direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D5×C8⋊C4, D10.17C42, C42.181D10, D10.12M4(2), Dic5.17C42, (C8×D5)⋊9C4, C8⋊15(C4×D5), C40⋊26(C2×C4), C40⋊8C4⋊24C2, C2.9(D5×C42), (C2×C8).269D10, C2.1(D5×M4(2)), C10.28(C2×C42), (C4×Dic5).17C4, (D5×C42).13C2, (C2×C40).225C22, (C4×C20).226C22, (C2×C20).810C23, C20.184(C22×C4), C42.D5⋊17C2, C10.49(C2×M4(2)), (C4×Dic5).297C22, C5⋊4(C2×C8⋊C4), C4.99(C2×C4×D5), (D5×C2×C8).26C2, (C2×C4×D5).18C4, (C5×C8⋊C4)⋊6C2, C5⋊2C8⋊31(C2×C4), C22.39(C2×C4×D5), (C4×D5).99(C2×C4), (C2×C4).126(C4×D5), (C2×C20).318(C2×C4), (C2×C4×D5).418C22, (C2×C4).752(C22×D5), (C2×C10).166(C22×C4), (C2×C5⋊2C8).302C22, (C2×Dic5).203(C2×C4), (C22×D5).138(C2×C4), SmallGroup(320,331)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D5×C8⋊C4
G = < a,b,c,d | a5=b2=c8=d4=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c5 >
Subgroups: 398 in 146 conjugacy classes, 83 normal (19 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, C2×C4, C23, D5, C10, C10, C42, C42, C2×C8, C2×C8, C22×C4, Dic5, Dic5, C20, C20, D10, C2×C10, C8⋊C4, C8⋊C4, C2×C42, C22×C8, C5⋊2C8, C40, C4×D5, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C2×C8⋊C4, C8×D5, C2×C5⋊2C8, C4×Dic5, C4×Dic5, C4×C20, C2×C40, C2×C4×D5, C2×C4×D5, C42.D5, C40⋊8C4, C5×C8⋊C4, D5×C42, D5×C2×C8, D5×C8⋊C4
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C42, M4(2), C22×C4, D10, C8⋊C4, C2×C42, C2×M4(2), C4×D5, C22×D5, C2×C8⋊C4, C2×C4×D5, D5×C42, D5×M4(2), D5×C8⋊C4
►On 160 points
Generators in S160
(1 106 26 57 70)(2 107 27 58 71)(3 108 28 59 72)(4 109 29 60 65)(5 110 30 61 66)(6 111 31 62 67)(7 112 32 63 68)(8 105 25 64 69)(9 92 24 52 79)(10 93 17 53 80)(11 94 18 54 73)(12 95 19 55 74)(13 96 20 56 75)(14 89 21 49 76)(15 90 22 50 77)(16 91 23 51 78)(33 134 138 99 147)(34 135 139 100 148)(35 136 140 101 149)(36 129 141 102 150)(37 130 142 103 151)(38 131 143 104 152)(39 132 144 97 145)(40 133 137 98 146)(41 116 123 81 154)(42 117 124 82 155)(43 118 125 83 156)(44 119 126 84 157)(45 120 127 85 158)(46 113 128 86 159)(47 114 121 87 160)(48 115 122 88 153)
(1 140)(2 141)(3 142)(4 143)(5 144)(6 137)(7 138)(8 139)(9 125)(10 126)(11 127)(12 128)(13 121)(14 122)(15 123)(16 124)(17 44)(18 45)(19 46)(20 47)(21 48)(22 41)(23 42)(24 43)(25 34)(26 35)(27 36)(28 37)(29 38)(30 39)(31 40)(32 33)(49 153)(50 154)(51 155)(52 156)(53 157)(54 158)(55 159)(56 160)(57 149)(58 150)(59 151)(60 152)(61 145)(62 146)(63 147)(64 148)(65 104)(66 97)(67 98)(68 99)(69 100)(70 101)(71 102)(72 103)(73 85)(74 86)(75 87)(76 88)(77 81)(78 82)(79 83)(80 84)(89 115)(90 116)(91 117)(92 118)(93 119)(94 120)(95 113)(96 114)(105 135)(106 136)(107 129)(108 130)(109 131)(110 132)(111 133)(112 134)
(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 11 97 81)(2 16 98 86)(3 13 99 83)(4 10 100 88)(5 15 101 85)(6 12 102 82)(7 9 103 87)(8 14 104 84)(17 34 48 29)(18 39 41 26)(19 36 42 31)(20 33 43 28)(21 38 44 25)(22 35 45 30)(23 40 46 27)(24 37 47 32)(49 131 119 64)(50 136 120 61)(51 133 113 58)(52 130 114 63)(53 135 115 60)(54 132 116 57)(55 129 117 62)(56 134 118 59)(65 80 139 122)(66 77 140 127)(67 74 141 124)(68 79 142 121)(69 76 143 126)(70 73 144 123)(71 78 137 128)(72 75 138 125)(89 152 157 105)(90 149 158 110)(91 146 159 107)(92 151 160 112)(93 148 153 109)(94 145 154 106)(95 150 155 111)(96 147 156 108)
G:=sub<Sym(160)| (1,106,26,57,70)(2,107,27,58,71)(3,108,28,59,72)(4,109,29,60,65)(5,110,30,61,66)(6,111,31,62,67)(7,112,32,63,68)(8,105,25,64,69)(9,92,24,52,79)(10,93,17,53,80)(11,94,18,54,73)(12,95,19,55,74)(13,96,20,56,75)(14,89,21,49,76)(15,90,22,50,77)(16,91,23,51,78)(33,134,138,99,147)(34,135,139,100,148)(35,136,140,101,149)(36,129,141,102,150)(37,130,142,103,151)(38,131,143,104,152)(39,132,144,97,145)(40,133,137,98,146)(41,116,123,81,154)(42,117,124,82,155)(43,118,125,83,156)(44,119,126,84,157)(45,120,127,85,158)(46,113,128,86,159)(47,114,121,87,160)(48,115,122,88,153), (1,140)(2,141)(3,142)(4,143)(5,144)(6,137)(7,138)(8,139)(9,125)(10,126)(11,127)(12,128)(13,121)(14,122)(15,123)(16,124)(17,44)(18,45)(19,46)(20,47)(21,48)(22,41)(23,42)(24,43)(25,34)(26,35)(27,36)(28,37)(29,38)(30,39)(31,40)(32,33)(49,153)(50,154)(51,155)(52,156)(53,157)(54,158)(55,159)(56,160)(57,149)(58,150)(59,151)(60,152)(61,145)(62,146)(63,147)(64,148)(65,104)(66,97)(67,98)(68,99)(69,100)(70,101)(71,102)(72,103)(73,85)(74,86)(75,87)(76,88)(77,81)(78,82)(79,83)(80,84)(89,115)(90,116)(91,117)(92,118)(93,119)(94,120)(95,113)(96,114)(105,135)(106,136)(107,129)(108,130)(109,131)(110,132)(111,133)(112,134), (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), (1,11,97,81)(2,16,98,86)(3,13,99,83)(4,10,100,88)(5,15,101,85)(6,12,102,82)(7,9,103,87)(8,14,104,84)(17,34,48,29)(18,39,41,26)(19,36,42,31)(20,33,43,28)(21,38,44,25)(22,35,45,30)(23,40,46,27)(24,37,47,32)(49,131,119,64)(50,136,120,61)(51,133,113,58)(52,130,114,63)(53,135,115,60)(54,132,116,57)(55,129,117,62)(56,134,118,59)(65,80,139,122)(66,77,140,127)(67,74,141,124)(68,79,142,121)(69,76,143,126)(70,73,144,123)(71,78,137,128)(72,75,138,125)(89,152,157,105)(90,149,158,110)(91,146,159,107)(92,151,160,112)(93,148,153,109)(94,145,154,106)(95,150,155,111)(96,147,156,108)>;
G:=Group( (1,106,26,57,70)(2,107,27,58,71)(3,108,28,59,72)(4,109,29,60,65)(5,110,30,61,66)(6,111,31,62,67)(7,112,32,63,68)(8,105,25,64,69)(9,92,24,52,79)(10,93,17,53,80)(11,94,18,54,73)(12,95,19,55,74)(13,96,20,56,75)(14,89,21,49,76)(15,90,22,50,77)(16,91,23,51,78)(33,134,138,99,147)(34,135,139,100,148)(35,136,140,101,149)(36,129,141,102,150)(37,130,142,103,151)(38,131,143,104,152)(39,132,144,97,145)(40,133,137,98,146)(41,116,123,81,154)(42,117,124,82,155)(43,118,125,83,156)(44,119,126,84,157)(45,120,127,85,158)(46,113,128,86,159)(47,114,121,87,160)(48,115,122,88,153), (1,140)(2,141)(3,142)(4,143)(5,144)(6,137)(7,138)(8,139)(9,125)(10,126)(11,127)(12,128)(13,121)(14,122)(15,123)(16,124)(17,44)(18,45)(19,46)(20,47)(21,48)(22,41)(23,42)(24,43)(25,34)(26,35)(27,36)(28,37)(29,38)(30,39)(31,40)(32,33)(49,153)(50,154)(51,155)(52,156)(53,157)(54,158)(55,159)(56,160)(57,149)(58,150)(59,151)(60,152)(61,145)(62,146)(63,147)(64,148)(65,104)(66,97)(67,98)(68,99)(69,100)(70,101)(71,102)(72,103)(73,85)(74,86)(75,87)(76,88)(77,81)(78,82)(79,83)(80,84)(89,115)(90,116)(91,117)(92,118)(93,119)(94,120)(95,113)(96,114)(105,135)(106,136)(107,129)(108,130)(109,131)(110,132)(111,133)(112,134), (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), (1,11,97,81)(2,16,98,86)(3,13,99,83)(4,10,100,88)(5,15,101,85)(6,12,102,82)(7,9,103,87)(8,14,104,84)(17,34,48,29)(18,39,41,26)(19,36,42,31)(20,33,43,28)(21,38,44,25)(22,35,45,30)(23,40,46,27)(24,37,47,32)(49,131,119,64)(50,136,120,61)(51,133,113,58)(52,130,114,63)(53,135,115,60)(54,132,116,57)(55,129,117,62)(56,134,118,59)(65,80,139,122)(66,77,140,127)(67,74,141,124)(68,79,142,121)(69,76,143,126)(70,73,144,123)(71,78,137,128)(72,75,138,125)(89,152,157,105)(90,149,158,110)(91,146,159,107)(92,151,160,112)(93,148,153,109)(94,145,154,106)(95,150,155,111)(96,147,156,108) );
G=PermutationGroup([[(1,106,26,57,70),(2,107,27,58,71),(3,108,28,59,72),(4,109,29,60,65),(5,110,30,61,66),(6,111,31,62,67),(7,112,32,63,68),(8,105,25,64,69),(9,92,24,52,79),(10,93,17,53,80),(11,94,18,54,73),(12,95,19,55,74),(13,96,20,56,75),(14,89,21,49,76),(15,90,22,50,77),(16,91,23,51,78),(33,134,138,99,147),(34,135,139,100,148),(35,136,140,101,149),(36,129,141,102,150),(37,130,142,103,151),(38,131,143,104,152),(39,132,144,97,145),(40,133,137,98,146),(41,116,123,81,154),(42,117,124,82,155),(43,118,125,83,156),(44,119,126,84,157),(45,120,127,85,158),(46,113,128,86,159),(47,114,121,87,160),(48,115,122,88,153)], [(1,140),(2,141),(3,142),(4,143),(5,144),(6,137),(7,138),(8,139),(9,125),(10,126),(11,127),(12,128),(13,121),(14,122),(15,123),(16,124),(17,44),(18,45),(19,46),(20,47),(21,48),(22,41),(23,42),(24,43),(25,34),(26,35),(27,36),(28,37),(29,38),(30,39),(31,40),(32,33),(49,153),(50,154),(51,155),(52,156),(53,157),(54,158),(55,159),(56,160),(57,149),(58,150),(59,151),(60,152),(61,145),(62,146),(63,147),(64,148),(65,104),(66,97),(67,98),(68,99),(69,100),(70,101),(71,102),(72,103),(73,85),(74,86),(75,87),(76,88),(77,81),(78,82),(79,83),(80,84),(89,115),(90,116),(91,117),(92,118),(93,119),(94,120),(95,113),(96,114),(105,135),(106,136),(107,129),(108,130),(109,131),(110,132),(111,133),(112,134)], [(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)], [(1,11,97,81),(2,16,98,86),(3,13,99,83),(4,10,100,88),(5,15,101,85),(6,12,102,82),(7,9,103,87),(8,14,104,84),(17,34,48,29),(18,39,41,26),(19,36,42,31),(20,33,43,28),(21,38,44,25),(22,35,45,30),(23,40,46,27),(24,37,47,32),(49,131,119,64),(50,136,120,61),(51,133,113,58),(52,130,114,63),(53,135,115,60),(54,132,116,57),(55,129,117,62),(56,134,118,59),(65,80,139,122),(66,77,140,127),(67,74,141,124),(68,79,142,121),(69,76,143,126),(70,73,144,123),(71,78,137,128),(72,75,138,125),(89,152,157,105),(90,149,158,110),(91,146,159,107),(92,151,160,112),(93,148,153,109),(94,145,154,106),(95,150,155,111),(96,147,156,108)]])
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 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D5 | M4(2) | D10 | D10 | C4×D5 | C4×D5 | D5×M4(2) |
| kernel | D5×C8⋊C4 | C42.D5 | C40⋊8C4 | C5×C8⋊C4 | D5×C42 | D5×C2×C8 | C8×D5 | C4×Dic5 | C2×C4×D5 | C8⋊C4 | D10 | C42 | C2×C8 | C8 | C2×C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 16 | 4 | 4 | 2 | 8 | 2 | 4 | 16 | 8 | 8 |
Matrix representation of D5×C8⋊C4 ►in GL4(𝔽41) generated by
| 40 | 1 | 0 | 0 |
| 5 | 35 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 36 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 9 | 7 |
| 0 | 0 | 19 | 32 |
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 34 | 40 |
| 0 | 0 | 9 | 7 |
G:=sub<GL(4,GF(41))| [40,5,0,0,1,35,0,0,0,0,1,0,0,0,0,1],[1,36,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,9,19,0,0,7,32],[9,0,0,0,0,9,0,0,0,0,34,9,0,0,40,7] >;
D5×C8⋊C4 in GAP, Magma, Sage, TeX
D_5\times C_8\rtimes C_4
% in TeX
G:=Group("D5xC8:C4"); // GroupNames label
G:=SmallGroup(320,331);
// by ID
G=gap.SmallGroup(320,331);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,387,58,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^2=c^8=d^4=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^5>;
// generators/relations