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, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×D4, C24, Dic5, C20, C20, D10, C2×C10, C2×C10, D4⋊C4, C2×C4⋊C4, C22×C8, C22×D4, C40, D20, D20, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×C10, C2×D4⋊C4, C4⋊Dic5, C4⋊Dic5, C2×C40, C2×C40, C2×D20, C2×D20, C22×Dic5, C22×C20, C23×D5, D20⋊5C4, C2×C4⋊Dic5, C22×C40, C22×D20, C2×D20⋊5C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, D8, SD16, C22×C4, C2×D4, D10, D4⋊C4, C2×C22⋊C4, C2×D8, C2×SD16, C4×D5, D20, C5⋊D4, C22×D5, C2×D4⋊C4, C40⋊C2, D40, D10⋊C4, C2×C4×D5, C2×D20, C2×C5⋊D4, D20⋊5C4, C2×C40⋊C2, C2×D40, C2×D10⋊C4, C2×D20⋊5C4
►On 160 points
Generators in S160
(1 46)(2 47)(3 48)(4 49)(5 50)(6 51)(7 52)(8 53)(9 54)(10 55)(11 56)(12 57)(13 58)(14 59)(15 60)(16 41)(17 42)(18 43)(19 44)(20 45)(21 110)(22 111)(23 112)(24 113)(25 114)(26 115)(27 116)(28 117)(29 118)(30 119)(31 120)(32 101)(33 102)(34 103)(35 104)(36 105)(37 106)(38 107)(39 108)(40 109)(61 88)(62 89)(63 90)(64 91)(65 92)(66 93)(67 94)(68 95)(69 96)(70 97)(71 98)(72 99)(73 100)(74 81)(75 82)(76 83)(77 84)(78 85)(79 86)(80 87)(121 157)(122 158)(123 159)(124 160)(125 141)(126 142)(127 143)(128 144)(129 145)(130 146)(131 147)(132 148)(133 149)(134 150)(135 151)(136 152)(137 153)(138 154)(139 155)(140 156)
(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 45)(2 44)(3 43)(4 42)(5 41)(6 60)(7 59)(8 58)(9 57)(10 56)(11 55)(12 54)(13 53)(14 52)(15 51)(16 50)(17 49)(18 48)(19 47)(20 46)(21 107)(22 106)(23 105)(24 104)(25 103)(26 102)(27 101)(28 120)(29 119)(30 118)(31 117)(32 116)(33 115)(34 114)(35 113)(36 112)(37 111)(38 110)(39 109)(40 108)(61 96)(62 95)(63 94)(64 93)(65 92)(66 91)(67 90)(68 89)(69 88)(70 87)(71 86)(72 85)(73 84)(74 83)(75 82)(76 81)(77 100)(78 99)(79 98)(80 97)(121 157)(122 156)(123 155)(124 154)(125 153)(126 152)(127 151)(128 150)(129 149)(130 148)(131 147)(132 146)(133 145)(134 144)(135 143)(136 142)(137 141)(138 160)(139 159)(140 158)
(1 139 119 90)(2 138 120 89)(3 137 101 88)(4 136 102 87)(5 135 103 86)(6 134 104 85)(7 133 105 84)(8 132 106 83)(9 131 107 82)(10 130 108 81)(11 129 109 100)(12 128 110 99)(13 127 111 98)(14 126 112 97)(15 125 113 96)(16 124 114 95)(17 123 115 94)(18 122 116 93)(19 121 117 92)(20 140 118 91)(21 72 57 144)(22 71 58 143)(23 70 59 142)(24 69 60 141)(25 68 41 160)(26 67 42 159)(27 66 43 158)(28 65 44 157)(29 64 45 156)(30 63 46 155)(31 62 47 154)(32 61 48 153)(33 80 49 152)(34 79 50 151)(35 78 51 150)(36 77 52 149)(37 76 53 148)(38 75 54 147)(39 74 55 146)(40 73 56 145)
G:=sub<Sym(160)| (1,46)(2,47)(3,48)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,55)(11,56)(12,57)(13,58)(14,59)(15,60)(16,41)(17,42)(18,43)(19,44)(20,45)(21,110)(22,111)(23,112)(24,113)(25,114)(26,115)(27,116)(28,117)(29,118)(30,119)(31,120)(32,101)(33,102)(34,103)(35,104)(36,105)(37,106)(38,107)(39,108)(40,109)(61,88)(62,89)(63,90)(64,91)(65,92)(66,93)(67,94)(68,95)(69,96)(70,97)(71,98)(72,99)(73,100)(74,81)(75,82)(76,83)(77,84)(78,85)(79,86)(80,87)(121,157)(122,158)(123,159)(124,160)(125,141)(126,142)(127,143)(128,144)(129,145)(130,146)(131,147)(132,148)(133,149)(134,150)(135,151)(136,152)(137,153)(138,154)(139,155)(140,156), (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), (1,45)(2,44)(3,43)(4,42)(5,41)(6,60)(7,59)(8,58)(9,57)(10,56)(11,55)(12,54)(13,53)(14,52)(15,51)(16,50)(17,49)(18,48)(19,47)(20,46)(21,107)(22,106)(23,105)(24,104)(25,103)(26,102)(27,101)(28,120)(29,119)(30,118)(31,117)(32,116)(33,115)(34,114)(35,113)(36,112)(37,111)(38,110)(39,109)(40,108)(61,96)(62,95)(63,94)(64,93)(65,92)(66,91)(67,90)(68,89)(69,88)(70,87)(71,86)(72,85)(73,84)(74,83)(75,82)(76,81)(77,100)(78,99)(79,98)(80,97)(121,157)(122,156)(123,155)(124,154)(125,153)(126,152)(127,151)(128,150)(129,149)(130,148)(131,147)(132,146)(133,145)(134,144)(135,143)(136,142)(137,141)(138,160)(139,159)(140,158), (1,139,119,90)(2,138,120,89)(3,137,101,88)(4,136,102,87)(5,135,103,86)(6,134,104,85)(7,133,105,84)(8,132,106,83)(9,131,107,82)(10,130,108,81)(11,129,109,100)(12,128,110,99)(13,127,111,98)(14,126,112,97)(15,125,113,96)(16,124,114,95)(17,123,115,94)(18,122,116,93)(19,121,117,92)(20,140,118,91)(21,72,57,144)(22,71,58,143)(23,70,59,142)(24,69,60,141)(25,68,41,160)(26,67,42,159)(27,66,43,158)(28,65,44,157)(29,64,45,156)(30,63,46,155)(31,62,47,154)(32,61,48,153)(33,80,49,152)(34,79,50,151)(35,78,51,150)(36,77,52,149)(37,76,53,148)(38,75,54,147)(39,74,55,146)(40,73,56,145)>;
G:=Group( (1,46)(2,47)(3,48)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,55)(11,56)(12,57)(13,58)(14,59)(15,60)(16,41)(17,42)(18,43)(19,44)(20,45)(21,110)(22,111)(23,112)(24,113)(25,114)(26,115)(27,116)(28,117)(29,118)(30,119)(31,120)(32,101)(33,102)(34,103)(35,104)(36,105)(37,106)(38,107)(39,108)(40,109)(61,88)(62,89)(63,90)(64,91)(65,92)(66,93)(67,94)(68,95)(69,96)(70,97)(71,98)(72,99)(73,100)(74,81)(75,82)(76,83)(77,84)(78,85)(79,86)(80,87)(121,157)(122,158)(123,159)(124,160)(125,141)(126,142)(127,143)(128,144)(129,145)(130,146)(131,147)(132,148)(133,149)(134,150)(135,151)(136,152)(137,153)(138,154)(139,155)(140,156), (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), (1,45)(2,44)(3,43)(4,42)(5,41)(6,60)(7,59)(8,58)(9,57)(10,56)(11,55)(12,54)(13,53)(14,52)(15,51)(16,50)(17,49)(18,48)(19,47)(20,46)(21,107)(22,106)(23,105)(24,104)(25,103)(26,102)(27,101)(28,120)(29,119)(30,118)(31,117)(32,116)(33,115)(34,114)(35,113)(36,112)(37,111)(38,110)(39,109)(40,108)(61,96)(62,95)(63,94)(64,93)(65,92)(66,91)(67,90)(68,89)(69,88)(70,87)(71,86)(72,85)(73,84)(74,83)(75,82)(76,81)(77,100)(78,99)(79,98)(80,97)(121,157)(122,156)(123,155)(124,154)(125,153)(126,152)(127,151)(128,150)(129,149)(130,148)(131,147)(132,146)(133,145)(134,144)(135,143)(136,142)(137,141)(138,160)(139,159)(140,158), (1,139,119,90)(2,138,120,89)(3,137,101,88)(4,136,102,87)(5,135,103,86)(6,134,104,85)(7,133,105,84)(8,132,106,83)(9,131,107,82)(10,130,108,81)(11,129,109,100)(12,128,110,99)(13,127,111,98)(14,126,112,97)(15,125,113,96)(16,124,114,95)(17,123,115,94)(18,122,116,93)(19,121,117,92)(20,140,118,91)(21,72,57,144)(22,71,58,143)(23,70,59,142)(24,69,60,141)(25,68,41,160)(26,67,42,159)(27,66,43,158)(28,65,44,157)(29,64,45,156)(30,63,46,155)(31,62,47,154)(32,61,48,153)(33,80,49,152)(34,79,50,151)(35,78,51,150)(36,77,52,149)(37,76,53,148)(38,75,54,147)(39,74,55,146)(40,73,56,145) );
G=PermutationGroup([[(1,46),(2,47),(3,48),(4,49),(5,50),(6,51),(7,52),(8,53),(9,54),(10,55),(11,56),(12,57),(13,58),(14,59),(15,60),(16,41),(17,42),(18,43),(19,44),(20,45),(21,110),(22,111),(23,112),(24,113),(25,114),(26,115),(27,116),(28,117),(29,118),(30,119),(31,120),(32,101),(33,102),(34,103),(35,104),(36,105),(37,106),(38,107),(39,108),(40,109),(61,88),(62,89),(63,90),(64,91),(65,92),(66,93),(67,94),(68,95),(69,96),(70,97),(71,98),(72,99),(73,100),(74,81),(75,82),(76,83),(77,84),(78,85),(79,86),(80,87),(121,157),(122,158),(123,159),(124,160),(125,141),(126,142),(127,143),(128,144),(129,145),(130,146),(131,147),(132,148),(133,149),(134,150),(135,151),(136,152),(137,153),(138,154),(139,155),(140,156)], [(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)], [(1,45),(2,44),(3,43),(4,42),(5,41),(6,60),(7,59),(8,58),(9,57),(10,56),(11,55),(12,54),(13,53),(14,52),(15,51),(16,50),(17,49),(18,48),(19,47),(20,46),(21,107),(22,106),(23,105),(24,104),(25,103),(26,102),(27,101),(28,120),(29,119),(30,118),(31,117),(32,116),(33,115),(34,114),(35,113),(36,112),(37,111),(38,110),(39,109),(40,108),(61,96),(62,95),(63,94),(64,93),(65,92),(66,91),(67,90),(68,89),(69,88),(70,87),(71,86),(72,85),(73,84),(74,83),(75,82),(76,81),(77,100),(78,99),(79,98),(80,97),(121,157),(122,156),(123,155),(124,154),(125,153),(126,152),(127,151),(128,150),(129,149),(130,148),(131,147),(132,146),(133,145),(134,144),(135,143),(136,142),(137,141),(138,160),(139,159),(140,158)], [(1,139,119,90),(2,138,120,89),(3,137,101,88),(4,136,102,87),(5,135,103,86),(6,134,104,85),(7,133,105,84),(8,132,106,83),(9,131,107,82),(10,130,108,81),(11,129,109,100),(12,128,110,99),(13,127,111,98),(14,126,112,97),(15,125,113,96),(16,124,114,95),(17,123,115,94),(18,122,116,93),(19,121,117,92),(20,140,118,91),(21,72,57,144),(22,71,58,143),(23,70,59,142),(24,69,60,141),(25,68,41,160),(26,67,42,159),(27,66,43,158),(28,65,44,157),(29,64,45,156),(30,63,46,155),(31,62,47,154),(32,61,48,153),(33,80,49,152),(34,79,50,151),(35,78,51,150),(36,77,52,149),(37,76,53,148),(38,75,54,147),(39,74,55,146),(40,73,56,145)]])
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