direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D20⋊5C4, C23.58D20, C22.16D40, (C2×C8)⋊32D10, C2.3(C2×D40), (C22×C8)⋊4D5, D20⋊27(C2×C4), (C2×D20)⋊19C4, (C22×C40)⋊3C2, (C2×C10).23D8, C10.16(C2×D8), (C2×C4).96D20, (C2×C40)⋊41C22, (C2×C20).474D4, C20.410(C2×D4), C10⋊3(D4⋊C4), C4⋊Dic5⋊47C22, C10.16(C2×SD16), (C2×C10).22SD16, (C22×D20).6C2, C22.53(C2×D20), C20.99(C22⋊C4), C20.171(C22×C4), (C2×C20).766C23, (C22×C10).138D4, (C22×C4).428D10, C4.27(D10⋊C4), (C2×D20).204C22, C22.12(C40⋊C2), (C22×C20).517C22, C22.49(D10⋊C4), C4.70(C2×C4×D5), C5⋊4(C2×D4⋊C4), C2.4(C2×C40⋊C2), (C2×C4⋊Dic5)⋊15C2, (C2×C4).116(C4×D5), C4.103(C2×C5⋊D4), (C2×C20).402(C2×C4), (C2×C10).156(C2×D4), C10.93(C2×C22⋊C4), C2.24(C2×D10⋊C4), (C2×C4).254(C5⋊D4), (C2×C4).714(C22×D5), (C2×C10).127(C22⋊C4), SmallGroup(320,739)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D20⋊5C4
G = < a,b,c,d | a2=b20=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=dbd-1=b-1, dcd-1=b3c >
Subgroups: 1054 in 202 conjugacy classes, 79 normal (27 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×6], C22 [×16], C5, C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×4], D4 [×10], C23, C23 [×10], D5 [×4], C10 [×3], C10 [×4], C4⋊C4 [×3], C2×C8 [×2], C2×C8 [×2], C22×C4, C22×C4, C2×D4 [×9], C24, Dic5 [×2], C20 [×2], C20 [×2], D10 [×16], C2×C10, C2×C10 [×6], D4⋊C4 [×4], C2×C4⋊C4, C22×C8, C22×D4, C40 [×2], D20 [×4], D20 [×6], C2×Dic5 [×4], C2×C20 [×2], C2×C20 [×4], C22×D5 [×10], C22×C10, C2×D4⋊C4, C4⋊Dic5 [×2], C4⋊Dic5, C2×C40 [×2], C2×C40 [×2], C2×D20 [×6], C2×D20 [×3], C22×Dic5, C22×C20, C23×D5, D20⋊5C4 [×4], C2×C4⋊Dic5, C22×C40, C22×D20, C2×D20⋊5C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], D8 [×2], SD16 [×2], C22×C4, C2×D4 [×2], D10 [×3], D4⋊C4 [×4], C2×C22⋊C4, C2×D8, C2×SD16, C4×D5 [×2], D20 [×2], C5⋊D4 [×2], C22×D5, C2×D4⋊C4, C40⋊C2 [×2], D40 [×2], D10⋊C4 [×4], C2×C4×D5, C2×D20, C2×C5⋊D4, D20⋊5C4 [×4], C2×C40⋊C2, C2×D40, C2×D10⋊C4, C2×D20⋊5C4
►On 160 points
Generators in S160
(1 104)(2 105)(3 106)(4 107)(5 108)(6 109)(7 110)(8 111)(9 112)(10 113)(11 114)(12 115)(13 116)(14 117)(15 118)(16 119)(17 120)(18 101)(19 102)(20 103)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(31 41)(32 42)(33 43)(34 44)(35 45)(36 46)(37 47)(38 48)(39 49)(40 50)(61 92)(62 93)(63 94)(64 95)(65 96)(66 97)(67 98)(68 99)(69 100)(70 81)(71 82)(72 83)(73 84)(74 85)(75 86)(76 87)(77 88)(78 89)(79 90)(80 91)(121 158)(122 159)(123 160)(124 141)(125 142)(126 143)(127 144)(128 145)(129 146)(130 147)(131 148)(132 149)(133 150)(134 151)(135 152)(136 153)(137 154)(138 155)(139 156)(140 157)
(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)
(1 103)(2 102)(3 101)(4 120)(5 119)(6 118)(7 117)(8 116)(9 115)(10 114)(11 113)(12 112)(13 111)(14 110)(15 109)(16 108)(17 107)(18 106)(19 105)(20 104)(21 51)(22 50)(23 49)(24 48)(25 47)(26 46)(27 45)(28 44)(29 43)(30 42)(31 41)(32 60)(33 59)(34 58)(35 57)(36 56)(37 55)(38 54)(39 53)(40 52)(61 89)(62 88)(63 87)(64 86)(65 85)(66 84)(67 83)(68 82)(69 81)(70 100)(71 99)(72 98)(73 97)(74 96)(75 95)(76 94)(77 93)(78 92)(79 91)(80 90)(121 150)(122 149)(123 148)(124 147)(125 146)(126 145)(127 144)(128 143)(129 142)(130 141)(131 160)(132 159)(133 158)(134 157)(135 156)(136 155)(137 154)(138 153)(139 152)(140 151)
(1 152 91 49)(2 151 92 48)(3 150 93 47)(4 149 94 46)(5 148 95 45)(6 147 96 44)(7 146 97 43)(8 145 98 42)(9 144 99 41)(10 143 100 60)(11 142 81 59)(12 141 82 58)(13 160 83 57)(14 159 84 56)(15 158 85 55)(16 157 86 54)(17 156 87 53)(18 155 88 52)(19 154 89 51)(20 153 90 50)(21 102 137 78)(22 101 138 77)(23 120 139 76)(24 119 140 75)(25 118 121 74)(26 117 122 73)(27 116 123 72)(28 115 124 71)(29 114 125 70)(30 113 126 69)(31 112 127 68)(32 111 128 67)(33 110 129 66)(34 109 130 65)(35 108 131 64)(36 107 132 63)(37 106 133 62)(38 105 134 61)(39 104 135 80)(40 103 136 79)
G:=sub<Sym(160)| (1,104)(2,105)(3,106)(4,107)(5,108)(6,109)(7,110)(8,111)(9,112)(10,113)(11,114)(12,115)(13,116)(14,117)(15,118)(16,119)(17,120)(18,101)(19,102)(20,103)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(31,41)(32,42)(33,43)(34,44)(35,45)(36,46)(37,47)(38,48)(39,49)(40,50)(61,92)(62,93)(63,94)(64,95)(65,96)(66,97)(67,98)(68,99)(69,100)(70,81)(71,82)(72,83)(73,84)(74,85)(75,86)(76,87)(77,88)(78,89)(79,90)(80,91)(121,158)(122,159)(123,160)(124,141)(125,142)(126,143)(127,144)(128,145)(129,146)(130,147)(131,148)(132,149)(133,150)(134,151)(135,152)(136,153)(137,154)(138,155)(139,156)(140,157), (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), (1,103)(2,102)(3,101)(4,120)(5,119)(6,118)(7,117)(8,116)(9,115)(10,114)(11,113)(12,112)(13,111)(14,110)(15,109)(16,108)(17,107)(18,106)(19,105)(20,104)(21,51)(22,50)(23,49)(24,48)(25,47)(26,46)(27,45)(28,44)(29,43)(30,42)(31,41)(32,60)(33,59)(34,58)(35,57)(36,56)(37,55)(38,54)(39,53)(40,52)(61,89)(62,88)(63,87)(64,86)(65,85)(66,84)(67,83)(68,82)(69,81)(70,100)(71,99)(72,98)(73,97)(74,96)(75,95)(76,94)(77,93)(78,92)(79,91)(80,90)(121,150)(122,149)(123,148)(124,147)(125,146)(126,145)(127,144)(128,143)(129,142)(130,141)(131,160)(132,159)(133,158)(134,157)(135,156)(136,155)(137,154)(138,153)(139,152)(140,151), (1,152,91,49)(2,151,92,48)(3,150,93,47)(4,149,94,46)(5,148,95,45)(6,147,96,44)(7,146,97,43)(8,145,98,42)(9,144,99,41)(10,143,100,60)(11,142,81,59)(12,141,82,58)(13,160,83,57)(14,159,84,56)(15,158,85,55)(16,157,86,54)(17,156,87,53)(18,155,88,52)(19,154,89,51)(20,153,90,50)(21,102,137,78)(22,101,138,77)(23,120,139,76)(24,119,140,75)(25,118,121,74)(26,117,122,73)(27,116,123,72)(28,115,124,71)(29,114,125,70)(30,113,126,69)(31,112,127,68)(32,111,128,67)(33,110,129,66)(34,109,130,65)(35,108,131,64)(36,107,132,63)(37,106,133,62)(38,105,134,61)(39,104,135,80)(40,103,136,79)>;
G:=Group( (1,104)(2,105)(3,106)(4,107)(5,108)(6,109)(7,110)(8,111)(9,112)(10,113)(11,114)(12,115)(13,116)(14,117)(15,118)(16,119)(17,120)(18,101)(19,102)(20,103)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(31,41)(32,42)(33,43)(34,44)(35,45)(36,46)(37,47)(38,48)(39,49)(40,50)(61,92)(62,93)(63,94)(64,95)(65,96)(66,97)(67,98)(68,99)(69,100)(70,81)(71,82)(72,83)(73,84)(74,85)(75,86)(76,87)(77,88)(78,89)(79,90)(80,91)(121,158)(122,159)(123,160)(124,141)(125,142)(126,143)(127,144)(128,145)(129,146)(130,147)(131,148)(132,149)(133,150)(134,151)(135,152)(136,153)(137,154)(138,155)(139,156)(140,157), (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), (1,103)(2,102)(3,101)(4,120)(5,119)(6,118)(7,117)(8,116)(9,115)(10,114)(11,113)(12,112)(13,111)(14,110)(15,109)(16,108)(17,107)(18,106)(19,105)(20,104)(21,51)(22,50)(23,49)(24,48)(25,47)(26,46)(27,45)(28,44)(29,43)(30,42)(31,41)(32,60)(33,59)(34,58)(35,57)(36,56)(37,55)(38,54)(39,53)(40,52)(61,89)(62,88)(63,87)(64,86)(65,85)(66,84)(67,83)(68,82)(69,81)(70,100)(71,99)(72,98)(73,97)(74,96)(75,95)(76,94)(77,93)(78,92)(79,91)(80,90)(121,150)(122,149)(123,148)(124,147)(125,146)(126,145)(127,144)(128,143)(129,142)(130,141)(131,160)(132,159)(133,158)(134,157)(135,156)(136,155)(137,154)(138,153)(139,152)(140,151), (1,152,91,49)(2,151,92,48)(3,150,93,47)(4,149,94,46)(5,148,95,45)(6,147,96,44)(7,146,97,43)(8,145,98,42)(9,144,99,41)(10,143,100,60)(11,142,81,59)(12,141,82,58)(13,160,83,57)(14,159,84,56)(15,158,85,55)(16,157,86,54)(17,156,87,53)(18,155,88,52)(19,154,89,51)(20,153,90,50)(21,102,137,78)(22,101,138,77)(23,120,139,76)(24,119,140,75)(25,118,121,74)(26,117,122,73)(27,116,123,72)(28,115,124,71)(29,114,125,70)(30,113,126,69)(31,112,127,68)(32,111,128,67)(33,110,129,66)(34,109,130,65)(35,108,131,64)(36,107,132,63)(37,106,133,62)(38,105,134,61)(39,104,135,80)(40,103,136,79) );
G=PermutationGroup([(1,104),(2,105),(3,106),(4,107),(5,108),(6,109),(7,110),(8,111),(9,112),(10,113),(11,114),(12,115),(13,116),(14,117),(15,118),(16,119),(17,120),(18,101),(19,102),(20,103),(21,51),(22,52),(23,53),(24,54),(25,55),(26,56),(27,57),(28,58),(29,59),(30,60),(31,41),(32,42),(33,43),(34,44),(35,45),(36,46),(37,47),(38,48),(39,49),(40,50),(61,92),(62,93),(63,94),(64,95),(65,96),(66,97),(67,98),(68,99),(69,100),(70,81),(71,82),(72,83),(73,84),(74,85),(75,86),(76,87),(77,88),(78,89),(79,90),(80,91),(121,158),(122,159),(123,160),(124,141),(125,142),(126,143),(127,144),(128,145),(129,146),(130,147),(131,148),(132,149),(133,150),(134,151),(135,152),(136,153),(137,154),(138,155),(139,156),(140,157)], [(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)], [(1,103),(2,102),(3,101),(4,120),(5,119),(6,118),(7,117),(8,116),(9,115),(10,114),(11,113),(12,112),(13,111),(14,110),(15,109),(16,108),(17,107),(18,106),(19,105),(20,104),(21,51),(22,50),(23,49),(24,48),(25,47),(26,46),(27,45),(28,44),(29,43),(30,42),(31,41),(32,60),(33,59),(34,58),(35,57),(36,56),(37,55),(38,54),(39,53),(40,52),(61,89),(62,88),(63,87),(64,86),(65,85),(66,84),(67,83),(68,82),(69,81),(70,100),(71,99),(72,98),(73,97),(74,96),(75,95),(76,94),(77,93),(78,92),(79,91),(80,90),(121,150),(122,149),(123,148),(124,147),(125,146),(126,145),(127,144),(128,143),(129,142),(130,141),(131,160),(132,159),(133,158),(134,157),(135,156),(136,155),(137,154),(138,153),(139,152),(140,151)], [(1,152,91,49),(2,151,92,48),(3,150,93,47),(4,149,94,46),(5,148,95,45),(6,147,96,44),(7,146,97,43),(8,145,98,42),(9,144,99,41),(10,143,100,60),(11,142,81,59),(12,141,82,58),(13,160,83,57),(14,159,84,56),(15,158,85,55),(16,157,86,54),(17,156,87,53),(18,155,88,52),(19,154,89,51),(20,153,90,50),(21,102,137,78),(22,101,138,77),(23,120,139,76),(24,119,140,75),(25,118,121,74),(26,117,122,73),(27,116,123,72),(28,115,124,71),(29,114,125,70),(30,113,126,69),(31,112,127,68),(32,111,128,67),(33,110,129,66),(34,109,130,65),(35,108,131,64),(36,107,132,63),(37,106,133,62),(38,105,134,61),(39,104,135,80),(40,103,136,79)])
92 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | ··· | 8H | 10A | ··· | 10N | 20A | ··· | 20P | 40A | ··· | 40AF |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | ··· | 1 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
92 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D5 | D8 | SD16 | D10 | D10 | C4×D5 | D20 | C5⋊D4 | D20 | C40⋊C2 | D40 |
| kernel | C2×D20⋊5C4 | D20⋊5C4 | C2×C4⋊Dic5 | C22×C40 | C22×D20 | C2×D20 | C2×C20 | C22×C10 | C22×C8 | C2×C10 | C2×C10 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C22 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 3 | 1 | 2 | 4 | 4 | 4 | 2 | 8 | 4 | 8 | 4 | 16 | 16 |
Matrix representation of C2×D20⋊5C4 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 11 | 39 |
| 0 | 0 | 16 | 27 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 9 | 25 |
| 0 | 0 | 5 | 32 |
| 32 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 39 | 3 |
| 0 | 0 | 40 | 2 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,11,16,0,0,39,27],[1,0,0,0,0,1,0,0,0,0,9,5,0,0,25,32],[32,0,0,0,0,1,0,0,0,0,39,40,0,0,3,2] >;
C2×D20⋊5C4 in GAP, Magma, Sage, TeX
C_2\times D_{20}\rtimes_5C_4 % in TeX
G:=Group("C2xD20:5C4"); // GroupNames label
G:=SmallGroup(320,739);
// by ID
G=gap.SmallGroup(320,739);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,142,1123,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^20=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=d*b*d^-1=b^-1,d*c*d^-1=b^3*c>;
// generators/relations