direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C20⋊7D4, C23⋊4D20, C24.71D10, C20⋊15(C2×D4), (C2×C20)⋊37D4, (C23×C4)⋊5D5, (C23×C20)⋊8C2, C22⋊2(C2×D20), C10⋊3(C4⋊D4), (C22×C4)⋊44D10, (C22×C10)⋊15D4, (C22×D20)⋊12C2, (C2×D20)⋊50C22, C4⋊Dic5⋊64C22, C2.33(C22×D20), (C2×C10).288C24, (C2×C20).705C23, (C22×C20)⋊60C22, C10.134(C22×D4), D10⋊C4⋊42C22, C22.83(C4○D20), (C23×D5).75C22, C22.303(C23×D5), C23.234(C22×D5), (C22×C10).417C23, (C23×C10).110C22, (C2×Dic5).150C23, (C22×D5).126C23, (C22×Dic5).162C22, C5⋊4(C2×C4⋊D4), C4⋊4(C2×C5⋊D4), (C2×C10)⋊11(C2×D4), (C2×C4)⋊16(C5⋊D4), (C2×C4⋊Dic5)⋊29C2, C2.71(C2×C4○D20), C10.63(C2×C4○D4), C2.7(C22×C5⋊D4), (C22×C5⋊D4)⋊11C2, (C2×C5⋊D4)⋊42C22, (C2×D10⋊C4)⋊14C2, (C2×C4).658(C22×D5), C22.104(C2×C5⋊D4), (C2×C10).114(C4○D4), SmallGroup(320,1462)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C20⋊7D4
G = < a,b,c,d | a2=b20=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 1646 in 426 conjugacy classes, 143 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, C23, D5, C10, C10, C10, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C24, C24, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C10, C2×C22⋊C4, C2×C4⋊C4, C4⋊D4, C23×C4, C22×D4, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C22×C10, C22×C10, C2×C4⋊D4, C4⋊Dic5, D10⋊C4, C2×D20, C2×D20, C22×Dic5, C2×C5⋊D4, C2×C5⋊D4, C22×C20, C22×C20, C22×C20, C23×D5, C23×C10, C2×C4⋊Dic5, C2×D10⋊C4, C20⋊7D4, C22×D20, C22×C5⋊D4, C23×C20, C2×C20⋊7D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, C24, D10, C4⋊D4, C22×D4, C2×C4○D4, D20, C5⋊D4, C22×D5, C2×C4⋊D4, C2×D20, C4○D20, C2×C5⋊D4, C23×D5, C20⋊7D4, C22×D20, C2×C4○D20, C22×C5⋊D4, C2×C20⋊7D4
►On 160 points
Generators in S160
(1 155)(2 156)(3 157)(4 158)(5 159)(6 160)(7 141)(8 142)(9 143)(10 144)(11 145)(12 146)(13 147)(14 148)(15 149)(16 150)(17 151)(18 152)(19 153)(20 154)(21 123)(22 124)(23 125)(24 126)(25 127)(26 128)(27 129)(28 130)(29 131)(30 132)(31 133)(32 134)(33 135)(34 136)(35 137)(36 138)(37 139)(38 140)(39 121)(40 122)(41 118)(42 119)(43 120)(44 101)(45 102)(46 103)(47 104)(48 105)(49 106)(50 107)(51 108)(52 109)(53 110)(54 111)(55 112)(56 113)(57 114)(58 115)(59 116)(60 117)(61 100)(62 81)(63 82)(64 83)(65 84)(66 85)(67 86)(68 87)(69 88)(70 89)(71 90)(72 91)(73 92)(74 93)(75 94)(76 95)(77 96)(78 97)(79 98)(80 99)
(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 77 109 25)(2 76 110 24)(3 75 111 23)(4 74 112 22)(5 73 113 21)(6 72 114 40)(7 71 115 39)(8 70 116 38)(9 69 117 37)(10 68 118 36)(11 67 119 35)(12 66 120 34)(13 65 101 33)(14 64 102 32)(15 63 103 31)(16 62 104 30)(17 61 105 29)(18 80 106 28)(19 79 107 27)(20 78 108 26)(41 138 144 87)(42 137 145 86)(43 136 146 85)(44 135 147 84)(45 134 148 83)(46 133 149 82)(47 132 150 81)(48 131 151 100)(49 130 152 99)(50 129 153 98)(51 128 154 97)(52 127 155 96)(53 126 156 95)(54 125 157 94)(55 124 158 93)(56 123 159 92)(57 122 160 91)(58 121 141 90)(59 140 142 89)(60 139 143 88)
(1 42)(2 41)(3 60)(4 59)(5 58)(6 57)(7 56)(8 55)(9 54)(10 53)(11 52)(12 51)(13 50)(14 49)(15 48)(16 47)(17 46)(18 45)(19 44)(20 43)(21 121)(22 140)(23 139)(24 138)(25 137)(26 136)(27 135)(28 134)(29 133)(30 132)(31 131)(32 130)(33 129)(34 128)(35 127)(36 126)(37 125)(38 124)(39 123)(40 122)(61 82)(62 81)(63 100)(64 99)(65 98)(66 97)(67 96)(68 95)(69 94)(70 93)(71 92)(72 91)(73 90)(74 89)(75 88)(76 87)(77 86)(78 85)(79 84)(80 83)(101 153)(102 152)(103 151)(104 150)(105 149)(106 148)(107 147)(108 146)(109 145)(110 144)(111 143)(112 142)(113 141)(114 160)(115 159)(116 158)(117 157)(118 156)(119 155)(120 154)
G:=sub<Sym(160)| (1,155)(2,156)(3,157)(4,158)(5,159)(6,160)(7,141)(8,142)(9,143)(10,144)(11,145)(12,146)(13,147)(14,148)(15,149)(16,150)(17,151)(18,152)(19,153)(20,154)(21,123)(22,124)(23,125)(24,126)(25,127)(26,128)(27,129)(28,130)(29,131)(30,132)(31,133)(32,134)(33,135)(34,136)(35,137)(36,138)(37,139)(38,140)(39,121)(40,122)(41,118)(42,119)(43,120)(44,101)(45,102)(46,103)(47,104)(48,105)(49,106)(50,107)(51,108)(52,109)(53,110)(54,111)(55,112)(56,113)(57,114)(58,115)(59,116)(60,117)(61,100)(62,81)(63,82)(64,83)(65,84)(66,85)(67,86)(68,87)(69,88)(70,89)(71,90)(72,91)(73,92)(74,93)(75,94)(76,95)(77,96)(78,97)(79,98)(80,99), (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), (1,77,109,25)(2,76,110,24)(3,75,111,23)(4,74,112,22)(5,73,113,21)(6,72,114,40)(7,71,115,39)(8,70,116,38)(9,69,117,37)(10,68,118,36)(11,67,119,35)(12,66,120,34)(13,65,101,33)(14,64,102,32)(15,63,103,31)(16,62,104,30)(17,61,105,29)(18,80,106,28)(19,79,107,27)(20,78,108,26)(41,138,144,87)(42,137,145,86)(43,136,146,85)(44,135,147,84)(45,134,148,83)(46,133,149,82)(47,132,150,81)(48,131,151,100)(49,130,152,99)(50,129,153,98)(51,128,154,97)(52,127,155,96)(53,126,156,95)(54,125,157,94)(55,124,158,93)(56,123,159,92)(57,122,160,91)(58,121,141,90)(59,140,142,89)(60,139,143,88), (1,42)(2,41)(3,60)(4,59)(5,58)(6,57)(7,56)(8,55)(9,54)(10,53)(11,52)(12,51)(13,50)(14,49)(15,48)(16,47)(17,46)(18,45)(19,44)(20,43)(21,121)(22,140)(23,139)(24,138)(25,137)(26,136)(27,135)(28,134)(29,133)(30,132)(31,131)(32,130)(33,129)(34,128)(35,127)(36,126)(37,125)(38,124)(39,123)(40,122)(61,82)(62,81)(63,100)(64,99)(65,98)(66,97)(67,96)(68,95)(69,94)(70,93)(71,92)(72,91)(73,90)(74,89)(75,88)(76,87)(77,86)(78,85)(79,84)(80,83)(101,153)(102,152)(103,151)(104,150)(105,149)(106,148)(107,147)(108,146)(109,145)(110,144)(111,143)(112,142)(113,141)(114,160)(115,159)(116,158)(117,157)(118,156)(119,155)(120,154)>;
G:=Group( (1,155)(2,156)(3,157)(4,158)(5,159)(6,160)(7,141)(8,142)(9,143)(10,144)(11,145)(12,146)(13,147)(14,148)(15,149)(16,150)(17,151)(18,152)(19,153)(20,154)(21,123)(22,124)(23,125)(24,126)(25,127)(26,128)(27,129)(28,130)(29,131)(30,132)(31,133)(32,134)(33,135)(34,136)(35,137)(36,138)(37,139)(38,140)(39,121)(40,122)(41,118)(42,119)(43,120)(44,101)(45,102)(46,103)(47,104)(48,105)(49,106)(50,107)(51,108)(52,109)(53,110)(54,111)(55,112)(56,113)(57,114)(58,115)(59,116)(60,117)(61,100)(62,81)(63,82)(64,83)(65,84)(66,85)(67,86)(68,87)(69,88)(70,89)(71,90)(72,91)(73,92)(74,93)(75,94)(76,95)(77,96)(78,97)(79,98)(80,99), (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), (1,77,109,25)(2,76,110,24)(3,75,111,23)(4,74,112,22)(5,73,113,21)(6,72,114,40)(7,71,115,39)(8,70,116,38)(9,69,117,37)(10,68,118,36)(11,67,119,35)(12,66,120,34)(13,65,101,33)(14,64,102,32)(15,63,103,31)(16,62,104,30)(17,61,105,29)(18,80,106,28)(19,79,107,27)(20,78,108,26)(41,138,144,87)(42,137,145,86)(43,136,146,85)(44,135,147,84)(45,134,148,83)(46,133,149,82)(47,132,150,81)(48,131,151,100)(49,130,152,99)(50,129,153,98)(51,128,154,97)(52,127,155,96)(53,126,156,95)(54,125,157,94)(55,124,158,93)(56,123,159,92)(57,122,160,91)(58,121,141,90)(59,140,142,89)(60,139,143,88), (1,42)(2,41)(3,60)(4,59)(5,58)(6,57)(7,56)(8,55)(9,54)(10,53)(11,52)(12,51)(13,50)(14,49)(15,48)(16,47)(17,46)(18,45)(19,44)(20,43)(21,121)(22,140)(23,139)(24,138)(25,137)(26,136)(27,135)(28,134)(29,133)(30,132)(31,131)(32,130)(33,129)(34,128)(35,127)(36,126)(37,125)(38,124)(39,123)(40,122)(61,82)(62,81)(63,100)(64,99)(65,98)(66,97)(67,96)(68,95)(69,94)(70,93)(71,92)(72,91)(73,90)(74,89)(75,88)(76,87)(77,86)(78,85)(79,84)(80,83)(101,153)(102,152)(103,151)(104,150)(105,149)(106,148)(107,147)(108,146)(109,145)(110,144)(111,143)(112,142)(113,141)(114,160)(115,159)(116,158)(117,157)(118,156)(119,155)(120,154) );
G=PermutationGroup([[(1,155),(2,156),(3,157),(4,158),(5,159),(6,160),(7,141),(8,142),(9,143),(10,144),(11,145),(12,146),(13,147),(14,148),(15,149),(16,150),(17,151),(18,152),(19,153),(20,154),(21,123),(22,124),(23,125),(24,126),(25,127),(26,128),(27,129),(28,130),(29,131),(30,132),(31,133),(32,134),(33,135),(34,136),(35,137),(36,138),(37,139),(38,140),(39,121),(40,122),(41,118),(42,119),(43,120),(44,101),(45,102),(46,103),(47,104),(48,105),(49,106),(50,107),(51,108),(52,109),(53,110),(54,111),(55,112),(56,113),(57,114),(58,115),(59,116),(60,117),(61,100),(62,81),(63,82),(64,83),(65,84),(66,85),(67,86),(68,87),(69,88),(70,89),(71,90),(72,91),(73,92),(74,93),(75,94),(76,95),(77,96),(78,97),(79,98),(80,99)], [(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)], [(1,77,109,25),(2,76,110,24),(3,75,111,23),(4,74,112,22),(5,73,113,21),(6,72,114,40),(7,71,115,39),(8,70,116,38),(9,69,117,37),(10,68,118,36),(11,67,119,35),(12,66,120,34),(13,65,101,33),(14,64,102,32),(15,63,103,31),(16,62,104,30),(17,61,105,29),(18,80,106,28),(19,79,107,27),(20,78,108,26),(41,138,144,87),(42,137,145,86),(43,136,146,85),(44,135,147,84),(45,134,148,83),(46,133,149,82),(47,132,150,81),(48,131,151,100),(49,130,152,99),(50,129,153,98),(51,128,154,97),(52,127,155,96),(53,126,156,95),(54,125,157,94),(55,124,158,93),(56,123,159,92),(57,122,160,91),(58,121,141,90),(59,140,142,89),(60,139,143,88)], [(1,42),(2,41),(3,60),(4,59),(5,58),(6,57),(7,56),(8,55),(9,54),(10,53),(11,52),(12,51),(13,50),(14,49),(15,48),(16,47),(17,46),(18,45),(19,44),(20,43),(21,121),(22,140),(23,139),(24,138),(25,137),(26,136),(27,135),(28,134),(29,133),(30,132),(31,131),(32,130),(33,129),(34,128),(35,127),(36,126),(37,125),(38,124),(39,123),(40,122),(61,82),(62,81),(63,100),(64,99),(65,98),(66,97),(67,96),(68,95),(69,94),(70,93),(71,92),(72,91),(73,90),(74,89),(75,88),(76,87),(77,86),(78,85),(79,84),(80,83),(101,153),(102,152),(103,151),(104,150),(105,149),(106,148),(107,147),(108,146),(109,145),(110,144),(111,143),(112,142),(113,141),(114,160),(115,159),(116,158),(117,157),(118,156),(119,155),(120,154)]])
92 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 10A | ··· | 10AD | 20A | ··· | 20AF |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
92 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | C4○D4 | D10 | D10 | C5⋊D4 | D20 | C4○D20 |
| kernel | C2×C20⋊7D4 | C2×C4⋊Dic5 | C2×D10⋊C4 | C20⋊7D4 | C22×D20 | C22×C5⋊D4 | C23×C20 | C2×C20 | C22×C10 | C23×C4 | C2×C10 | C22×C4 | C24 | C2×C4 | C23 | C22 |
| # reps | 1 | 1 | 2 | 8 | 1 | 2 | 1 | 4 | 4 | 2 | 4 | 12 | 2 | 16 | 16 | 16 |
Matrix representation of C2×C20⋊7D4 ►in GL5(𝔽41)
| 40 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 40 | 8 | 0 | 0 |
| 0 | 40 | 7 | 0 | 0 |
| 0 | 0 | 0 | 13 | 2 |
| 0 | 0 | 0 | 39 | 25 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 38 | 20 | 0 | 0 |
| 0 | 20 | 3 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 35 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 7 | 33 | 0 | 0 |
| 0 | 6 | 34 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 35 | 1 |
G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[40,0,0,0,0,0,40,40,0,0,0,8,7,0,0,0,0,0,13,39,0,0,0,2,25],[40,0,0,0,0,0,38,20,0,0,0,20,3,0,0,0,0,0,40,35,0,0,0,0,1],[1,0,0,0,0,0,7,6,0,0,0,33,34,0,0,0,0,0,40,35,0,0,0,0,1] >;
C2×C20⋊7D4 in GAP, Magma, Sage, TeX
C_2\times C_{20}\rtimes_7D_4 % in TeX
G:=Group("C2xC20:7D4"); // GroupNames label
G:=SmallGroup(320,1462);
// by ID
G=gap.SmallGroup(320,1462);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,184,675,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^20=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations