metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×D20)⋊22C4, C10.89(C4×D4), C20⋊5(C22⋊C4), (C2×Dic5)⋊11D4, (C2×C4).144D20, (C2×C20).142D4, C4⋊1(D10⋊C4), C2.3(C20⋊D4), C2.5(C4⋊D20), C22.111(D4×D5), C22.48(C2×D20), C10.54(C4⋊D4), C10.15(C4⋊1D4), (C22×D20).12C2, (C22×C4).335D10, C2.18(D20⋊8C4), C2.2(C20.23D4), C10.51(C4.4D4), C5⋊4(C24.3C22), (C23×D5).18C22, C23.296(C22×D5), (C22×C10).352C23, (C22×C20).144C22, C22.28(Q8⋊2D5), (C22×Dic5).215C22, (C2×C4⋊C4)⋊6D5, (C10×C4⋊C4)⋊6C2, (C2×C4×Dic5)⋊1C2, (C2×C4).79(C4×D5), C22.137(C2×C4×D5), (C2×C20).259(C2×C4), (C2×C10).452(C2×D4), C10.84(C2×C22⋊C4), (C2×D10⋊C4)⋊10C2, C22.67(C2×C5⋊D4), C2.16(C2×D10⋊C4), (C2×C4).129(C5⋊D4), (C22×D5).26(C2×C4), (C2×C10).189(C4○D4), (C2×C10).221(C22×C4), SmallGroup(320,615)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×D20)⋊22C4
G = < a,b,c,d | a2=b20=c2=d4=1, ab=ba, dcd-1=ac=ca, ad=da, cbc=b-1, dbd-1=b11 >
Subgroups: 1182 in 258 conjugacy classes, 83 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C24, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C22×D4, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C24.3C22, C4×Dic5, D10⋊C4, C5×C4⋊C4, C2×D20, C2×D20, C22×Dic5, C22×C20, C22×C20, C23×D5, C2×C4×Dic5, C2×D10⋊C4, C10×C4⋊C4, C22×D20, (C2×D20)⋊22C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, C4○D4, D10, C2×C22⋊C4, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C4×D5, D20, C5⋊D4, C22×D5, C24.3C22, D10⋊C4, C2×C4×D5, C2×D20, D4×D5, Q8⋊2D5, C2×C5⋊D4, D20⋊8C4, C4⋊D20, C2×D10⋊C4, C20⋊D4, C20.23D4, (C2×D20)⋊22C4
►On 160 points
Generators in S160
(1 83)(2 84)(3 85)(4 86)(5 87)(6 88)(7 89)(8 90)(9 91)(10 92)(11 93)(12 94)(13 95)(14 96)(15 97)(16 98)(17 99)(18 100)(19 81)(20 82)(21 43)(22 44)(23 45)(24 46)(25 47)(26 48)(27 49)(28 50)(29 51)(30 52)(31 53)(32 54)(33 55)(34 56)(35 57)(36 58)(37 59)(38 60)(39 41)(40 42)(61 150)(62 151)(63 152)(64 153)(65 154)(66 155)(67 156)(68 157)(69 158)(70 159)(71 160)(72 141)(73 142)(74 143)(75 144)(76 145)(77 146)(78 147)(79 148)(80 149)(101 132)(102 133)(103 134)(104 135)(105 136)(106 137)(107 138)(108 139)(109 140)(110 121)(111 122)(112 123)(113 124)(114 125)(115 126)(116 127)(117 128)(118 129)(119 130)(120 131)
(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 53)(22 52)(23 51)(24 50)(25 49)(26 48)(27 47)(28 46)(29 45)(30 44)(31 43)(32 42)(33 41)(34 60)(35 59)(36 58)(37 57)(38 56)(39 55)(40 54)(61 71)(62 70)(63 69)(64 68)(65 67)(72 80)(73 79)(74 78)(75 77)(81 99)(82 98)(83 97)(84 96)(85 95)(86 94)(87 93)(88 92)(89 91)(101 122)(102 121)(103 140)(104 139)(105 138)(106 137)(107 136)(108 135)(109 134)(110 133)(111 132)(112 131)(113 130)(114 129)(115 128)(116 127)(117 126)(118 125)(119 124)(120 123)(141 149)(142 148)(143 147)(144 146)(150 160)(151 159)(152 158)(153 157)(154 156)
(1 109 158 41)(2 120 159 52)(3 111 160 43)(4 102 141 54)(5 113 142 45)(6 104 143 56)(7 115 144 47)(8 106 145 58)(9 117 146 49)(10 108 147 60)(11 119 148 51)(12 110 149 42)(13 101 150 53)(14 112 151 44)(15 103 152 55)(16 114 153 46)(17 105 154 57)(18 116 155 48)(19 107 156 59)(20 118 157 50)(21 85 122 71)(22 96 123 62)(23 87 124 73)(24 98 125 64)(25 89 126 75)(26 100 127 66)(27 91 128 77)(28 82 129 68)(29 93 130 79)(30 84 131 70)(31 95 132 61)(32 86 133 72)(33 97 134 63)(34 88 135 74)(35 99 136 65)(36 90 137 76)(37 81 138 67)(38 92 139 78)(39 83 140 69)(40 94 121 80)
G:=sub<Sym(160)| (1,83)(2,84)(3,85)(4,86)(5,87)(6,88)(7,89)(8,90)(9,91)(10,92)(11,93)(12,94)(13,95)(14,96)(15,97)(16,98)(17,99)(18,100)(19,81)(20,82)(21,43)(22,44)(23,45)(24,46)(25,47)(26,48)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,55)(34,56)(35,57)(36,58)(37,59)(38,60)(39,41)(40,42)(61,150)(62,151)(63,152)(64,153)(65,154)(66,155)(67,156)(68,157)(69,158)(70,159)(71,160)(72,141)(73,142)(74,143)(75,144)(76,145)(77,146)(78,147)(79,148)(80,149)(101,132)(102,133)(103,134)(104,135)(105,136)(106,137)(107,138)(108,139)(109,140)(110,121)(111,122)(112,123)(113,124)(114,125)(115,126)(116,127)(117,128)(118,129)(119,130)(120,131), (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)(21,53)(22,52)(23,51)(24,50)(25,49)(26,48)(27,47)(28,46)(29,45)(30,44)(31,43)(32,42)(33,41)(34,60)(35,59)(36,58)(37,57)(38,56)(39,55)(40,54)(61,71)(62,70)(63,69)(64,68)(65,67)(72,80)(73,79)(74,78)(75,77)(81,99)(82,98)(83,97)(84,96)(85,95)(86,94)(87,93)(88,92)(89,91)(101,122)(102,121)(103,140)(104,139)(105,138)(106,137)(107,136)(108,135)(109,134)(110,133)(111,132)(112,131)(113,130)(114,129)(115,128)(116,127)(117,126)(118,125)(119,124)(120,123)(141,149)(142,148)(143,147)(144,146)(150,160)(151,159)(152,158)(153,157)(154,156), (1,109,158,41)(2,120,159,52)(3,111,160,43)(4,102,141,54)(5,113,142,45)(6,104,143,56)(7,115,144,47)(8,106,145,58)(9,117,146,49)(10,108,147,60)(11,119,148,51)(12,110,149,42)(13,101,150,53)(14,112,151,44)(15,103,152,55)(16,114,153,46)(17,105,154,57)(18,116,155,48)(19,107,156,59)(20,118,157,50)(21,85,122,71)(22,96,123,62)(23,87,124,73)(24,98,125,64)(25,89,126,75)(26,100,127,66)(27,91,128,77)(28,82,129,68)(29,93,130,79)(30,84,131,70)(31,95,132,61)(32,86,133,72)(33,97,134,63)(34,88,135,74)(35,99,136,65)(36,90,137,76)(37,81,138,67)(38,92,139,78)(39,83,140,69)(40,94,121,80)>;
G:=Group( (1,83)(2,84)(3,85)(4,86)(5,87)(6,88)(7,89)(8,90)(9,91)(10,92)(11,93)(12,94)(13,95)(14,96)(15,97)(16,98)(17,99)(18,100)(19,81)(20,82)(21,43)(22,44)(23,45)(24,46)(25,47)(26,48)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,55)(34,56)(35,57)(36,58)(37,59)(38,60)(39,41)(40,42)(61,150)(62,151)(63,152)(64,153)(65,154)(66,155)(67,156)(68,157)(69,158)(70,159)(71,160)(72,141)(73,142)(74,143)(75,144)(76,145)(77,146)(78,147)(79,148)(80,149)(101,132)(102,133)(103,134)(104,135)(105,136)(106,137)(107,138)(108,139)(109,140)(110,121)(111,122)(112,123)(113,124)(114,125)(115,126)(116,127)(117,128)(118,129)(119,130)(120,131), (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)(21,53)(22,52)(23,51)(24,50)(25,49)(26,48)(27,47)(28,46)(29,45)(30,44)(31,43)(32,42)(33,41)(34,60)(35,59)(36,58)(37,57)(38,56)(39,55)(40,54)(61,71)(62,70)(63,69)(64,68)(65,67)(72,80)(73,79)(74,78)(75,77)(81,99)(82,98)(83,97)(84,96)(85,95)(86,94)(87,93)(88,92)(89,91)(101,122)(102,121)(103,140)(104,139)(105,138)(106,137)(107,136)(108,135)(109,134)(110,133)(111,132)(112,131)(113,130)(114,129)(115,128)(116,127)(117,126)(118,125)(119,124)(120,123)(141,149)(142,148)(143,147)(144,146)(150,160)(151,159)(152,158)(153,157)(154,156), (1,109,158,41)(2,120,159,52)(3,111,160,43)(4,102,141,54)(5,113,142,45)(6,104,143,56)(7,115,144,47)(8,106,145,58)(9,117,146,49)(10,108,147,60)(11,119,148,51)(12,110,149,42)(13,101,150,53)(14,112,151,44)(15,103,152,55)(16,114,153,46)(17,105,154,57)(18,116,155,48)(19,107,156,59)(20,118,157,50)(21,85,122,71)(22,96,123,62)(23,87,124,73)(24,98,125,64)(25,89,126,75)(26,100,127,66)(27,91,128,77)(28,82,129,68)(29,93,130,79)(30,84,131,70)(31,95,132,61)(32,86,133,72)(33,97,134,63)(34,88,135,74)(35,99,136,65)(36,90,137,76)(37,81,138,67)(38,92,139,78)(39,83,140,69)(40,94,121,80) );
G=PermutationGroup([[(1,83),(2,84),(3,85),(4,86),(5,87),(6,88),(7,89),(8,90),(9,91),(10,92),(11,93),(12,94),(13,95),(14,96),(15,97),(16,98),(17,99),(18,100),(19,81),(20,82),(21,43),(22,44),(23,45),(24,46),(25,47),(26,48),(27,49),(28,50),(29,51),(30,52),(31,53),(32,54),(33,55),(34,56),(35,57),(36,58),(37,59),(38,60),(39,41),(40,42),(61,150),(62,151),(63,152),(64,153),(65,154),(66,155),(67,156),(68,157),(69,158),(70,159),(71,160),(72,141),(73,142),(74,143),(75,144),(76,145),(77,146),(78,147),(79,148),(80,149),(101,132),(102,133),(103,134),(104,135),(105,136),(106,137),(107,138),(108,139),(109,140),(110,121),(111,122),(112,123),(113,124),(114,125),(115,126),(116,127),(117,128),(118,129),(119,130),(120,131)], [(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),(21,53),(22,52),(23,51),(24,50),(25,49),(26,48),(27,47),(28,46),(29,45),(30,44),(31,43),(32,42),(33,41),(34,60),(35,59),(36,58),(37,57),(38,56),(39,55),(40,54),(61,71),(62,70),(63,69),(64,68),(65,67),(72,80),(73,79),(74,78),(75,77),(81,99),(82,98),(83,97),(84,96),(85,95),(86,94),(87,93),(88,92),(89,91),(101,122),(102,121),(103,140),(104,139),(105,138),(106,137),(107,136),(108,135),(109,134),(110,133),(111,132),(112,131),(113,130),(114,129),(115,128),(116,127),(117,126),(118,125),(119,124),(120,123),(141,149),(142,148),(143,147),(144,146),(150,160),(151,159),(152,158),(153,157),(154,156)], [(1,109,158,41),(2,120,159,52),(3,111,160,43),(4,102,141,54),(5,113,142,45),(6,104,143,56),(7,115,144,47),(8,106,145,58),(9,117,146,49),(10,108,147,60),(11,119,148,51),(12,110,149,42),(13,101,150,53),(14,112,151,44),(15,103,152,55),(16,114,153,46),(17,105,154,57),(18,116,155,48),(19,107,156,59),(20,118,157,50),(21,85,122,71),(22,96,123,62),(23,87,124,73),(24,98,125,64),(25,89,126,75),(26,100,127,66),(27,91,128,77),(28,82,129,68),(29,93,130,79),(30,84,131,70),(31,95,132,61),(32,86,133,72),(33,97,134,63),(34,88,135,74),(35,99,136,65),(36,90,137,76),(37,81,138,67),(38,92,139,78),(39,83,140,69),(40,94,121,80)]])
68 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20X |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D5 | C4○D4 | D10 | C4×D5 | D20 | C5⋊D4 | D4×D5 | Q8⋊2D5 |
| kernel | (C2×D20)⋊22C4 | C2×C4×Dic5 | C2×D10⋊C4 | C10×C4⋊C4 | C22×D20 | C2×D20 | C2×Dic5 | C2×C20 | C2×C4⋊C4 | C2×C10 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 4 | 1 | 1 | 8 | 4 | 4 | 2 | 4 | 6 | 8 | 8 | 8 | 4 | 4 |
Matrix representation of (C2×D20)⋊22C4 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 35 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 35 | 1 | 0 | 0 | 0 | 0 |
| 6 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 40 | 0 | 0 |
| 0 | 0 | 7 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 2 | 13 | 0 | 0 | 0 | 0 |
| 28 | 39 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 1 | 0 | 0 |
| 0 | 0 | 40 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[35,40,0,0,0,0,1,0,0,0,0,0,0,0,7,40,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[35,6,0,0,0,0,1,6,0,0,0,0,0,0,34,7,0,0,0,0,40,7,0,0,0,0,0,0,40,0,0,0,0,0,0,1],[2,28,0,0,0,0,13,39,0,0,0,0,0,0,24,40,0,0,0,0,1,17,0,0,0,0,0,0,9,0,0,0,0,0,0,32] >;
(C2×D20)⋊22C4 in GAP, Magma, Sage, TeX
(C_2\times D_{20})\rtimes_{22}C_4 % in TeX
G:=Group("(C2xD20):22C4"); // GroupNames label
G:=SmallGroup(320,615);
// by ID
G=gap.SmallGroup(320,615);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,232,422,387,58,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^20=c^2=d^4=1,a*b=b*a,d*c*d^-1=a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^11>;
// generators/relations