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, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, Dic5, C20, D10, C2×C10, C2×C10, C2.C42, C2×C22⋊C4, C2×C4⋊C4, C22×D4, D20, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C23.10D4, D10⋊C4, C5×C4⋊C4, C2×D20, C22×Dic5, C22×C20, C23×D5, C10.10C42, C2×D10⋊C4, C10×C4⋊C4, C22×D20, (C2×C4)⋊3D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C22≀C2, C4⋊D4, C22.D4, C4.4D4, D20, C5⋊D4, C22×D5, C23.10D4, C2×D20, C4○D20, D4×D5, Q8⋊2D5, C2×C5⋊D4, D10.13D4, C4⋊D20, C20⋊7D4, C23⋊D10, C20.23D4, (C2×C4)⋊3D20
►On 160 points
Generators in S160
(1 106)(2 107)(3 108)(4 109)(5 110)(6 111)(7 112)(8 113)(9 114)(10 115)(11 116)(12 117)(13 118)(14 119)(15 120)(16 101)(17 102)(18 103)(19 104)(20 105)(21 77)(22 78)(23 79)(24 80)(25 61)(26 62)(27 63)(28 64)(29 65)(30 66)(31 67)(32 68)(33 69)(34 70)(35 71)(36 72)(37 73)(38 74)(39 75)(40 76)(41 147)(42 148)(43 149)(44 150)(45 151)(46 152)(47 153)(48 154)(49 155)(50 156)(51 157)(52 158)(53 159)(54 160)(55 141)(56 142)(57 143)(58 144)(59 145)(60 146)(81 127)(82 128)(83 129)(84 130)(85 131)(86 132)(87 133)(88 134)(89 135)(90 136)(91 137)(92 138)(93 139)(94 140)(95 121)(96 122)(97 123)(98 124)(99 125)(100 126)
(1 97 34 49)(2 50 35 98)(3 99 36 51)(4 52 37 100)(5 81 38 53)(6 54 39 82)(7 83 40 55)(8 56 21 84)(9 85 22 57)(10 58 23 86)(11 87 24 59)(12 60 25 88)(13 89 26 41)(14 42 27 90)(15 91 28 43)(16 44 29 92)(17 93 30 45)(18 46 31 94)(19 95 32 47)(20 48 33 96)(61 134 117 146)(62 147 118 135)(63 136 119 148)(64 149 120 137)(65 138 101 150)(66 151 102 139)(67 140 103 152)(68 153 104 121)(69 122 105 154)(70 155 106 123)(71 124 107 156)(72 157 108 125)(73 126 109 158)(74 159 110 127)(75 128 111 160)(76 141 112 129)(77 130 113 142)(78 143 114 131)(79 132 115 144)(80 145 116 133)
(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)(22 40)(23 39)(24 38)(25 37)(26 36)(27 35)(28 34)(29 33)(30 32)(41 125)(42 124)(43 123)(44 122)(45 121)(46 140)(47 139)(48 138)(49 137)(50 136)(51 135)(52 134)(53 133)(54 132)(55 131)(56 130)(57 129)(58 128)(59 127)(60 126)(61 73)(62 72)(63 71)(64 70)(65 69)(66 68)(74 80)(75 79)(76 78)(81 145)(82 144)(83 143)(84 142)(85 141)(86 160)(87 159)(88 158)(89 157)(90 156)(91 155)(92 154)(93 153)(94 152)(95 151)(96 150)(97 149)(98 148)(99 147)(100 146)(101 105)(102 104)(106 120)(107 119)(108 118)(109 117)(110 116)(111 115)(112 114)
G:=sub<Sym(160)| (1,106)(2,107)(3,108)(4,109)(5,110)(6,111)(7,112)(8,113)(9,114)(10,115)(11,116)(12,117)(13,118)(14,119)(15,120)(16,101)(17,102)(18,103)(19,104)(20,105)(21,77)(22,78)(23,79)(24,80)(25,61)(26,62)(27,63)(28,64)(29,65)(30,66)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72)(37,73)(38,74)(39,75)(40,76)(41,147)(42,148)(43,149)(44,150)(45,151)(46,152)(47,153)(48,154)(49,155)(50,156)(51,157)(52,158)(53,159)(54,160)(55,141)(56,142)(57,143)(58,144)(59,145)(60,146)(81,127)(82,128)(83,129)(84,130)(85,131)(86,132)(87,133)(88,134)(89,135)(90,136)(91,137)(92,138)(93,139)(94,140)(95,121)(96,122)(97,123)(98,124)(99,125)(100,126), (1,97,34,49)(2,50,35,98)(3,99,36,51)(4,52,37,100)(5,81,38,53)(6,54,39,82)(7,83,40,55)(8,56,21,84)(9,85,22,57)(10,58,23,86)(11,87,24,59)(12,60,25,88)(13,89,26,41)(14,42,27,90)(15,91,28,43)(16,44,29,92)(17,93,30,45)(18,46,31,94)(19,95,32,47)(20,48,33,96)(61,134,117,146)(62,147,118,135)(63,136,119,148)(64,149,120,137)(65,138,101,150)(66,151,102,139)(67,140,103,152)(68,153,104,121)(69,122,105,154)(70,155,106,123)(71,124,107,156)(72,157,108,125)(73,126,109,158)(74,159,110,127)(75,128,111,160)(76,141,112,129)(77,130,113,142)(78,143,114,131)(79,132,115,144)(80,145,116,133), (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)(22,40)(23,39)(24,38)(25,37)(26,36)(27,35)(28,34)(29,33)(30,32)(41,125)(42,124)(43,123)(44,122)(45,121)(46,140)(47,139)(48,138)(49,137)(50,136)(51,135)(52,134)(53,133)(54,132)(55,131)(56,130)(57,129)(58,128)(59,127)(60,126)(61,73)(62,72)(63,71)(64,70)(65,69)(66,68)(74,80)(75,79)(76,78)(81,145)(82,144)(83,143)(84,142)(85,141)(86,160)(87,159)(88,158)(89,157)(90,156)(91,155)(92,154)(93,153)(94,152)(95,151)(96,150)(97,149)(98,148)(99,147)(100,146)(101,105)(102,104)(106,120)(107,119)(108,118)(109,117)(110,116)(111,115)(112,114)>;
G:=Group( (1,106)(2,107)(3,108)(4,109)(5,110)(6,111)(7,112)(8,113)(9,114)(10,115)(11,116)(12,117)(13,118)(14,119)(15,120)(16,101)(17,102)(18,103)(19,104)(20,105)(21,77)(22,78)(23,79)(24,80)(25,61)(26,62)(27,63)(28,64)(29,65)(30,66)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72)(37,73)(38,74)(39,75)(40,76)(41,147)(42,148)(43,149)(44,150)(45,151)(46,152)(47,153)(48,154)(49,155)(50,156)(51,157)(52,158)(53,159)(54,160)(55,141)(56,142)(57,143)(58,144)(59,145)(60,146)(81,127)(82,128)(83,129)(84,130)(85,131)(86,132)(87,133)(88,134)(89,135)(90,136)(91,137)(92,138)(93,139)(94,140)(95,121)(96,122)(97,123)(98,124)(99,125)(100,126), (1,97,34,49)(2,50,35,98)(3,99,36,51)(4,52,37,100)(5,81,38,53)(6,54,39,82)(7,83,40,55)(8,56,21,84)(9,85,22,57)(10,58,23,86)(11,87,24,59)(12,60,25,88)(13,89,26,41)(14,42,27,90)(15,91,28,43)(16,44,29,92)(17,93,30,45)(18,46,31,94)(19,95,32,47)(20,48,33,96)(61,134,117,146)(62,147,118,135)(63,136,119,148)(64,149,120,137)(65,138,101,150)(66,151,102,139)(67,140,103,152)(68,153,104,121)(69,122,105,154)(70,155,106,123)(71,124,107,156)(72,157,108,125)(73,126,109,158)(74,159,110,127)(75,128,111,160)(76,141,112,129)(77,130,113,142)(78,143,114,131)(79,132,115,144)(80,145,116,133), (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)(22,40)(23,39)(24,38)(25,37)(26,36)(27,35)(28,34)(29,33)(30,32)(41,125)(42,124)(43,123)(44,122)(45,121)(46,140)(47,139)(48,138)(49,137)(50,136)(51,135)(52,134)(53,133)(54,132)(55,131)(56,130)(57,129)(58,128)(59,127)(60,126)(61,73)(62,72)(63,71)(64,70)(65,69)(66,68)(74,80)(75,79)(76,78)(81,145)(82,144)(83,143)(84,142)(85,141)(86,160)(87,159)(88,158)(89,157)(90,156)(91,155)(92,154)(93,153)(94,152)(95,151)(96,150)(97,149)(98,148)(99,147)(100,146)(101,105)(102,104)(106,120)(107,119)(108,118)(109,117)(110,116)(111,115)(112,114) );
G=PermutationGroup([[(1,106),(2,107),(3,108),(4,109),(5,110),(6,111),(7,112),(8,113),(9,114),(10,115),(11,116),(12,117),(13,118),(14,119),(15,120),(16,101),(17,102),(18,103),(19,104),(20,105),(21,77),(22,78),(23,79),(24,80),(25,61),(26,62),(27,63),(28,64),(29,65),(30,66),(31,67),(32,68),(33,69),(34,70),(35,71),(36,72),(37,73),(38,74),(39,75),(40,76),(41,147),(42,148),(43,149),(44,150),(45,151),(46,152),(47,153),(48,154),(49,155),(50,156),(51,157),(52,158),(53,159),(54,160),(55,141),(56,142),(57,143),(58,144),(59,145),(60,146),(81,127),(82,128),(83,129),(84,130),(85,131),(86,132),(87,133),(88,134),(89,135),(90,136),(91,137),(92,138),(93,139),(94,140),(95,121),(96,122),(97,123),(98,124),(99,125),(100,126)], [(1,97,34,49),(2,50,35,98),(3,99,36,51),(4,52,37,100),(5,81,38,53),(6,54,39,82),(7,83,40,55),(8,56,21,84),(9,85,22,57),(10,58,23,86),(11,87,24,59),(12,60,25,88),(13,89,26,41),(14,42,27,90),(15,91,28,43),(16,44,29,92),(17,93,30,45),(18,46,31,94),(19,95,32,47),(20,48,33,96),(61,134,117,146),(62,147,118,135),(63,136,119,148),(64,149,120,137),(65,138,101,150),(66,151,102,139),(67,140,103,152),(68,153,104,121),(69,122,105,154),(70,155,106,123),(71,124,107,156),(72,157,108,125),(73,126,109,158),(74,159,110,127),(75,128,111,160),(76,141,112,129),(77,130,113,142),(78,143,114,131),(79,132,115,144),(80,145,116,133)], [(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),(22,40),(23,39),(24,38),(25,37),(26,36),(27,35),(28,34),(29,33),(30,32),(41,125),(42,124),(43,123),(44,122),(45,121),(46,140),(47,139),(48,138),(49,137),(50,136),(51,135),(52,134),(53,133),(54,132),(55,131),(56,130),(57,129),(58,128),(59,127),(60,126),(61,73),(62,72),(63,71),(64,70),(65,69),(66,68),(74,80),(75,79),(76,78),(81,145),(82,144),(83,143),(84,142),(85,141),(86,160),(87,159),(88,158),(89,157),(90,156),(91,155),(92,154),(93,153),(94,152),(95,151),(96,150),(97,149),(98,148),(99,147),(100,146),(101,105),(102,104),(106,120),(107,119),(108,118),(109,117),(110,116),(111,115),(112,114)]])
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