metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C20)⋊5D4, (C2×C4)⋊2D20, (C22×D5)⋊3D4, C10.4C22≀C2, (C22×D20)⋊1C2, C5⋊1(C23⋊2D4), C2.7(C4⋊D20), C10.1(C4⋊1D4), C2.3(C20⋊4D4), (C22×C4).70D10, C22.80(C2×D20), C22.155(D4×D5), C2.7(C22⋊D20), C10.35(C4⋊D4), C2.C42⋊10D5, (C23×D5).3C22, (C22×C20).46C22, C23.359(C22×D5), (C22×C10).296C23, C22.44(Q8⋊2D5), (C22×Dic5).18C22, (C2×C10).96(C2×D4), (C2×D10⋊C4)⋊15C2, (C5×C2.C42)⋊8C2, (C2×C10).184(C4○D4), SmallGroup(320,298)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C20)⋊5D4
G = < a,b,c,d | a2=b4=c20=d2=1, cbc-1=ab=ba, ac=ca, ad=da, dbd=ab-1, dcd=c-1 >
Subgroups: 1702 in 322 conjugacy classes, 69 normal (12 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C22⋊C4, C22×C4, C22×C4, C2×D4, C24, Dic5, C20, D10, C2×C10, C2×C10, C2.C42, C2×C22⋊C4, C22×D4, D20, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C23⋊2D4, D10⋊C4, C2×D20, C22×Dic5, C22×C20, C23×D5, C5×C2.C42, C2×D10⋊C4, C22×D20, (C2×C20)⋊5D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C22≀C2, C4⋊D4, C4⋊1D4, D20, C22×D5, C23⋊2D4, C2×D20, D4×D5, Q8⋊2D5, C20⋊4D4, C22⋊D20, C4⋊D20, (C2×C20)⋊5D4
►On 160 points
Generators in S160
(1 135)(2 136)(3 137)(4 138)(5 139)(6 140)(7 121)(8 122)(9 123)(10 124)(11 125)(12 126)(13 127)(14 128)(15 129)(16 130)(17 131)(18 132)(19 133)(20 134)(21 91)(22 92)(23 93)(24 94)(25 95)(26 96)(27 97)(28 98)(29 99)(30 100)(31 81)(32 82)(33 83)(34 84)(35 85)(36 86)(37 87)(38 88)(39 89)(40 90)(41 156)(42 157)(43 158)(44 159)(45 160)(46 141)(47 142)(48 143)(49 144)(50 145)(51 146)(52 147)(53 148)(54 149)(55 150)(56 151)(57 152)(58 153)(59 154)(60 155)(61 110)(62 111)(63 112)(64 113)(65 114)(66 115)(67 116)(68 117)(69 118)(70 119)(71 120)(72 101)(73 102)(74 103)(75 104)(76 105)(77 106)(78 107)(79 108)(80 109)
(1 53 104 28)(2 149 105 99)(3 55 106 30)(4 151 107 81)(5 57 108 32)(6 153 109 83)(7 59 110 34)(8 155 111 85)(9 41 112 36)(10 157 113 87)(11 43 114 38)(12 159 115 89)(13 45 116 40)(14 141 117 91)(15 47 118 22)(16 143 119 93)(17 49 120 24)(18 145 101 95)(19 51 102 26)(20 147 103 97)(21 128 46 68)(23 130 48 70)(25 132 50 72)(27 134 52 74)(29 136 54 76)(31 138 56 78)(33 140 58 80)(35 122 60 62)(37 124 42 64)(39 126 44 66)(61 84 121 154)(63 86 123 156)(65 88 125 158)(67 90 127 160)(69 92 129 142)(71 94 131 144)(73 96 133 146)(75 98 135 148)(77 100 137 150)(79 82 139 152)
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 59)(22 58)(23 57)(24 56)(25 55)(26 54)(27 53)(28 52)(29 51)(30 50)(31 49)(32 48)(33 47)(34 46)(35 45)(36 44)(37 43)(38 42)(39 41)(40 60)(61 68)(62 67)(63 66)(64 65)(69 80)(70 79)(71 78)(72 77)(73 76)(74 75)(81 144)(82 143)(83 142)(84 141)(85 160)(86 159)(87 158)(88 157)(89 156)(90 155)(91 154)(92 153)(93 152)(94 151)(95 150)(96 149)(97 148)(98 147)(99 146)(100 145)(101 106)(102 105)(103 104)(107 120)(108 119)(109 118)(110 117)(111 116)(112 115)(113 114)(121 128)(122 127)(123 126)(124 125)(129 140)(130 139)(131 138)(132 137)(133 136)(134 135)
G:=sub<Sym(160)| (1,135)(2,136)(3,137)(4,138)(5,139)(6,140)(7,121)(8,122)(9,123)(10,124)(11,125)(12,126)(13,127)(14,128)(15,129)(16,130)(17,131)(18,132)(19,133)(20,134)(21,91)(22,92)(23,93)(24,94)(25,95)(26,96)(27,97)(28,98)(29,99)(30,100)(31,81)(32,82)(33,83)(34,84)(35,85)(36,86)(37,87)(38,88)(39,89)(40,90)(41,156)(42,157)(43,158)(44,159)(45,160)(46,141)(47,142)(48,143)(49,144)(50,145)(51,146)(52,147)(53,148)(54,149)(55,150)(56,151)(57,152)(58,153)(59,154)(60,155)(61,110)(62,111)(63,112)(64,113)(65,114)(66,115)(67,116)(68,117)(69,118)(70,119)(71,120)(72,101)(73,102)(74,103)(75,104)(76,105)(77,106)(78,107)(79,108)(80,109), (1,53,104,28)(2,149,105,99)(3,55,106,30)(4,151,107,81)(5,57,108,32)(6,153,109,83)(7,59,110,34)(8,155,111,85)(9,41,112,36)(10,157,113,87)(11,43,114,38)(12,159,115,89)(13,45,116,40)(14,141,117,91)(15,47,118,22)(16,143,119,93)(17,49,120,24)(18,145,101,95)(19,51,102,26)(20,147,103,97)(21,128,46,68)(23,130,48,70)(25,132,50,72)(27,134,52,74)(29,136,54,76)(31,138,56,78)(33,140,58,80)(35,122,60,62)(37,124,42,64)(39,126,44,66)(61,84,121,154)(63,86,123,156)(65,88,125,158)(67,90,127,160)(69,92,129,142)(71,94,131,144)(73,96,133,146)(75,98,135,148)(77,100,137,150)(79,82,139,152), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,59)(22,58)(23,57)(24,56)(25,55)(26,54)(27,53)(28,52)(29,51)(30,50)(31,49)(32,48)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)(40,60)(61,68)(62,67)(63,66)(64,65)(69,80)(70,79)(71,78)(72,77)(73,76)(74,75)(81,144)(82,143)(83,142)(84,141)(85,160)(86,159)(87,158)(88,157)(89,156)(90,155)(91,154)(92,153)(93,152)(94,151)(95,150)(96,149)(97,148)(98,147)(99,146)(100,145)(101,106)(102,105)(103,104)(107,120)(108,119)(109,118)(110,117)(111,116)(112,115)(113,114)(121,128)(122,127)(123,126)(124,125)(129,140)(130,139)(131,138)(132,137)(133,136)(134,135)>;
G:=Group( (1,135)(2,136)(3,137)(4,138)(5,139)(6,140)(7,121)(8,122)(9,123)(10,124)(11,125)(12,126)(13,127)(14,128)(15,129)(16,130)(17,131)(18,132)(19,133)(20,134)(21,91)(22,92)(23,93)(24,94)(25,95)(26,96)(27,97)(28,98)(29,99)(30,100)(31,81)(32,82)(33,83)(34,84)(35,85)(36,86)(37,87)(38,88)(39,89)(40,90)(41,156)(42,157)(43,158)(44,159)(45,160)(46,141)(47,142)(48,143)(49,144)(50,145)(51,146)(52,147)(53,148)(54,149)(55,150)(56,151)(57,152)(58,153)(59,154)(60,155)(61,110)(62,111)(63,112)(64,113)(65,114)(66,115)(67,116)(68,117)(69,118)(70,119)(71,120)(72,101)(73,102)(74,103)(75,104)(76,105)(77,106)(78,107)(79,108)(80,109), (1,53,104,28)(2,149,105,99)(3,55,106,30)(4,151,107,81)(5,57,108,32)(6,153,109,83)(7,59,110,34)(8,155,111,85)(9,41,112,36)(10,157,113,87)(11,43,114,38)(12,159,115,89)(13,45,116,40)(14,141,117,91)(15,47,118,22)(16,143,119,93)(17,49,120,24)(18,145,101,95)(19,51,102,26)(20,147,103,97)(21,128,46,68)(23,130,48,70)(25,132,50,72)(27,134,52,74)(29,136,54,76)(31,138,56,78)(33,140,58,80)(35,122,60,62)(37,124,42,64)(39,126,44,66)(61,84,121,154)(63,86,123,156)(65,88,125,158)(67,90,127,160)(69,92,129,142)(71,94,131,144)(73,96,133,146)(75,98,135,148)(77,100,137,150)(79,82,139,152), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,59)(22,58)(23,57)(24,56)(25,55)(26,54)(27,53)(28,52)(29,51)(30,50)(31,49)(32,48)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)(40,60)(61,68)(62,67)(63,66)(64,65)(69,80)(70,79)(71,78)(72,77)(73,76)(74,75)(81,144)(82,143)(83,142)(84,141)(85,160)(86,159)(87,158)(88,157)(89,156)(90,155)(91,154)(92,153)(93,152)(94,151)(95,150)(96,149)(97,148)(98,147)(99,146)(100,145)(101,106)(102,105)(103,104)(107,120)(108,119)(109,118)(110,117)(111,116)(112,115)(113,114)(121,128)(122,127)(123,126)(124,125)(129,140)(130,139)(131,138)(132,137)(133,136)(134,135) );
G=PermutationGroup([[(1,135),(2,136),(3,137),(4,138),(5,139),(6,140),(7,121),(8,122),(9,123),(10,124),(11,125),(12,126),(13,127),(14,128),(15,129),(16,130),(17,131),(18,132),(19,133),(20,134),(21,91),(22,92),(23,93),(24,94),(25,95),(26,96),(27,97),(28,98),(29,99),(30,100),(31,81),(32,82),(33,83),(34,84),(35,85),(36,86),(37,87),(38,88),(39,89),(40,90),(41,156),(42,157),(43,158),(44,159),(45,160),(46,141),(47,142),(48,143),(49,144),(50,145),(51,146),(52,147),(53,148),(54,149),(55,150),(56,151),(57,152),(58,153),(59,154),(60,155),(61,110),(62,111),(63,112),(64,113),(65,114),(66,115),(67,116),(68,117),(69,118),(70,119),(71,120),(72,101),(73,102),(74,103),(75,104),(76,105),(77,106),(78,107),(79,108),(80,109)], [(1,53,104,28),(2,149,105,99),(3,55,106,30),(4,151,107,81),(5,57,108,32),(6,153,109,83),(7,59,110,34),(8,155,111,85),(9,41,112,36),(10,157,113,87),(11,43,114,38),(12,159,115,89),(13,45,116,40),(14,141,117,91),(15,47,118,22),(16,143,119,93),(17,49,120,24),(18,145,101,95),(19,51,102,26),(20,147,103,97),(21,128,46,68),(23,130,48,70),(25,132,50,72),(27,134,52,74),(29,136,54,76),(31,138,56,78),(33,140,58,80),(35,122,60,62),(37,124,42,64),(39,126,44,66),(61,84,121,154),(63,86,123,156),(65,88,125,158),(67,90,127,160),(69,92,129,142),(71,94,131,144),(73,96,133,146),(75,98,135,148),(77,100,137,150),(79,82,139,152)], [(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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(21,59),(22,58),(23,57),(24,56),(25,55),(26,54),(27,53),(28,52),(29,51),(30,50),(31,49),(32,48),(33,47),(34,46),(35,45),(36,44),(37,43),(38,42),(39,41),(40,60),(61,68),(62,67),(63,66),(64,65),(69,80),(70,79),(71,78),(72,77),(73,76),(74,75),(81,144),(82,143),(83,142),(84,141),(85,160),(86,159),(87,158),(88,157),(89,156),(90,155),(91,154),(92,153),(93,152),(94,151),(95,150),(96,149),(97,148),(98,147),(99,146),(100,145),(101,106),(102,105),(103,104),(107,120),(108,119),(109,118),(110,117),(111,116),(112,115),(113,114),(121,128),(122,127),(123,126),(124,125),(129,140),(130,139),(131,138),(132,137),(133,136),(134,135)]])
62 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2M | 4A | ··· | 4F | 4G | 4H | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20X |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 20 | ··· | 20 | 4 | ··· | 4 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | D4 | D4 | D5 | C4○D4 | D10 | D20 | D4×D5 | Q8⋊2D5 |
| kernel | (C2×C20)⋊5D4 | C5×C2.C42 | C2×D10⋊C4 | C22×D20 | C2×C20 | C22×D5 | C2.C42 | C2×C10 | C22×C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 3 | 3 | 6 | 6 | 2 | 2 | 6 | 24 | 6 | 2 |
Matrix representation of (C2×C20)⋊5D4 ►in GL6(𝔽41)
| 1 | 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 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 9 | 0 | 0 |
| 0 | 0 | 32 | 30 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 37 | 40 |
| 28 | 39 | 0 | 0 | 0 | 0 |
| 2 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 8 |
| 0 | 0 | 0 | 0 | 14 | 25 |
| 28 | 39 | 0 | 0 | 0 | 0 |
| 2 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 8 |
| 0 | 0 | 0 | 0 | 4 | 25 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,32,0,0,0,0,9,30,0,0,0,0,0,0,1,37,0,0,0,0,0,40],[28,2,0,0,0,0,39,16,0,0,0,0,0,0,0,1,0,0,0,0,40,7,0,0,0,0,0,0,16,14,0,0,0,0,8,25],[28,2,0,0,0,0,39,13,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,0,0,0,16,4,0,0,0,0,8,25] >;
(C2×C20)⋊5D4 in GAP, Magma, Sage, TeX
(C_2\times C_{20})\rtimes_5D_4 % in TeX
G:=Group("(C2xC20):5D4"); // GroupNames label
G:=SmallGroup(320,298);
// by ID
G=gap.SmallGroup(320,298);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,254,387,58,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^20=d^2=1,c*b*c^-1=a*b=b*a,a*c=c*a,a*d=d*a,d*b*d=a*b^-1,d*c*d=c^-1>;
// generators/relations