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, C2, C4, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C10, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C10, C22⋊C8, C22×C8, C2×M4(2), C2×C4○D4, C5⋊2C8, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, (C22×C8)⋊C2, C2×C5⋊2C8, C2×C5⋊2C8, C2×C40, C5×M4(2), C2×Dic10, C2×C4×D5, C2×D20, C4○D20, C2×C5⋊D4, C22×C20, D10⋊1C8, C22×C5⋊2C8, C10×M4(2), C2×C4○D20, C4.89(C2×D20)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, D10, C2×C22⋊C4, C8○D4, C4×D5, D20, C5⋊D4, C22×D5, (C22×C8)⋊C2, D10⋊C4, C2×C4×D5, C2×D20, C2×C5⋊D4, D20.2C4, C2×D10⋊C4, C4.89(C2×D20)
►On 160 points
Generators in S160
(1 47 21 67)(2 48 22 68)(3 49 23 69)(4 50 24 70)(5 51 25 71)(6 52 26 72)(7 53 27 73)(8 54 28 74)(9 55 29 75)(10 56 30 76)(11 57 31 77)(12 58 32 78)(13 59 33 79)(14 60 34 80)(15 61 35 41)(16 62 36 42)(17 63 37 43)(18 64 38 44)(19 65 39 45)(20 66 40 46)(81 123 101 143)(82 124 102 144)(83 125 103 145)(84 126 104 146)(85 127 105 147)(86 128 106 148)(87 129 107 149)(88 130 108 150)(89 131 109 151)(90 132 110 152)(91 133 111 153)(92 134 112 154)(93 135 113 155)(94 136 114 156)(95 137 115 157)(96 138 116 158)(97 139 117 159)(98 140 118 160)(99 141 119 121)(100 142 120 122)
(1 89)(2 110)(3 91)(4 112)(5 93)(6 114)(7 95)(8 116)(9 97)(10 118)(11 99)(12 120)(13 101)(14 82)(15 103)(16 84)(17 105)(18 86)(19 107)(20 88)(21 109)(22 90)(23 111)(24 92)(25 113)(26 94)(27 115)(28 96)(29 117)(30 98)(31 119)(32 100)(33 81)(34 102)(35 83)(36 104)(37 85)(38 106)(39 87)(40 108)(41 125)(42 146)(43 127)(44 148)(45 129)(46 150)(47 131)(48 152)(49 133)(50 154)(51 135)(52 156)(53 137)(54 158)(55 139)(56 160)(57 141)(58 122)(59 143)(60 124)(61 145)(62 126)(63 147)(64 128)(65 149)(66 130)(67 151)(68 132)(69 153)(70 134)(71 155)(72 136)(73 157)(74 138)(75 159)(76 140)(77 121)(78 142)(79 123)(80 144)
(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 46 47 20 21 66 67 40)(2 19 48 65 22 39 68 45)(3 64 49 38 23 44 69 18)(4 37 50 43 24 17 70 63)(5 42 51 16 25 62 71 36)(6 15 52 61 26 35 72 41)(7 60 53 34 27 80 73 14)(8 33 54 79 28 13 74 59)(9 78 55 12 29 58 75 32)(10 11 56 57 30 31 76 77)(81 158 123 96 101 138 143 116)(82 95 124 137 102 115 144 157)(83 136 125 114 103 156 145 94)(84 113 126 155 104 93 146 135)(85 154 127 92 105 134 147 112)(86 91 128 133 106 111 148 153)(87 132 129 110 107 152 149 90)(88 109 130 151 108 89 150 131)(97 142 139 120 117 122 159 100)(98 119 140 121 118 99 160 141)
G:=sub<Sym(160)| (1,47,21,67)(2,48,22,68)(3,49,23,69)(4,50,24,70)(5,51,25,71)(6,52,26,72)(7,53,27,73)(8,54,28,74)(9,55,29,75)(10,56,30,76)(11,57,31,77)(12,58,32,78)(13,59,33,79)(14,60,34,80)(15,61,35,41)(16,62,36,42)(17,63,37,43)(18,64,38,44)(19,65,39,45)(20,66,40,46)(81,123,101,143)(82,124,102,144)(83,125,103,145)(84,126,104,146)(85,127,105,147)(86,128,106,148)(87,129,107,149)(88,130,108,150)(89,131,109,151)(90,132,110,152)(91,133,111,153)(92,134,112,154)(93,135,113,155)(94,136,114,156)(95,137,115,157)(96,138,116,158)(97,139,117,159)(98,140,118,160)(99,141,119,121)(100,142,120,122), (1,89)(2,110)(3,91)(4,112)(5,93)(6,114)(7,95)(8,116)(9,97)(10,118)(11,99)(12,120)(13,101)(14,82)(15,103)(16,84)(17,105)(18,86)(19,107)(20,88)(21,109)(22,90)(23,111)(24,92)(25,113)(26,94)(27,115)(28,96)(29,117)(30,98)(31,119)(32,100)(33,81)(34,102)(35,83)(36,104)(37,85)(38,106)(39,87)(40,108)(41,125)(42,146)(43,127)(44,148)(45,129)(46,150)(47,131)(48,152)(49,133)(50,154)(51,135)(52,156)(53,137)(54,158)(55,139)(56,160)(57,141)(58,122)(59,143)(60,124)(61,145)(62,126)(63,147)(64,128)(65,149)(66,130)(67,151)(68,132)(69,153)(70,134)(71,155)(72,136)(73,157)(74,138)(75,159)(76,140)(77,121)(78,142)(79,123)(80,144), (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), (1,46,47,20,21,66,67,40)(2,19,48,65,22,39,68,45)(3,64,49,38,23,44,69,18)(4,37,50,43,24,17,70,63)(5,42,51,16,25,62,71,36)(6,15,52,61,26,35,72,41)(7,60,53,34,27,80,73,14)(8,33,54,79,28,13,74,59)(9,78,55,12,29,58,75,32)(10,11,56,57,30,31,76,77)(81,158,123,96,101,138,143,116)(82,95,124,137,102,115,144,157)(83,136,125,114,103,156,145,94)(84,113,126,155,104,93,146,135)(85,154,127,92,105,134,147,112)(86,91,128,133,106,111,148,153)(87,132,129,110,107,152,149,90)(88,109,130,151,108,89,150,131)(97,142,139,120,117,122,159,100)(98,119,140,121,118,99,160,141)>;
G:=Group( (1,47,21,67)(2,48,22,68)(3,49,23,69)(4,50,24,70)(5,51,25,71)(6,52,26,72)(7,53,27,73)(8,54,28,74)(9,55,29,75)(10,56,30,76)(11,57,31,77)(12,58,32,78)(13,59,33,79)(14,60,34,80)(15,61,35,41)(16,62,36,42)(17,63,37,43)(18,64,38,44)(19,65,39,45)(20,66,40,46)(81,123,101,143)(82,124,102,144)(83,125,103,145)(84,126,104,146)(85,127,105,147)(86,128,106,148)(87,129,107,149)(88,130,108,150)(89,131,109,151)(90,132,110,152)(91,133,111,153)(92,134,112,154)(93,135,113,155)(94,136,114,156)(95,137,115,157)(96,138,116,158)(97,139,117,159)(98,140,118,160)(99,141,119,121)(100,142,120,122), (1,89)(2,110)(3,91)(4,112)(5,93)(6,114)(7,95)(8,116)(9,97)(10,118)(11,99)(12,120)(13,101)(14,82)(15,103)(16,84)(17,105)(18,86)(19,107)(20,88)(21,109)(22,90)(23,111)(24,92)(25,113)(26,94)(27,115)(28,96)(29,117)(30,98)(31,119)(32,100)(33,81)(34,102)(35,83)(36,104)(37,85)(38,106)(39,87)(40,108)(41,125)(42,146)(43,127)(44,148)(45,129)(46,150)(47,131)(48,152)(49,133)(50,154)(51,135)(52,156)(53,137)(54,158)(55,139)(56,160)(57,141)(58,122)(59,143)(60,124)(61,145)(62,126)(63,147)(64,128)(65,149)(66,130)(67,151)(68,132)(69,153)(70,134)(71,155)(72,136)(73,157)(74,138)(75,159)(76,140)(77,121)(78,142)(79,123)(80,144), (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), (1,46,47,20,21,66,67,40)(2,19,48,65,22,39,68,45)(3,64,49,38,23,44,69,18)(4,37,50,43,24,17,70,63)(5,42,51,16,25,62,71,36)(6,15,52,61,26,35,72,41)(7,60,53,34,27,80,73,14)(8,33,54,79,28,13,74,59)(9,78,55,12,29,58,75,32)(10,11,56,57,30,31,76,77)(81,158,123,96,101,138,143,116)(82,95,124,137,102,115,144,157)(83,136,125,114,103,156,145,94)(84,113,126,155,104,93,146,135)(85,154,127,92,105,134,147,112)(86,91,128,133,106,111,148,153)(87,132,129,110,107,152,149,90)(88,109,130,151,108,89,150,131)(97,142,139,120,117,122,159,100)(98,119,140,121,118,99,160,141) );
G=PermutationGroup([[(1,47,21,67),(2,48,22,68),(3,49,23,69),(4,50,24,70),(5,51,25,71),(6,52,26,72),(7,53,27,73),(8,54,28,74),(9,55,29,75),(10,56,30,76),(11,57,31,77),(12,58,32,78),(13,59,33,79),(14,60,34,80),(15,61,35,41),(16,62,36,42),(17,63,37,43),(18,64,38,44),(19,65,39,45),(20,66,40,46),(81,123,101,143),(82,124,102,144),(83,125,103,145),(84,126,104,146),(85,127,105,147),(86,128,106,148),(87,129,107,149),(88,130,108,150),(89,131,109,151),(90,132,110,152),(91,133,111,153),(92,134,112,154),(93,135,113,155),(94,136,114,156),(95,137,115,157),(96,138,116,158),(97,139,117,159),(98,140,118,160),(99,141,119,121),(100,142,120,122)], [(1,89),(2,110),(3,91),(4,112),(5,93),(6,114),(7,95),(8,116),(9,97),(10,118),(11,99),(12,120),(13,101),(14,82),(15,103),(16,84),(17,105),(18,86),(19,107),(20,88),(21,109),(22,90),(23,111),(24,92),(25,113),(26,94),(27,115),(28,96),(29,117),(30,98),(31,119),(32,100),(33,81),(34,102),(35,83),(36,104),(37,85),(38,106),(39,87),(40,108),(41,125),(42,146),(43,127),(44,148),(45,129),(46,150),(47,131),(48,152),(49,133),(50,154),(51,135),(52,156),(53,137),(54,158),(55,139),(56,160),(57,141),(58,122),(59,143),(60,124),(61,145),(62,126),(63,147),(64,128),(65,149),(66,130),(67,151),(68,132),(69,153),(70,134),(71,155),(72,136),(73,157),(74,138),(75,159),(76,140),(77,121),(78,142),(79,123),(80,144)], [(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)], [(1,46,47,20,21,66,67,40),(2,19,48,65,22,39,68,45),(3,64,49,38,23,44,69,18),(4,37,50,43,24,17,70,63),(5,42,51,16,25,62,71,36),(6,15,52,61,26,35,72,41),(7,60,53,34,27,80,73,14),(8,33,54,79,28,13,74,59),(9,78,55,12,29,58,75,32),(10,11,56,57,30,31,76,77),(81,158,123,96,101,138,143,116),(82,95,124,137,102,115,144,157),(83,136,125,114,103,156,145,94),(84,113,126,155,104,93,146,135),(85,154,127,92,105,134,147,112),(86,91,128,133,106,111,148,153),(87,132,129,110,107,152,149,90),(88,109,130,151,108,89,150,131),(97,142,139,120,117,122,159,100),(98,119,140,121,118,99,160,141)]])
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