direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C22⋊C4×C20, C22⋊(C4×C20), C2.1(D4×C20), (C2×C42)⋊1C10, (C22×C4)⋊6C20, (C2×C10)⋊5C42, (C22×C20)⋊23C4, (C2×C20).533D4, C10.133(C4×D4), (C23×C4).2C10, (C23×C20).5C2, C10.51(C2×C42), C23.14(C2×C20), C24.22(C2×C10), C22.27(D4×C10), C2.C42⋊13C10, C22.14(C22×C20), C23.50(C22×C10), (C23×C10).82C22, C10.70(C42⋊C2), (C22×C20).488C22, (C22×C10).441C23, (C2×C4×C20)⋊2C2, C2.3(C2×C4×C20), (C2×C4)⋊6(C2×C20), (C2×C20)⋊41(C2×C4), (C2×C4).143(C5×D4), C2.2(C10×C22⋊C4), (C2×C10).594(C2×D4), C2.2(C5×C42⋊C2), C22.13(C5×C4○D4), (C2×C22⋊C4).14C10, (C10×C22⋊C4).34C2, C10.131(C2×C22⋊C4), (C22×C4).82(C2×C10), (C2×C10).203(C4○D4), (C22×C10).177(C2×C4), (C5×C2.C42)⋊29C2, (C2×C10).314(C22×C4), SmallGroup(320,878)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22⋊C4×C20
G = < a,b,c,d | a20=b2=c2=d4=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >
Subgroups: 370 in 258 conjugacy classes, 146 normal (22 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C23, C23, C23, C10, C10, C10, C42, C22⋊C4, C22×C4, C22×C4, C22×C4, C24, C20, C20, C2×C10, C2×C10, C2×C10, C2.C42, C2×C42, C2×C22⋊C4, C23×C4, C2×C20, C2×C20, C22×C10, C22×C10, C22×C10, C4×C22⋊C4, C4×C20, C5×C22⋊C4, C22×C20, C22×C20, C22×C20, C23×C10, C5×C2.C42, C2×C4×C20, C10×C22⋊C4, C23×C20, C22⋊C4×C20
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, C23, C10, C42, C22⋊C4, C22×C4, C2×D4, C4○D4, C20, C2×C10, C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4, C2×C20, C5×D4, C22×C10, C4×C22⋊C4, C4×C20, C5×C22⋊C4, C22×C20, D4×C10, C5×C4○D4, C2×C4×C20, C10×C22⋊C4, C5×C42⋊C2, D4×C20, C22⋊C4×C20
►On 160 points
Generators in S160
(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)
(21 100)(22 81)(23 82)(24 83)(25 84)(26 85)(27 86)(28 87)(29 88)(30 89)(31 90)(32 91)(33 92)(34 93)(35 94)(36 95)(37 96)(38 97)(39 98)(40 99)(41 123)(42 124)(43 125)(44 126)(45 127)(46 128)(47 129)(48 130)(49 131)(50 132)(51 133)(52 134)(53 135)(54 136)(55 137)(56 138)(57 139)(58 140)(59 121)(60 122)
(1 111)(2 112)(3 113)(4 114)(5 115)(6 116)(7 117)(8 118)(9 119)(10 120)(11 101)(12 102)(13 103)(14 104)(15 105)(16 106)(17 107)(18 108)(19 109)(20 110)(21 100)(22 81)(23 82)(24 83)(25 84)(26 85)(27 86)(28 87)(29 88)(30 89)(31 90)(32 91)(33 92)(34 93)(35 94)(36 95)(37 96)(38 97)(39 98)(40 99)(41 123)(42 124)(43 125)(44 126)(45 127)(46 128)(47 129)(48 130)(49 131)(50 132)(51 133)(52 134)(53 135)(54 136)(55 137)(56 138)(57 139)(58 140)(59 121)(60 122)(61 148)(62 149)(63 150)(64 151)(65 152)(66 153)(67 154)(68 155)(69 156)(70 157)(71 158)(72 159)(73 160)(74 141)(75 142)(76 143)(77 144)(78 145)(79 146)(80 147)
(1 41 160 95)(2 42 141 96)(3 43 142 97)(4 44 143 98)(5 45 144 99)(6 46 145 100)(7 47 146 81)(8 48 147 82)(9 49 148 83)(10 50 149 84)(11 51 150 85)(12 52 151 86)(13 53 152 87)(14 54 153 88)(15 55 154 89)(16 56 155 90)(17 57 156 91)(18 58 157 92)(19 59 158 93)(20 60 159 94)(21 116 128 78)(22 117 129 79)(23 118 130 80)(24 119 131 61)(25 120 132 62)(26 101 133 63)(27 102 134 64)(28 103 135 65)(29 104 136 66)(30 105 137 67)(31 106 138 68)(32 107 139 69)(33 108 140 70)(34 109 121 71)(35 110 122 72)(36 111 123 73)(37 112 124 74)(38 113 125 75)(39 114 126 76)(40 115 127 77)
G:=sub<Sym(160)| (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), (21,100)(22,81)(23,82)(24,83)(25,84)(26,85)(27,86)(28,87)(29,88)(30,89)(31,90)(32,91)(33,92)(34,93)(35,94)(36,95)(37,96)(38,97)(39,98)(40,99)(41,123)(42,124)(43,125)(44,126)(45,127)(46,128)(47,129)(48,130)(49,131)(50,132)(51,133)(52,134)(53,135)(54,136)(55,137)(56,138)(57,139)(58,140)(59,121)(60,122), (1,111)(2,112)(3,113)(4,114)(5,115)(6,116)(7,117)(8,118)(9,119)(10,120)(11,101)(12,102)(13,103)(14,104)(15,105)(16,106)(17,107)(18,108)(19,109)(20,110)(21,100)(22,81)(23,82)(24,83)(25,84)(26,85)(27,86)(28,87)(29,88)(30,89)(31,90)(32,91)(33,92)(34,93)(35,94)(36,95)(37,96)(38,97)(39,98)(40,99)(41,123)(42,124)(43,125)(44,126)(45,127)(46,128)(47,129)(48,130)(49,131)(50,132)(51,133)(52,134)(53,135)(54,136)(55,137)(56,138)(57,139)(58,140)(59,121)(60,122)(61,148)(62,149)(63,150)(64,151)(65,152)(66,153)(67,154)(68,155)(69,156)(70,157)(71,158)(72,159)(73,160)(74,141)(75,142)(76,143)(77,144)(78,145)(79,146)(80,147), (1,41,160,95)(2,42,141,96)(3,43,142,97)(4,44,143,98)(5,45,144,99)(6,46,145,100)(7,47,146,81)(8,48,147,82)(9,49,148,83)(10,50,149,84)(11,51,150,85)(12,52,151,86)(13,53,152,87)(14,54,153,88)(15,55,154,89)(16,56,155,90)(17,57,156,91)(18,58,157,92)(19,59,158,93)(20,60,159,94)(21,116,128,78)(22,117,129,79)(23,118,130,80)(24,119,131,61)(25,120,132,62)(26,101,133,63)(27,102,134,64)(28,103,135,65)(29,104,136,66)(30,105,137,67)(31,106,138,68)(32,107,139,69)(33,108,140,70)(34,109,121,71)(35,110,122,72)(36,111,123,73)(37,112,124,74)(38,113,125,75)(39,114,126,76)(40,115,127,77)>;
G:=Group( (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), (21,100)(22,81)(23,82)(24,83)(25,84)(26,85)(27,86)(28,87)(29,88)(30,89)(31,90)(32,91)(33,92)(34,93)(35,94)(36,95)(37,96)(38,97)(39,98)(40,99)(41,123)(42,124)(43,125)(44,126)(45,127)(46,128)(47,129)(48,130)(49,131)(50,132)(51,133)(52,134)(53,135)(54,136)(55,137)(56,138)(57,139)(58,140)(59,121)(60,122), (1,111)(2,112)(3,113)(4,114)(5,115)(6,116)(7,117)(8,118)(9,119)(10,120)(11,101)(12,102)(13,103)(14,104)(15,105)(16,106)(17,107)(18,108)(19,109)(20,110)(21,100)(22,81)(23,82)(24,83)(25,84)(26,85)(27,86)(28,87)(29,88)(30,89)(31,90)(32,91)(33,92)(34,93)(35,94)(36,95)(37,96)(38,97)(39,98)(40,99)(41,123)(42,124)(43,125)(44,126)(45,127)(46,128)(47,129)(48,130)(49,131)(50,132)(51,133)(52,134)(53,135)(54,136)(55,137)(56,138)(57,139)(58,140)(59,121)(60,122)(61,148)(62,149)(63,150)(64,151)(65,152)(66,153)(67,154)(68,155)(69,156)(70,157)(71,158)(72,159)(73,160)(74,141)(75,142)(76,143)(77,144)(78,145)(79,146)(80,147), (1,41,160,95)(2,42,141,96)(3,43,142,97)(4,44,143,98)(5,45,144,99)(6,46,145,100)(7,47,146,81)(8,48,147,82)(9,49,148,83)(10,50,149,84)(11,51,150,85)(12,52,151,86)(13,53,152,87)(14,54,153,88)(15,55,154,89)(16,56,155,90)(17,57,156,91)(18,58,157,92)(19,59,158,93)(20,60,159,94)(21,116,128,78)(22,117,129,79)(23,118,130,80)(24,119,131,61)(25,120,132,62)(26,101,133,63)(27,102,134,64)(28,103,135,65)(29,104,136,66)(30,105,137,67)(31,106,138,68)(32,107,139,69)(33,108,140,70)(34,109,121,71)(35,110,122,72)(36,111,123,73)(37,112,124,74)(38,113,125,75)(39,114,126,76)(40,115,127,77) );
G=PermutationGroup([[(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)], [(21,100),(22,81),(23,82),(24,83),(25,84),(26,85),(27,86),(28,87),(29,88),(30,89),(31,90),(32,91),(33,92),(34,93),(35,94),(36,95),(37,96),(38,97),(39,98),(40,99),(41,123),(42,124),(43,125),(44,126),(45,127),(46,128),(47,129),(48,130),(49,131),(50,132),(51,133),(52,134),(53,135),(54,136),(55,137),(56,138),(57,139),(58,140),(59,121),(60,122)], [(1,111),(2,112),(3,113),(4,114),(5,115),(6,116),(7,117),(8,118),(9,119),(10,120),(11,101),(12,102),(13,103),(14,104),(15,105),(16,106),(17,107),(18,108),(19,109),(20,110),(21,100),(22,81),(23,82),(24,83),(25,84),(26,85),(27,86),(28,87),(29,88),(30,89),(31,90),(32,91),(33,92),(34,93),(35,94),(36,95),(37,96),(38,97),(39,98),(40,99),(41,123),(42,124),(43,125),(44,126),(45,127),(46,128),(47,129),(48,130),(49,131),(50,132),(51,133),(52,134),(53,135),(54,136),(55,137),(56,138),(57,139),(58,140),(59,121),(60,122),(61,148),(62,149),(63,150),(64,151),(65,152),(66,153),(67,154),(68,155),(69,156),(70,157),(71,158),(72,159),(73,160),(74,141),(75,142),(76,143),(77,144),(78,145),(79,146),(80,147)], [(1,41,160,95),(2,42,141,96),(3,43,142,97),(4,44,143,98),(5,45,144,99),(6,46,145,100),(7,47,146,81),(8,48,147,82),(9,49,148,83),(10,50,149,84),(11,51,150,85),(12,52,151,86),(13,53,152,87),(14,54,153,88),(15,55,154,89),(16,56,155,90),(17,57,156,91),(18,58,157,92),(19,59,158,93),(20,60,159,94),(21,116,128,78),(22,117,129,79),(23,118,130,80),(24,119,131,61),(25,120,132,62),(26,101,133,63),(27,102,134,64),(28,103,135,65),(29,104,136,66),(30,105,137,67),(31,106,138,68),(32,107,139,69),(33,108,140,70),(34,109,121,71),(35,110,122,72),(36,111,123,73),(37,112,124,74),(38,113,125,75),(39,114,126,76),(40,115,127,77)]])
200 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4AB | 5A | 5B | 5C | 5D | 10A | ··· | 10AB | 10AC | ··· | 10AR | 20A | ··· | 20AF | 20AG | ··· | 20DH |
| order | 1 | 2 | ··· | 2 | 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 | 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 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C5 | C10 | C10 | C10 | C10 | C20 | C20 | D4 | C4○D4 | C5×D4 | C5×C4○D4 |
| kernel | C22⋊C4×C20 | C5×C2.C42 | C2×C4×C20 | C10×C22⋊C4 | C23×C20 | C5×C22⋊C4 | C22×C20 | C4×C22⋊C4 | C2.C42 | C2×C42 | C2×C22⋊C4 | C23×C4 | C22⋊C4 | C22×C4 | C2×C20 | C2×C10 | C2×C4 | C22 |
| # reps | 1 | 2 | 2 | 2 | 1 | 16 | 8 | 4 | 8 | 8 | 8 | 4 | 64 | 32 | 4 | 4 | 16 | 16 |
Matrix representation of C22⋊C4×C20 ►in GL4(𝔽41) generated by
| 32 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 5 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 9 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 9 |
| 0 | 0 | 9 | 0 |
G:=sub<GL(4,GF(41))| [32,0,0,0,0,1,0,0,0,0,5,0,0,0,0,5],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[9,0,0,0,0,32,0,0,0,0,0,9,0,0,9,0] >;
C22⋊C4×C20 in GAP, Magma, Sage, TeX
C_2^2\rtimes C_4\times C_{20} % in TeX
G:=Group("C2^2:C4xC20"); // GroupNames label
G:=SmallGroup(320,878);
// by ID
G=gap.SmallGroup(320,878);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,1128,436]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,c*d=d*c>;
// generators/relations