direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D5×C4⋊C8, C42.199D10, D10.14M4(2), C4⋊3(C8×D5), C20⋊5(C2×C8), (C4×D5)⋊1C8, C4.54(Q8×D5), Dic5⋊4(C2×C8), C20⋊3C8⋊11C2, C4.204(D4×D5), (C4×D5).34Q8, (C4×D5).123D4, D10.20(C2×C8), C20.363(C2×D4), (C2×C8).214D10, C20.112(C2×Q8), (C4×Dic5).8C4, (D5×C42).1C2, C2.5(D5×M4(2)), D10.35(C4⋊C4), (C4×C20).58C22, C10.33(C22×C8), C20.8Q8⋊22C2, Dic5.37(C4⋊C4), (C2×C40).208C22, (C2×C20).829C23, C10.59(C2×M4(2)), (C4×Dic5).306C22, C5⋊4(C2×C4⋊C8), (C5×C4⋊C8)⋊13C2, C2.3(D5×C4⋊C4), (C2×C4×D5).9C4, C2.11(D5×C2×C8), (D5×C2×C8).17C2, C10.30(C2×C4⋊C4), C22.46(C2×C4×D5), (C2×C4).144(C4×D5), (C2×C20).241(C2×C4), (C2×C4×D5).420C22, (C2×C4).771(C22×D5), (C2×C10).185(C22×C4), (C2×C5⋊2C8).312C22, (C2×Dic5).142(C2×C4), (C22×D5).139(C2×C4), SmallGroup(320,459)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D5×C4⋊C8
G = < a,b,c,d | a5=b2=c4=d8=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 398 in 138 conjugacy classes, 73 normal (37 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, D5, C10, C42, C42, C2×C8, C2×C8, C22×C4, Dic5, Dic5, C20, C20, C20, D10, C2×C10, C4⋊C8, C4⋊C8, C2×C42, C22×C8, C5⋊2C8, C40, C4×D5, C4×D5, C2×Dic5, C2×C20, C22×D5, C2×C4⋊C8, C8×D5, C2×C5⋊2C8, C4×Dic5, C4×C20, C2×C40, C2×C4×D5, C20⋊3C8, C20.8Q8, C5×C4⋊C8, D5×C42, D5×C2×C8, D5×C4⋊C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, Q8, C23, D5, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, D10, C4⋊C8, C2×C4⋊C4, C22×C8, C2×M4(2), C4×D5, C22×D5, C2×C4⋊C8, C8×D5, C2×C4×D5, D4×D5, Q8×D5, D5×C4⋊C4, D5×C2×C8, D5×M4(2), D5×C4⋊C8
►On 160 points
Generators in S160
(1 73 97 121 54)(2 74 98 122 55)(3 75 99 123 56)(4 76 100 124 49)(5 77 101 125 50)(6 78 102 126 51)(7 79 103 127 52)(8 80 104 128 53)(9 34 152 116 81)(10 35 145 117 82)(11 36 146 118 83)(12 37 147 119 84)(13 38 148 120 85)(14 39 149 113 86)(15 40 150 114 87)(16 33 151 115 88)(17 72 91 130 48)(18 65 92 131 41)(19 66 93 132 42)(20 67 94 133 43)(21 68 95 134 44)(22 69 96 135 45)(23 70 89 136 46)(24 71 90 129 47)(25 62 108 140 154)(26 63 109 141 155)(27 64 110 142 156)(28 57 111 143 157)(29 58 112 144 158)(30 59 105 137 159)(31 60 106 138 160)(32 61 107 139 153)
(1 50)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 49)(9 13)(10 14)(11 15)(12 16)(17 44)(18 45)(19 46)(20 47)(21 48)(22 41)(23 42)(24 43)(25 29)(26 30)(27 31)(28 32)(33 84)(34 85)(35 86)(36 87)(37 88)(38 81)(39 82)(40 83)(57 153)(58 154)(59 155)(60 156)(61 157)(62 158)(63 159)(64 160)(65 135)(66 136)(67 129)(68 130)(69 131)(70 132)(71 133)(72 134)(73 125)(74 126)(75 127)(76 128)(77 121)(78 122)(79 123)(80 124)(89 93)(90 94)(91 95)(92 96)(97 101)(98 102)(99 103)(100 104)(105 141)(106 142)(107 143)(108 144)(109 137)(110 138)(111 139)(112 140)(113 145)(114 146)(115 147)(116 148)(117 149)(118 150)(119 151)(120 152)
(1 140 119 22)(2 23 120 141)(3 142 113 24)(4 17 114 143)(5 144 115 18)(6 19 116 137)(7 138 117 20)(8 21 118 139)(9 30 102 93)(10 94 103 31)(11 32 104 95)(12 96 97 25)(13 26 98 89)(14 90 99 27)(15 28 100 91)(16 92 101 29)(33 131 125 58)(34 59 126 132)(35 133 127 60)(36 61 128 134)(37 135 121 62)(38 63 122 136)(39 129 123 64)(40 57 124 130)(41 50 112 151)(42 152 105 51)(43 52 106 145)(44 146 107 53)(45 54 108 147)(46 148 109 55)(47 56 110 149)(48 150 111 49)(65 77 158 88)(66 81 159 78)(67 79 160 82)(68 83 153 80)(69 73 154 84)(70 85 155 74)(71 75 156 86)(72 87 157 76)
(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)
G:=sub<Sym(160)| (1,73,97,121,54)(2,74,98,122,55)(3,75,99,123,56)(4,76,100,124,49)(5,77,101,125,50)(6,78,102,126,51)(7,79,103,127,52)(8,80,104,128,53)(9,34,152,116,81)(10,35,145,117,82)(11,36,146,118,83)(12,37,147,119,84)(13,38,148,120,85)(14,39,149,113,86)(15,40,150,114,87)(16,33,151,115,88)(17,72,91,130,48)(18,65,92,131,41)(19,66,93,132,42)(20,67,94,133,43)(21,68,95,134,44)(22,69,96,135,45)(23,70,89,136,46)(24,71,90,129,47)(25,62,108,140,154)(26,63,109,141,155)(27,64,110,142,156)(28,57,111,143,157)(29,58,112,144,158)(30,59,105,137,159)(31,60,106,138,160)(32,61,107,139,153), (1,50)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,49)(9,13)(10,14)(11,15)(12,16)(17,44)(18,45)(19,46)(20,47)(21,48)(22,41)(23,42)(24,43)(25,29)(26,30)(27,31)(28,32)(33,84)(34,85)(35,86)(36,87)(37,88)(38,81)(39,82)(40,83)(57,153)(58,154)(59,155)(60,156)(61,157)(62,158)(63,159)(64,160)(65,135)(66,136)(67,129)(68,130)(69,131)(70,132)(71,133)(72,134)(73,125)(74,126)(75,127)(76,128)(77,121)(78,122)(79,123)(80,124)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(105,141)(106,142)(107,143)(108,144)(109,137)(110,138)(111,139)(112,140)(113,145)(114,146)(115,147)(116,148)(117,149)(118,150)(119,151)(120,152), (1,140,119,22)(2,23,120,141)(3,142,113,24)(4,17,114,143)(5,144,115,18)(6,19,116,137)(7,138,117,20)(8,21,118,139)(9,30,102,93)(10,94,103,31)(11,32,104,95)(12,96,97,25)(13,26,98,89)(14,90,99,27)(15,28,100,91)(16,92,101,29)(33,131,125,58)(34,59,126,132)(35,133,127,60)(36,61,128,134)(37,135,121,62)(38,63,122,136)(39,129,123,64)(40,57,124,130)(41,50,112,151)(42,152,105,51)(43,52,106,145)(44,146,107,53)(45,54,108,147)(46,148,109,55)(47,56,110,149)(48,150,111,49)(65,77,158,88)(66,81,159,78)(67,79,160,82)(68,83,153,80)(69,73,154,84)(70,85,155,74)(71,75,156,86)(72,87,157,76), (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)>;
G:=Group( (1,73,97,121,54)(2,74,98,122,55)(3,75,99,123,56)(4,76,100,124,49)(5,77,101,125,50)(6,78,102,126,51)(7,79,103,127,52)(8,80,104,128,53)(9,34,152,116,81)(10,35,145,117,82)(11,36,146,118,83)(12,37,147,119,84)(13,38,148,120,85)(14,39,149,113,86)(15,40,150,114,87)(16,33,151,115,88)(17,72,91,130,48)(18,65,92,131,41)(19,66,93,132,42)(20,67,94,133,43)(21,68,95,134,44)(22,69,96,135,45)(23,70,89,136,46)(24,71,90,129,47)(25,62,108,140,154)(26,63,109,141,155)(27,64,110,142,156)(28,57,111,143,157)(29,58,112,144,158)(30,59,105,137,159)(31,60,106,138,160)(32,61,107,139,153), (1,50)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,49)(9,13)(10,14)(11,15)(12,16)(17,44)(18,45)(19,46)(20,47)(21,48)(22,41)(23,42)(24,43)(25,29)(26,30)(27,31)(28,32)(33,84)(34,85)(35,86)(36,87)(37,88)(38,81)(39,82)(40,83)(57,153)(58,154)(59,155)(60,156)(61,157)(62,158)(63,159)(64,160)(65,135)(66,136)(67,129)(68,130)(69,131)(70,132)(71,133)(72,134)(73,125)(74,126)(75,127)(76,128)(77,121)(78,122)(79,123)(80,124)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(105,141)(106,142)(107,143)(108,144)(109,137)(110,138)(111,139)(112,140)(113,145)(114,146)(115,147)(116,148)(117,149)(118,150)(119,151)(120,152), (1,140,119,22)(2,23,120,141)(3,142,113,24)(4,17,114,143)(5,144,115,18)(6,19,116,137)(7,138,117,20)(8,21,118,139)(9,30,102,93)(10,94,103,31)(11,32,104,95)(12,96,97,25)(13,26,98,89)(14,90,99,27)(15,28,100,91)(16,92,101,29)(33,131,125,58)(34,59,126,132)(35,133,127,60)(36,61,128,134)(37,135,121,62)(38,63,122,136)(39,129,123,64)(40,57,124,130)(41,50,112,151)(42,152,105,51)(43,52,106,145)(44,146,107,53)(45,54,108,147)(46,148,109,55)(47,56,110,149)(48,150,111,49)(65,77,158,88)(66,81,159,78)(67,79,160,82)(68,83,153,80)(69,73,154,84)(70,85,155,74)(71,75,156,86)(72,87,157,76), (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) );
G=PermutationGroup([[(1,73,97,121,54),(2,74,98,122,55),(3,75,99,123,56),(4,76,100,124,49),(5,77,101,125,50),(6,78,102,126,51),(7,79,103,127,52),(8,80,104,128,53),(9,34,152,116,81),(10,35,145,117,82),(11,36,146,118,83),(12,37,147,119,84),(13,38,148,120,85),(14,39,149,113,86),(15,40,150,114,87),(16,33,151,115,88),(17,72,91,130,48),(18,65,92,131,41),(19,66,93,132,42),(20,67,94,133,43),(21,68,95,134,44),(22,69,96,135,45),(23,70,89,136,46),(24,71,90,129,47),(25,62,108,140,154),(26,63,109,141,155),(27,64,110,142,156),(28,57,111,143,157),(29,58,112,144,158),(30,59,105,137,159),(31,60,106,138,160),(32,61,107,139,153)], [(1,50),(2,51),(3,52),(4,53),(5,54),(6,55),(7,56),(8,49),(9,13),(10,14),(11,15),(12,16),(17,44),(18,45),(19,46),(20,47),(21,48),(22,41),(23,42),(24,43),(25,29),(26,30),(27,31),(28,32),(33,84),(34,85),(35,86),(36,87),(37,88),(38,81),(39,82),(40,83),(57,153),(58,154),(59,155),(60,156),(61,157),(62,158),(63,159),(64,160),(65,135),(66,136),(67,129),(68,130),(69,131),(70,132),(71,133),(72,134),(73,125),(74,126),(75,127),(76,128),(77,121),(78,122),(79,123),(80,124),(89,93),(90,94),(91,95),(92,96),(97,101),(98,102),(99,103),(100,104),(105,141),(106,142),(107,143),(108,144),(109,137),(110,138),(111,139),(112,140),(113,145),(114,146),(115,147),(116,148),(117,149),(118,150),(119,151),(120,152)], [(1,140,119,22),(2,23,120,141),(3,142,113,24),(4,17,114,143),(5,144,115,18),(6,19,116,137),(7,138,117,20),(8,21,118,139),(9,30,102,93),(10,94,103,31),(11,32,104,95),(12,96,97,25),(13,26,98,89),(14,90,99,27),(15,28,100,91),(16,92,101,29),(33,131,125,58),(34,59,126,132),(35,133,127,60),(36,61,128,134),(37,135,121,62),(38,63,122,136),(39,129,123,64),(40,57,124,130),(41,50,112,151),(42,152,105,51),(43,52,106,145),(44,146,107,53),(45,54,108,147),(46,148,109,55),(47,56,110,149),(48,150,111,49),(65,77,158,88),(66,81,159,78),(67,79,160,82),(68,83,153,80),(69,73,154,84),(70,85,155,74),(71,75,156,86),(72,87,157,76)], [(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)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 5A | 5B | 8A | ··· | 8H | 8I | ··· | 8P | 10A | ··· | 10F | 20A | ··· | 20H | 20I | ··· | 20P | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C8 | D4 | Q8 | D5 | M4(2) | D10 | D10 | C4×D5 | C8×D5 | D4×D5 | Q8×D5 | D5×M4(2) |
| kernel | D5×C4⋊C8 | C20⋊3C8 | C20.8Q8 | C5×C4⋊C8 | D5×C42 | D5×C2×C8 | C4×Dic5 | C2×C4×D5 | C4×D5 | C4×D5 | C4×D5 | C4⋊C8 | D10 | C42 | C2×C8 | C2×C4 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 4 | 4 | 16 | 2 | 2 | 2 | 4 | 2 | 4 | 8 | 16 | 2 | 2 | 4 |
Matrix representation of D5×C4⋊C8 ►in GL4(𝔽41) generated by
| 6 | 1 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 40 | 35 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 31 | 28 |
| 0 | 0 | 33 | 10 |
| 27 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 |
| 0 | 0 | 9 | 39 |
| 0 | 0 | 0 | 32 |
G:=sub<GL(4,GF(41))| [6,40,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,35,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,31,33,0,0,28,10],[27,0,0,0,0,27,0,0,0,0,9,0,0,0,39,32] >;
D5×C4⋊C8 in GAP, Magma, Sage, TeX
D_5\times C_4\rtimes C_8
% in TeX
G:=Group("D5xC4:C8"); // GroupNames label
G:=SmallGroup(320,459);
// by ID
G=gap.SmallGroup(320,459);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,58,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^2=c^4=d^8=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations