direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×C4⋊D8, C20⋊9D8, C4⋊2(C5×D8), C4⋊C8⋊3C10, D4⋊1(C5×D4), (C2×D8)⋊3C10, (C5×D4)⋊12D4, (C4×D4)⋊3C10, C2.5(C10×D8), (C10×D8)⋊17C2, (D4×C20)⋊32C2, C4⋊1D4⋊2C10, C10.77(C2×D8), C4.31(D4×C10), D4⋊C4⋊6C10, C20.392(C2×D4), (C2×C20).321D4, C42.14(C2×C10), C22.83(D4×C10), C20.341(C4○D4), (C2×C40).256C22, (C4×C20).256C22, (C2×C20).918C23, C10.142(C4⋊D4), C10.134(C8⋊C22), (D4×C10).185C22, (C5×C4⋊C8)⋊22C2, (C2×C8).3(C2×C10), C4.40(C5×C4○D4), C2.9(C5×C8⋊C22), (C5×C4⋊1D4)⋊12C2, C4⋊C4.51(C2×C10), (C2×C4).127(C5×D4), C2.11(C5×C4⋊D4), (C5×D4⋊C4)⋊29C2, (C2×D4).55(C2×C10), (C2×C10).639(C2×D4), (C5×C4⋊C4).372C22, (C2×C4).93(C22×C10), SmallGroup(320,960)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C4⋊D8
G = < a,b,c,d | a5=b4=c8=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 314 in 140 conjugacy classes, 58 normal (34 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, D8, C22×C4, C2×D4, C2×D4, C2×D4, C20, C20, C20, C2×C10, C2×C10, D4⋊C4, C4⋊C8, C4×D4, C4⋊1D4, C2×D8, C40, C2×C20, C2×C20, C5×D4, C5×D4, C22×C10, C4⋊D8, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×C40, C5×D8, C22×C20, D4×C10, D4×C10, D4×C10, C5×D4⋊C4, C5×C4⋊C8, D4×C20, C5×C4⋊1D4, C10×D8, C5×C4⋊D8
Quotients: C1, C2, C22, C5, D4, C23, C10, D8, C2×D4, C4○D4, C2×C10, C4⋊D4, C2×D8, C8⋊C22, C5×D4, C22×C10, C4⋊D8, C5×D8, D4×C10, C5×C4○D4, C5×C4⋊D4, C10×D8, C5×C8⋊C22, C5×C4⋊D8
►On 160 points
Generators in S160
(1 46 91 50 83)(2 47 92 51 84)(3 48 93 52 85)(4 41 94 53 86)(5 42 95 54 87)(6 43 96 55 88)(7 44 89 56 81)(8 45 90 49 82)(9 115 139 17 131)(10 116 140 18 132)(11 117 141 19 133)(12 118 142 20 134)(13 119 143 21 135)(14 120 144 22 136)(15 113 137 23 129)(16 114 138 24 130)(25 124 155 33 147)(26 125 156 34 148)(27 126 157 35 149)(28 127 158 36 150)(29 128 159 37 151)(30 121 160 38 152)(31 122 153 39 145)(32 123 154 40 146)(57 74 107 66 99)(58 75 108 67 100)(59 76 109 68 101)(60 77 110 69 102)(61 78 111 70 103)(62 79 112 71 104)(63 80 105 72 97)(64 73 106 65 98)
(1 114 27 104)(2 97 28 115)(3 116 29 98)(4 99 30 117)(5 118 31 100)(6 101 32 119)(7 120 25 102)(8 103 26 113)(9 84 72 150)(10 151 65 85)(11 86 66 152)(12 145 67 87)(13 88 68 146)(14 147 69 81)(15 82 70 148)(16 149 71 83)(17 92 80 158)(18 159 73 93)(19 94 74 160)(20 153 75 95)(21 96 76 154)(22 155 77 89)(23 90 78 156)(24 157 79 91)(33 110 56 136)(34 129 49 111)(35 112 50 130)(36 131 51 105)(37 106 52 132)(38 133 53 107)(39 108 54 134)(40 135 55 109)(41 57 121 141)(42 142 122 58)(43 59 123 143)(44 144 124 60)(45 61 125 137)(46 138 126 62)(47 63 127 139)(48 140 128 64)
(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)
(1 104)(2 103)(3 102)(4 101)(5 100)(6 99)(7 98)(8 97)(9 148)(10 147)(11 146)(12 145)(13 152)(14 151)(15 150)(16 149)(17 156)(18 155)(19 154)(20 153)(21 160)(22 159)(23 158)(24 157)(25 116)(26 115)(27 114)(28 113)(29 120)(30 119)(31 118)(32 117)(33 132)(34 131)(35 130)(36 129)(37 136)(38 135)(39 134)(40 133)(41 59)(42 58)(43 57)(44 64)(45 63)(46 62)(47 61)(48 60)(49 105)(50 112)(51 111)(52 110)(53 109)(54 108)(55 107)(56 106)(65 81)(66 88)(67 87)(68 86)(69 85)(70 84)(71 83)(72 82)(73 89)(74 96)(75 95)(76 94)(77 93)(78 92)(79 91)(80 90)(121 143)(122 142)(123 141)(124 140)(125 139)(126 138)(127 137)(128 144)
G:=sub<Sym(160)| (1,46,91,50,83)(2,47,92,51,84)(3,48,93,52,85)(4,41,94,53,86)(5,42,95,54,87)(6,43,96,55,88)(7,44,89,56,81)(8,45,90,49,82)(9,115,139,17,131)(10,116,140,18,132)(11,117,141,19,133)(12,118,142,20,134)(13,119,143,21,135)(14,120,144,22,136)(15,113,137,23,129)(16,114,138,24,130)(25,124,155,33,147)(26,125,156,34,148)(27,126,157,35,149)(28,127,158,36,150)(29,128,159,37,151)(30,121,160,38,152)(31,122,153,39,145)(32,123,154,40,146)(57,74,107,66,99)(58,75,108,67,100)(59,76,109,68,101)(60,77,110,69,102)(61,78,111,70,103)(62,79,112,71,104)(63,80,105,72,97)(64,73,106,65,98), (1,114,27,104)(2,97,28,115)(3,116,29,98)(4,99,30,117)(5,118,31,100)(6,101,32,119)(7,120,25,102)(8,103,26,113)(9,84,72,150)(10,151,65,85)(11,86,66,152)(12,145,67,87)(13,88,68,146)(14,147,69,81)(15,82,70,148)(16,149,71,83)(17,92,80,158)(18,159,73,93)(19,94,74,160)(20,153,75,95)(21,96,76,154)(22,155,77,89)(23,90,78,156)(24,157,79,91)(33,110,56,136)(34,129,49,111)(35,112,50,130)(36,131,51,105)(37,106,52,132)(38,133,53,107)(39,108,54,134)(40,135,55,109)(41,57,121,141)(42,142,122,58)(43,59,123,143)(44,144,124,60)(45,61,125,137)(46,138,126,62)(47,63,127,139)(48,140,128,64), (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), (1,104)(2,103)(3,102)(4,101)(5,100)(6,99)(7,98)(8,97)(9,148)(10,147)(11,146)(12,145)(13,152)(14,151)(15,150)(16,149)(17,156)(18,155)(19,154)(20,153)(21,160)(22,159)(23,158)(24,157)(25,116)(26,115)(27,114)(28,113)(29,120)(30,119)(31,118)(32,117)(33,132)(34,131)(35,130)(36,129)(37,136)(38,135)(39,134)(40,133)(41,59)(42,58)(43,57)(44,64)(45,63)(46,62)(47,61)(48,60)(49,105)(50,112)(51,111)(52,110)(53,109)(54,108)(55,107)(56,106)(65,81)(66,88)(67,87)(68,86)(69,85)(70,84)(71,83)(72,82)(73,89)(74,96)(75,95)(76,94)(77,93)(78,92)(79,91)(80,90)(121,143)(122,142)(123,141)(124,140)(125,139)(126,138)(127,137)(128,144)>;
G:=Group( (1,46,91,50,83)(2,47,92,51,84)(3,48,93,52,85)(4,41,94,53,86)(5,42,95,54,87)(6,43,96,55,88)(7,44,89,56,81)(8,45,90,49,82)(9,115,139,17,131)(10,116,140,18,132)(11,117,141,19,133)(12,118,142,20,134)(13,119,143,21,135)(14,120,144,22,136)(15,113,137,23,129)(16,114,138,24,130)(25,124,155,33,147)(26,125,156,34,148)(27,126,157,35,149)(28,127,158,36,150)(29,128,159,37,151)(30,121,160,38,152)(31,122,153,39,145)(32,123,154,40,146)(57,74,107,66,99)(58,75,108,67,100)(59,76,109,68,101)(60,77,110,69,102)(61,78,111,70,103)(62,79,112,71,104)(63,80,105,72,97)(64,73,106,65,98), (1,114,27,104)(2,97,28,115)(3,116,29,98)(4,99,30,117)(5,118,31,100)(6,101,32,119)(7,120,25,102)(8,103,26,113)(9,84,72,150)(10,151,65,85)(11,86,66,152)(12,145,67,87)(13,88,68,146)(14,147,69,81)(15,82,70,148)(16,149,71,83)(17,92,80,158)(18,159,73,93)(19,94,74,160)(20,153,75,95)(21,96,76,154)(22,155,77,89)(23,90,78,156)(24,157,79,91)(33,110,56,136)(34,129,49,111)(35,112,50,130)(36,131,51,105)(37,106,52,132)(38,133,53,107)(39,108,54,134)(40,135,55,109)(41,57,121,141)(42,142,122,58)(43,59,123,143)(44,144,124,60)(45,61,125,137)(46,138,126,62)(47,63,127,139)(48,140,128,64), (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), (1,104)(2,103)(3,102)(4,101)(5,100)(6,99)(7,98)(8,97)(9,148)(10,147)(11,146)(12,145)(13,152)(14,151)(15,150)(16,149)(17,156)(18,155)(19,154)(20,153)(21,160)(22,159)(23,158)(24,157)(25,116)(26,115)(27,114)(28,113)(29,120)(30,119)(31,118)(32,117)(33,132)(34,131)(35,130)(36,129)(37,136)(38,135)(39,134)(40,133)(41,59)(42,58)(43,57)(44,64)(45,63)(46,62)(47,61)(48,60)(49,105)(50,112)(51,111)(52,110)(53,109)(54,108)(55,107)(56,106)(65,81)(66,88)(67,87)(68,86)(69,85)(70,84)(71,83)(72,82)(73,89)(74,96)(75,95)(76,94)(77,93)(78,92)(79,91)(80,90)(121,143)(122,142)(123,141)(124,140)(125,139)(126,138)(127,137)(128,144) );
G=PermutationGroup([[(1,46,91,50,83),(2,47,92,51,84),(3,48,93,52,85),(4,41,94,53,86),(5,42,95,54,87),(6,43,96,55,88),(7,44,89,56,81),(8,45,90,49,82),(9,115,139,17,131),(10,116,140,18,132),(11,117,141,19,133),(12,118,142,20,134),(13,119,143,21,135),(14,120,144,22,136),(15,113,137,23,129),(16,114,138,24,130),(25,124,155,33,147),(26,125,156,34,148),(27,126,157,35,149),(28,127,158,36,150),(29,128,159,37,151),(30,121,160,38,152),(31,122,153,39,145),(32,123,154,40,146),(57,74,107,66,99),(58,75,108,67,100),(59,76,109,68,101),(60,77,110,69,102),(61,78,111,70,103),(62,79,112,71,104),(63,80,105,72,97),(64,73,106,65,98)], [(1,114,27,104),(2,97,28,115),(3,116,29,98),(4,99,30,117),(5,118,31,100),(6,101,32,119),(7,120,25,102),(8,103,26,113),(9,84,72,150),(10,151,65,85),(11,86,66,152),(12,145,67,87),(13,88,68,146),(14,147,69,81),(15,82,70,148),(16,149,71,83),(17,92,80,158),(18,159,73,93),(19,94,74,160),(20,153,75,95),(21,96,76,154),(22,155,77,89),(23,90,78,156),(24,157,79,91),(33,110,56,136),(34,129,49,111),(35,112,50,130),(36,131,51,105),(37,106,52,132),(38,133,53,107),(39,108,54,134),(40,135,55,109),(41,57,121,141),(42,142,122,58),(43,59,123,143),(44,144,124,60),(45,61,125,137),(46,138,126,62),(47,63,127,139),(48,140,128,64)], [(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)], [(1,104),(2,103),(3,102),(4,101),(5,100),(6,99),(7,98),(8,97),(9,148),(10,147),(11,146),(12,145),(13,152),(14,151),(15,150),(16,149),(17,156),(18,155),(19,154),(20,153),(21,160),(22,159),(23,158),(24,157),(25,116),(26,115),(27,114),(28,113),(29,120),(30,119),(31,118),(32,117),(33,132),(34,131),(35,130),(36,129),(37,136),(38,135),(39,134),(40,133),(41,59),(42,58),(43,57),(44,64),(45,63),(46,62),(47,61),(48,60),(49,105),(50,112),(51,111),(52,110),(53,109),(54,108),(55,107),(56,106),(65,81),(66,88),(67,87),(68,86),(69,85),(70,84),(71,83),(72,82),(73,89),(74,96),(75,95),(76,94),(77,93),(78,92),(79,91),(80,90),(121,143),(122,142),(123,141),(124,140),(125,139),(126,138),(127,137),(128,144)]])
95 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 10A | ··· | 10L | 10M | ··· | 10T | 10U | ··· | 10AB | 20A | ··· | 20P | 20Q | ··· | 20AB | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 8 | ··· | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
95 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | D4 | D4 | D8 | C4○D4 | C5×D4 | C5×D4 | C5×D8 | C5×C4○D4 | C8⋊C22 | C5×C8⋊C22 |
| kernel | C5×C4⋊D8 | C5×D4⋊C4 | C5×C4⋊C8 | D4×C20 | C5×C4⋊1D4 | C10×D8 | C4⋊D8 | D4⋊C4 | C4⋊C8 | C4×D4 | C4⋊1D4 | C2×D8 | C2×C20 | C5×D4 | C20 | C20 | C2×C4 | D4 | C4 | C4 | C10 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 4 | 8 | 4 | 4 | 4 | 8 | 2 | 2 | 4 | 2 | 8 | 8 | 16 | 8 | 1 | 4 |
Matrix representation of C5×C4⋊D8 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 18 |
| 40 | 39 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 40 | 39 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 24 |
| 0 | 0 | 29 | 24 |
| 40 | 39 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 40 | 1 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,18,0,0,0,0,18],[40,1,0,0,39,1,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,39,1,0,0,0,0,0,29,0,0,24,24],[40,0,0,0,39,1,0,0,0,0,40,40,0,0,0,1] >;
C5×C4⋊D8 in GAP, Magma, Sage, TeX
C_5\times C_4\rtimes D_8
% in TeX
G:=Group("C5xC4:D8"); // GroupNames label
G:=SmallGroup(320,960);
// by ID
G=gap.SmallGroup(320,960);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,589,288,1766,10085,2539,124]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^4=c^8=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