,133,146,52,65,153,126,72)(46,51,147,152,66,71,127,132)(47,151,148,70,67,131,128,50)(48,69,149,130,68,49,129,150)(57,121,158,80,77,141,138,60)(58,79,159,140,78,59,139,160)>;
G:=Group( (1,117,21,97)(2,118,22,98)(3,119,23,99)(4,120,24,100)(5,81,25,101)(6,82,26,102)(7,83,27,103)(8,84,28,104)(9,85,29,105)(10,86,30,106)(11,87,31,107)(12,88,32,108)(13,89,33,109)(14,90,34,110)(15,91,35,111)(16,92,36,112)(17,93,37,113)(18,94,38,114)(19,95,39,115)(20,96,40,116)(41,142,61,122)(42,143,62,123)(43,144,63,124)(44,145,64,125)(45,146,65,126)(46,147,66,127)(47,148,67,128)(48,149,68,129)(49,150,69,130)(50,151,70,131)(51,152,71,132)(52,153,72,133)(53,154,73,134)(54,155,74,135)(55,156,75,136)(56,157,76,137)(57,158,77,138)(58,159,78,139)(59,160,79,140)(60,121,80,141), (1,130)(2,151)(3,132)(4,153)(5,134)(6,155)(7,136)(8,157)(9,138)(10,159)(11,140)(12,121)(13,142)(14,123)(15,144)(16,125)(17,146)(18,127)(19,148)(20,129)(21,150)(22,131)(23,152)(24,133)(25,154)(26,135)(27,156)(28,137)(29,158)(30,139)(31,160)(32,141)(33,122)(34,143)(35,124)(36,145)(37,126)(38,147)(39,128)(40,149)(41,109)(42,90)(43,111)(44,92)(45,113)(46,94)(47,115)(48,96)(49,117)(50,98)(51,119)(52,100)(53,81)(54,102)(55,83)(56,104)(57,85)(58,106)(59,87)(60,108)(61,89)(62,110)(63,91)(64,112)(65,93)(66,114)(67,95)(68,116)(69,97)(70,118)(71,99)(72,120)(73,101)(74,82)(75,103)(76,84)(77,105)(78,86)(79,107)(80,88), (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), (1,116,117,20,21,96,97,40)(2,19,118,95,22,39,98,115)(3,94,119,38,23,114,99,18)(4,37,120,113,24,17,100,93)(5,112,81,16,25,92,101,36)(6,15,82,91,26,35,102,111)(7,90,83,34,27,110,103,14)(8,33,84,109,28,13,104,89)(9,108,85,12,29,88,105,32)(10,11,86,87,30,31,106,107)(41,137,142,56,61,157,122,76)(42,55,143,156,62,75,123,136)(43,155,144,74,63,135,124,54)(44,73,145,134,64,53,125,154)(45,133,146,52,65,153,126,72)(46,51,147,152,66,71,127,132)(47,151,148,70,67,131,128,50)(48,69,149,130,68,49,129,150)(57,121,158,80,77,141,138,60)(58,79,159,140,78,59,139,160) );
G=PermutationGroup([(1,117,21,97),(2,118,22,98),(3,119,23,99),(4,120,24,100),(5,81,25,101),(6,82,26,102),(7,83,27,103),(8,84,28,104),(9,85,29,105),(10,86,30,106),(11,87,31,107),(12,88,32,108),(13,89,33,109),(14,90,34,110),(15,91,35,111),(16,92,36,112),(17,93,37,113),(18,94,38,114),(19,95,39,115),(20,96,40,116),(41,142,61,122),(42,143,62,123),(43,144,63,124),(44,145,64,125),(45,146,65,126),(46,147,66,127),(47,148,67,128),(48,149,68,129),(49,150,69,130),(50,151,70,131),(51,152,71,132),(52,153,72,133),(53,154,73,134),(54,155,74,135),(55,156,75,136),(56,157,76,137),(57,158,77,138),(58,159,78,139),(59,160,79,140),(60,121,80,141)], [(1,130),(2,151),(3,132),(4,153),(5,134),(6,155),(7,136),(8,157),(9,138),(10,159),(11,140),(12,121),(13,142),(14,123),(15,144),(16,125),(17,146),(18,127),(19,148),(20,129),(21,150),(22,131),(23,152),(24,133),(25,154),(26,135),(27,156),(28,137),(29,158),(30,139),(31,160),(32,141),(33,122),(34,143),(35,124),(36,145),(37,126),(38,147),(39,128),(40,149),(41,109),(42,90),(43,111),(44,92),(45,113),(46,94),(47,115),(48,96),(49,117),(50,98),(51,119),(52,100),(53,81),(54,102),(55,83),(56,104),(57,85),(58,106),(59,87),(60,108),(61,89),(62,110),(63,91),(64,112),(65,93),(66,114),(67,95),(68,116),(69,97),(70,118),(71,99),(72,120),(73,101),(74,82),(75,103),(76,84),(77,105),(78,86),(79,107),(80,88)], [(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)], [(1,116,117,20,21,96,97,40),(2,19,118,95,22,39,98,115),(3,94,119,38,23,114,99,18),(4,37,120,113,24,17,100,93),(5,112,81,16,25,92,101,36),(6,15,82,91,26,35,102,111),(7,90,83,34,27,110,103,14),(8,33,84,109,28,13,104,89),(9,108,85,12,29,88,105,32),(10,11,86,87,30,31,106,107),(41,137,142,56,61,157,122,76),(42,55,143,156,62,75,123,136),(43,155,144,74,63,135,124,54),(44,73,145,134,64,53,125,154),(45,133,146,52,65,153,126,72),(46,51,147,152,66,71,127,132),(47,151,148,70,67,131,128,50),(48,69,149,130,68,49,129,150),(57,121,158,80,77,141,138,60),(58,79,159,140,78,59,139,160)])
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | 8B | 8C | 8D | 8E | ··· | 8L | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 20 | 20 | 1 | 1 | 1 | 1 | 2 | 2 | 20 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D5 | D10 | D10 | C8○D4 | C4×D5 | D20 | C5⋊D4 | C4×D5 | D20.2C4 |
| kernel | C4.89(C2×D20) | D10⋊1C8 | C22×C5⋊2C8 | C10×M4(2) | C2×C4○D20 | C2×Dic10 | C2×D20 | C2×C5⋊D4 | C2×C20 | C2×M4(2) | C2×C8 | C22×C4 | C10 | C2×C4 | C2×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 4 | 2 | 8 | 4 | 8 | 8 | 4 | 8 |
Matrix representation of C4.89(C2×D20) ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 9 | 2 |
| 0 | 0 | 1 | 32 |
| 21 | 17 | 0 | 0 |
| 3 | 18 | 0 | 0 |
| 0 | 0 | 38 | 0 |
| 0 | 0 | 27 | 3 |
| 6 | 38 | 0 | 0 |
| 26 | 35 | 0 | 0 |
| 0 | 0 | 38 | 0 |
| 0 | 0 | 0 | 38 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,9,0,0,0,0,9],[1,0,0,0,0,1,0,0,0,0,9,1,0,0,2,32],[21,3,0,0,17,18,0,0,0,0,38,27,0,0,0,3],[6,26,0,0,38,35,0,0,0,0,38,0,0,0,0,38] >;
C4.89(C2×D20) in GAP, Magma, Sage, TeX
C_4._{89}(C_2\times D_{20}) % in TeX
G:=Group("C4.89(C2xD20)"); // GroupNames label
G:=SmallGroup(320,756);
// by ID
G=gap.SmallGroup(320,756);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,422,387,58,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=1,c^20=a^2,d^2=a,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=a^-1*c^19>;
// generators/relations