metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C4)⋊3D20, (C2×C20)⋊22D4, C10.42C22≀C2, (C22×D5).41D4, (C22×D20).5C2, C22.247(D4×D5), C2.25(C4⋊D20), C2.10(C20⋊7D4), C10.62(C4⋊D4), (C22×C4).104D10, C22.129(C2×D20), C5⋊4(C23.10D4), C2.10(C23⋊D10), C2.6(C20.23D4), C10.53(C4.4D4), (C22×C20).68C22, (C23×D5).20C22, C23.380(C22×D5), C10.10C42⋊33C2, C22.108(C4○D20), (C22×C10).355C23, C22.51(Q8⋊2D5), C2.20(D10.13D4), C10.53(C22.D4), (C22×Dic5).60C22, (C2×C4⋊C4)⋊9D5, (C10×C4⋊C4)⋊22C2, (C2×C10).336(C2×D4), (C2×C4).41(C5⋊D4), (C2×D10⋊C4)⋊38C2, (C2×C10).86(C4○D4), C22.140(C2×C5⋊D4), SmallGroup(320,618)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C4)⋊3D20
G = < a,b,c,d | a2=b4=c20=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd=ab-1, dcd=c-1 >
Subgroups: 1158 in 238 conjugacy classes, 63 normal (27 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×7], C22 [×3], C22 [×4], C22 [×20], C5, C2×C4 [×4], C2×C4 [×13], D4 [×8], C23, C23 [×16], D5 [×4], C10 [×3], C10 [×4], C22⋊C4 [×8], C4⋊C4 [×2], C22×C4 [×3], C22×C4 [×2], C2×D4 [×6], C24 [×2], Dic5 [×2], C20 [×5], D10 [×20], C2×C10 [×3], C2×C10 [×4], C2.C42, C2×C22⋊C4 [×4], C2×C4⋊C4, C22×D4, D20 [×8], C2×Dic5 [×6], C2×C20 [×4], C2×C20 [×7], C22×D5 [×4], C22×D5 [×12], C22×C10, C23.10D4, D10⋊C4 [×8], C5×C4⋊C4 [×2], C2×D20 [×6], C22×Dic5 [×2], C22×C20 [×3], C23×D5 [×2], C10.10C42, C2×D10⋊C4 [×4], C10×C4⋊C4, C22×D20, (C2×C4)⋊3D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×8], C23, D5, C2×D4 [×4], C4○D4 [×3], D10 [×3], C22≀C2, C4⋊D4 [×3], C22.D4 [×2], C4.4D4, D20 [×2], C5⋊D4 [×2], C22×D5, C23.10D4, C2×D20, C4○D20, D4×D5 [×2], Q8⋊2D5 [×2], C2×C5⋊D4, D10.13D4 [×2], C4⋊D20 [×2], C20⋊7D4, C23⋊D10, C20.23D4, (C2×C4)⋊3D20
►On 160 points
Generators in S160
(1 99)(2 100)(3 81)(4 82)(5 83)(6 84)(7 85)(8 86)(9 87)(10 88)(11 89)(12 90)(13 91)(14 92)(15 93)(16 94)(17 95)(18 96)(19 97)(20 98)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)(41 145)(42 146)(43 147)(44 148)(45 149)(46 150)(47 151)(48 152)(49 153)(50 154)(51 155)(52 156)(53 157)(54 158)(55 159)(56 160)(57 141)(58 142)(59 143)(60 144)(101 127)(102 128)(103 129)(104 130)(105 131)(106 132)(107 133)(108 134)(109 135)(110 136)(111 137)(112 138)(113 139)(114 140)(115 121)(116 122)(117 123)(118 124)(119 125)(120 126)
(1 152 79 140)(2 121 80 153)(3 154 61 122)(4 123 62 155)(5 156 63 124)(6 125 64 157)(7 158 65 126)(8 127 66 159)(9 160 67 128)(10 129 68 141)(11 142 69 130)(12 131 70 143)(13 144 71 132)(14 133 72 145)(15 146 73 134)(16 135 74 147)(17 148 75 136)(18 137 76 149)(19 150 77 138)(20 139 78 151)(21 116 81 50)(22 51 82 117)(23 118 83 52)(24 53 84 119)(25 120 85 54)(26 55 86 101)(27 102 87 56)(28 57 88 103)(29 104 89 58)(30 59 90 105)(31 106 91 60)(32 41 92 107)(33 108 93 42)(34 43 94 109)(35 110 95 44)(36 45 96 111)(37 112 97 46)(38 47 98 113)(39 114 99 48)(40 49 100 115)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 31)(22 30)(23 29)(24 28)(25 27)(32 40)(33 39)(34 38)(35 37)(41 121)(42 140)(43 139)(44 138)(45 137)(46 136)(47 135)(48 134)(49 133)(50 132)(51 131)(52 130)(53 129)(54 128)(55 127)(56 126)(57 125)(58 124)(59 123)(60 122)(61 71)(62 70)(63 69)(64 68)(65 67)(72 80)(73 79)(74 78)(75 77)(81 91)(82 90)(83 89)(84 88)(85 87)(92 100)(93 99)(94 98)(95 97)(101 159)(102 158)(103 157)(104 156)(105 155)(106 154)(107 153)(108 152)(109 151)(110 150)(111 149)(112 148)(113 147)(114 146)(115 145)(116 144)(117 143)(118 142)(119 141)(120 160)
G:=sub<Sym(160)| (1,99)(2,100)(3,81)(4,82)(5,83)(6,84)(7,85)(8,86)(9,87)(10,88)(11,89)(12,90)(13,91)(14,92)(15,93)(16,94)(17,95)(18,96)(19,97)(20,98)(21,61)(22,62)(23,63)(24,64)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80)(41,145)(42,146)(43,147)(44,148)(45,149)(46,150)(47,151)(48,152)(49,153)(50,154)(51,155)(52,156)(53,157)(54,158)(55,159)(56,160)(57,141)(58,142)(59,143)(60,144)(101,127)(102,128)(103,129)(104,130)(105,131)(106,132)(107,133)(108,134)(109,135)(110,136)(111,137)(112,138)(113,139)(114,140)(115,121)(116,122)(117,123)(118,124)(119,125)(120,126), (1,152,79,140)(2,121,80,153)(3,154,61,122)(4,123,62,155)(5,156,63,124)(6,125,64,157)(7,158,65,126)(8,127,66,159)(9,160,67,128)(10,129,68,141)(11,142,69,130)(12,131,70,143)(13,144,71,132)(14,133,72,145)(15,146,73,134)(16,135,74,147)(17,148,75,136)(18,137,76,149)(19,150,77,138)(20,139,78,151)(21,116,81,50)(22,51,82,117)(23,118,83,52)(24,53,84,119)(25,120,85,54)(26,55,86,101)(27,102,87,56)(28,57,88,103)(29,104,89,58)(30,59,90,105)(31,106,91,60)(32,41,92,107)(33,108,93,42)(34,43,94,109)(35,110,95,44)(36,45,96,111)(37,112,97,46)(38,47,98,113)(39,114,99,48)(40,49,100,115), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,31)(22,30)(23,29)(24,28)(25,27)(32,40)(33,39)(34,38)(35,37)(41,121)(42,140)(43,139)(44,138)(45,137)(46,136)(47,135)(48,134)(49,133)(50,132)(51,131)(52,130)(53,129)(54,128)(55,127)(56,126)(57,125)(58,124)(59,123)(60,122)(61,71)(62,70)(63,69)(64,68)(65,67)(72,80)(73,79)(74,78)(75,77)(81,91)(82,90)(83,89)(84,88)(85,87)(92,100)(93,99)(94,98)(95,97)(101,159)(102,158)(103,157)(104,156)(105,155)(106,154)(107,153)(108,152)(109,151)(110,150)(111,149)(112,148)(113,147)(114,146)(115,145)(116,144)(117,143)(118,142)(119,141)(120,160)>;
G:=Group( (1,99)(2,100)(3,81)(4,82)(5,83)(6,84)(7,85)(8,86)(9,87)(10,88)(11,89)(12,90)(13,91)(14,92)(15,93)(16,94)(17,95)(18,96)(19,97)(20,98)(21,61)(22,62)(23,63)(24,64)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80)(41,145)(42,146)(43,147)(44,148)(45,149)(46,150)(47,151)(48,152)(49,153)(50,154)(51,155)(52,156)(53,157)(54,158)(55,159)(56,160)(57,141)(58,142)(59,143)(60,144)(101,127)(102,128)(103,129)(104,130)(105,131)(106,132)(107,133)(108,134)(109,135)(110,136)(111,137)(112,138)(113,139)(114,140)(115,121)(116,122)(117,123)(118,124)(119,125)(120,126), (1,152,79,140)(2,121,80,153)(3,154,61,122)(4,123,62,155)(5,156,63,124)(6,125,64,157)(7,158,65,126)(8,127,66,159)(9,160,67,128)(10,129,68,141)(11,142,69,130)(12,131,70,143)(13,144,71,132)(14,133,72,145)(15,146,73,134)(16,135,74,147)(17,148,75,136)(18,137,76,149)(19,150,77,138)(20,139,78,151)(21,116,81,50)(22,51,82,117)(23,118,83,52)(24,53,84,119)(25,120,85,54)(26,55,86,101)(27,102,87,56)(28,57,88,103)(29,104,89,58)(30,59,90,105)(31,106,91,60)(32,41,92,107)(33,108,93,42)(34,43,94,109)(35,110,95,44)(36,45,96,111)(37,112,97,46)(38,47,98,113)(39,114,99,48)(40,49,100,115), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,31)(22,30)(23,29)(24,28)(25,27)(32,40)(33,39)(34,38)(35,37)(41,121)(42,140)(43,139)(44,138)(45,137)(46,136)(47,135)(48,134)(49,133)(50,132)(51,131)(52,130)(53,129)(54,128)(55,127)(56,126)(57,125)(58,124)(59,123)(60,122)(61,71)(62,70)(63,69)(64,68)(65,67)(72,80)(73,79)(74,78)(75,77)(81,91)(82,90)(83,89)(84,88)(85,87)(92,100)(93,99)(94,98)(95,97)(101,159)(102,158)(103,157)(104,156)(105,155)(106,154)(107,153)(108,152)(109,151)(110,150)(111,149)(112,148)(113,147)(114,146)(115,145)(116,144)(117,143)(118,142)(119,141)(120,160) );
G=PermutationGroup([(1,99),(2,100),(3,81),(4,82),(5,83),(6,84),(7,85),(8,86),(9,87),(10,88),(11,89),(12,90),(13,91),(14,92),(15,93),(16,94),(17,95),(18,96),(19,97),(20,98),(21,61),(22,62),(23,63),(24,64),(25,65),(26,66),(27,67),(28,68),(29,69),(30,70),(31,71),(32,72),(33,73),(34,74),(35,75),(36,76),(37,77),(38,78),(39,79),(40,80),(41,145),(42,146),(43,147),(44,148),(45,149),(46,150),(47,151),(48,152),(49,153),(50,154),(51,155),(52,156),(53,157),(54,158),(55,159),(56,160),(57,141),(58,142),(59,143),(60,144),(101,127),(102,128),(103,129),(104,130),(105,131),(106,132),(107,133),(108,134),(109,135),(110,136),(111,137),(112,138),(113,139),(114,140),(115,121),(116,122),(117,123),(118,124),(119,125),(120,126)], [(1,152,79,140),(2,121,80,153),(3,154,61,122),(4,123,62,155),(5,156,63,124),(6,125,64,157),(7,158,65,126),(8,127,66,159),(9,160,67,128),(10,129,68,141),(11,142,69,130),(12,131,70,143),(13,144,71,132),(14,133,72,145),(15,146,73,134),(16,135,74,147),(17,148,75,136),(18,137,76,149),(19,150,77,138),(20,139,78,151),(21,116,81,50),(22,51,82,117),(23,118,83,52),(24,53,84,119),(25,120,85,54),(26,55,86,101),(27,102,87,56),(28,57,88,103),(29,104,89,58),(30,59,90,105),(31,106,91,60),(32,41,92,107),(33,108,93,42),(34,43,94,109),(35,110,95,44),(36,45,96,111),(37,112,97,46),(38,47,98,113),(39,114,99,48),(40,49,100,115)], [(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,20),(17,19),(21,31),(22,30),(23,29),(24,28),(25,27),(32,40),(33,39),(34,38),(35,37),(41,121),(42,140),(43,139),(44,138),(45,137),(46,136),(47,135),(48,134),(49,133),(50,132),(51,131),(52,130),(53,129),(54,128),(55,127),(56,126),(57,125),(58,124),(59,123),(60,122),(61,71),(62,70),(63,69),(64,68),(65,67),(72,80),(73,79),(74,78),(75,77),(81,91),(82,90),(83,89),(84,88),(85,87),(92,100),(93,99),(94,98),(95,97),(101,159),(102,158),(103,157),(104,156),(105,155),(106,154),(107,153),(108,152),(109,151),(110,150),(111,149),(112,148),(113,147),(114,146),(115,145),(116,144),(117,143),(118,142),(119,141),(120,160)])
62 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20X |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 20 | 20 | 20 | 20 | 4 | ··· | 4 | 20 | 20 | 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 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | C4○D4 | D10 | D20 | C5⋊D4 | C4○D20 | D4×D5 | Q8⋊2D5 |
| kernel | (C2×C4)⋊3D20 | C10.10C42 | C2×D10⋊C4 | C10×C4⋊C4 | C22×D20 | C2×C20 | C22×D5 | C2×C4⋊C4 | C2×C10 | C22×C4 | C2×C4 | C2×C4 | C22 | C22 | C22 |
| # reps | 1 | 1 | 4 | 1 | 1 | 4 | 4 | 2 | 6 | 6 | 8 | 8 | 8 | 4 | 4 |
Matrix representation of (C2×C4)⋊3D20 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 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 | 1 |
| 16 | 9 | 0 | 0 | 0 | 0 |
| 40 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 37 | 0 | 0 |
| 0 | 0 | 26 | 29 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 21 | 40 | 0 | 0 | 0 | 0 |
| 32 | 20 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 36 | 0 | 0 |
| 0 | 0 | 37 | 26 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 39 |
| 0 | 0 | 0 | 0 | 2 | 16 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 40 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 35 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 40 | 0 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,40,0,0,0,0,9,25,0,0,0,0,0,0,12,26,0,0,0,0,37,29,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[21,32,0,0,0,0,40,20,0,0,0,0,0,0,15,37,0,0,0,0,36,26,0,0,0,0,0,0,28,2,0,0,0,0,39,16],[40,40,0,0,0,0,0,1,0,0,0,0,0,0,40,35,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,40,0] >;
(C2×C4)⋊3D20 in GAP, Magma, Sage, TeX
(C_2\times C_4)\rtimes_3D_{20} % in TeX
G:=Group("(C2xC4):3D20"); // GroupNames label
G:=SmallGroup(320,618);
// by ID
G=gap.SmallGroup(320,618);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,254,387,184,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^20=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d=a*b^-1,d*c*d=c^-1>;
// generators/relations