direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×C4.D8, C20.60D8, C20.51SD16, C4⋊C8⋊2C10, C4.9(C5×D8), (C2×D4).2C20, (D4×C10).18C4, (C2×C20).503D4, C4⋊1D4.1C10, C42.3(C2×C10), C4.11(C5×SD16), (C4×C20).243C22, C10.49(D4⋊C4), C10.20(C4.D4), (C5×C4⋊C8)⋊4C2, (C2×C4).11(C2×C20), (C5×C4⋊1D4).8C2, (C2×C4).109(C5×D4), C2.4(C5×D4⋊C4), C2.4(C5×C4.D4), (C2×C20).351(C2×C4), C22.39(C5×C22⋊C4), (C2×C10).190(C22⋊C4), SmallGroup(320,136)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C4.D8
G = < a,b,c,d | a5=b4=c8=1, d2=b, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=bc-1 >
Subgroups: 210 in 84 conjugacy classes, 38 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, C10, C10, C10, C42, C2×C8, C2×D4, C2×D4, C20, C20, C2×C10, C2×C10, C4⋊C8, C4⋊1D4, C40, C2×C20, C2×C20, C5×D4, C22×C10, C4.D8, C4×C20, C2×C40, D4×C10, D4×C10, C5×C4⋊C8, C5×C4⋊1D4, C5×C4.D8
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, C10, C22⋊C4, D8, SD16, C20, C2×C10, C4.D4, D4⋊C4, C2×C20, C5×D4, C4.D8, C5×C22⋊C4, C5×D8, C5×SD16, C5×C4.D4, C5×D4⋊C4, C5×C4.D8
►On 160 points
Generators in S160
(1 51 41 35 25)(2 52 42 36 26)(3 53 43 37 27)(4 54 44 38 28)(5 55 45 39 29)(6 56 46 40 30)(7 49 47 33 31)(8 50 48 34 32)(9 153 146 137 130)(10 154 147 138 131)(11 155 148 139 132)(12 156 149 140 133)(13 157 150 141 134)(14 158 151 142 135)(15 159 152 143 136)(16 160 145 144 129)(17 123 116 107 100)(18 124 117 108 101)(19 125 118 109 102)(20 126 119 110 103)(21 127 120 111 104)(22 128 113 112 97)(23 121 114 105 98)(24 122 115 106 99)(57 89 85 73 67)(58 90 86 74 68)(59 91 87 75 69)(60 92 88 76 70)(61 93 81 77 71)(62 94 82 78 72)(63 95 83 79 65)(64 96 84 80 66)
(1 22 130 62)(2 63 131 23)(3 24 132 64)(4 57 133 17)(5 18 134 58)(6 59 135 19)(7 20 136 60)(8 61 129 21)(9 94 51 128)(10 121 52 95)(11 96 53 122)(12 123 54 89)(13 90 55 124)(14 125 56 91)(15 92 49 126)(16 127 50 93)(25 97 137 72)(26 65 138 98)(27 99 139 66)(28 67 140 100)(29 101 141 68)(30 69 142 102)(31 103 143 70)(32 71 144 104)(33 110 152 76)(34 77 145 111)(35 112 146 78)(36 79 147 105)(37 106 148 80)(38 73 149 107)(39 108 150 74)(40 75 151 109)(41 113 153 82)(42 83 154 114)(43 115 155 84)(44 85 156 116)(45 117 157 86)(46 87 158 118)(47 119 159 88)(48 81 160 120)
(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)
(1 129 22 21 130 8 62 61)(2 60 63 7 131 20 23 136)(3 135 24 19 132 6 64 59)(4 58 57 5 133 18 17 134)(9 50 94 93 51 16 128 127)(10 126 121 15 52 92 95 49)(11 56 96 91 53 14 122 125)(12 124 123 13 54 90 89 55)(25 144 97 104 137 32 72 71)(26 70 65 31 138 103 98 143)(27 142 99 102 139 30 66 69)(28 68 67 29 140 101 100 141)(33 147 110 105 152 36 76 79)(34 78 77 35 145 112 111 146)(37 151 106 109 148 40 80 75)(38 74 73 39 149 108 107 150)(41 160 113 120 153 48 82 81)(42 88 83 47 154 119 114 159)(43 158 115 118 155 46 84 87)(44 86 85 45 156 117 116 157)
G:=sub<Sym(160)| (1,51,41,35,25)(2,52,42,36,26)(3,53,43,37,27)(4,54,44,38,28)(5,55,45,39,29)(6,56,46,40,30)(7,49,47,33,31)(8,50,48,34,32)(9,153,146,137,130)(10,154,147,138,131)(11,155,148,139,132)(12,156,149,140,133)(13,157,150,141,134)(14,158,151,142,135)(15,159,152,143,136)(16,160,145,144,129)(17,123,116,107,100)(18,124,117,108,101)(19,125,118,109,102)(20,126,119,110,103)(21,127,120,111,104)(22,128,113,112,97)(23,121,114,105,98)(24,122,115,106,99)(57,89,85,73,67)(58,90,86,74,68)(59,91,87,75,69)(60,92,88,76,70)(61,93,81,77,71)(62,94,82,78,72)(63,95,83,79,65)(64,96,84,80,66), (1,22,130,62)(2,63,131,23)(3,24,132,64)(4,57,133,17)(5,18,134,58)(6,59,135,19)(7,20,136,60)(8,61,129,21)(9,94,51,128)(10,121,52,95)(11,96,53,122)(12,123,54,89)(13,90,55,124)(14,125,56,91)(15,92,49,126)(16,127,50,93)(25,97,137,72)(26,65,138,98)(27,99,139,66)(28,67,140,100)(29,101,141,68)(30,69,142,102)(31,103,143,70)(32,71,144,104)(33,110,152,76)(34,77,145,111)(35,112,146,78)(36,79,147,105)(37,106,148,80)(38,73,149,107)(39,108,150,74)(40,75,151,109)(41,113,153,82)(42,83,154,114)(43,115,155,84)(44,85,156,116)(45,117,157,86)(46,87,158,118)(47,119,159,88)(48,81,160,120), (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), (1,129,22,21,130,8,62,61)(2,60,63,7,131,20,23,136)(3,135,24,19,132,6,64,59)(4,58,57,5,133,18,17,134)(9,50,94,93,51,16,128,127)(10,126,121,15,52,92,95,49)(11,56,96,91,53,14,122,125)(12,124,123,13,54,90,89,55)(25,144,97,104,137,32,72,71)(26,70,65,31,138,103,98,143)(27,142,99,102,139,30,66,69)(28,68,67,29,140,101,100,141)(33,147,110,105,152,36,76,79)(34,78,77,35,145,112,111,146)(37,151,106,109,148,40,80,75)(38,74,73,39,149,108,107,150)(41,160,113,120,153,48,82,81)(42,88,83,47,154,119,114,159)(43,158,115,118,155,46,84,87)(44,86,85,45,156,117,116,157)>;
G:=Group( (1,51,41,35,25)(2,52,42,36,26)(3,53,43,37,27)(4,54,44,38,28)(5,55,45,39,29)(6,56,46,40,30)(7,49,47,33,31)(8,50,48,34,32)(9,153,146,137,130)(10,154,147,138,131)(11,155,148,139,132)(12,156,149,140,133)(13,157,150,141,134)(14,158,151,142,135)(15,159,152,143,136)(16,160,145,144,129)(17,123,116,107,100)(18,124,117,108,101)(19,125,118,109,102)(20,126,119,110,103)(21,127,120,111,104)(22,128,113,112,97)(23,121,114,105,98)(24,122,115,106,99)(57,89,85,73,67)(58,90,86,74,68)(59,91,87,75,69)(60,92,88,76,70)(61,93,81,77,71)(62,94,82,78,72)(63,95,83,79,65)(64,96,84,80,66), (1,22,130,62)(2,63,131,23)(3,24,132,64)(4,57,133,17)(5,18,134,58)(6,59,135,19)(7,20,136,60)(8,61,129,21)(9,94,51,128)(10,121,52,95)(11,96,53,122)(12,123,54,89)(13,90,55,124)(14,125,56,91)(15,92,49,126)(16,127,50,93)(25,97,137,72)(26,65,138,98)(27,99,139,66)(28,67,140,100)(29,101,141,68)(30,69,142,102)(31,103,143,70)(32,71,144,104)(33,110,152,76)(34,77,145,111)(35,112,146,78)(36,79,147,105)(37,106,148,80)(38,73,149,107)(39,108,150,74)(40,75,151,109)(41,113,153,82)(42,83,154,114)(43,115,155,84)(44,85,156,116)(45,117,157,86)(46,87,158,118)(47,119,159,88)(48,81,160,120), (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), (1,129,22,21,130,8,62,61)(2,60,63,7,131,20,23,136)(3,135,24,19,132,6,64,59)(4,58,57,5,133,18,17,134)(9,50,94,93,51,16,128,127)(10,126,121,15,52,92,95,49)(11,56,96,91,53,14,122,125)(12,124,123,13,54,90,89,55)(25,144,97,104,137,32,72,71)(26,70,65,31,138,103,98,143)(27,142,99,102,139,30,66,69)(28,68,67,29,140,101,100,141)(33,147,110,105,152,36,76,79)(34,78,77,35,145,112,111,146)(37,151,106,109,148,40,80,75)(38,74,73,39,149,108,107,150)(41,160,113,120,153,48,82,81)(42,88,83,47,154,119,114,159)(43,158,115,118,155,46,84,87)(44,86,85,45,156,117,116,157) );
G=PermutationGroup([[(1,51,41,35,25),(2,52,42,36,26),(3,53,43,37,27),(4,54,44,38,28),(5,55,45,39,29),(6,56,46,40,30),(7,49,47,33,31),(8,50,48,34,32),(9,153,146,137,130),(10,154,147,138,131),(11,155,148,139,132),(12,156,149,140,133),(13,157,150,141,134),(14,158,151,142,135),(15,159,152,143,136),(16,160,145,144,129),(17,123,116,107,100),(18,124,117,108,101),(19,125,118,109,102),(20,126,119,110,103),(21,127,120,111,104),(22,128,113,112,97),(23,121,114,105,98),(24,122,115,106,99),(57,89,85,73,67),(58,90,86,74,68),(59,91,87,75,69),(60,92,88,76,70),(61,93,81,77,71),(62,94,82,78,72),(63,95,83,79,65),(64,96,84,80,66)], [(1,22,130,62),(2,63,131,23),(3,24,132,64),(4,57,133,17),(5,18,134,58),(6,59,135,19),(7,20,136,60),(8,61,129,21),(9,94,51,128),(10,121,52,95),(11,96,53,122),(12,123,54,89),(13,90,55,124),(14,125,56,91),(15,92,49,126),(16,127,50,93),(25,97,137,72),(26,65,138,98),(27,99,139,66),(28,67,140,100),(29,101,141,68),(30,69,142,102),(31,103,143,70),(32,71,144,104),(33,110,152,76),(34,77,145,111),(35,112,146,78),(36,79,147,105),(37,106,148,80),(38,73,149,107),(39,108,150,74),(40,75,151,109),(41,113,153,82),(42,83,154,114),(43,115,155,84),(44,85,156,116),(45,117,157,86),(46,87,158,118),(47,119,159,88),(48,81,160,120)], [(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)], [(1,129,22,21,130,8,62,61),(2,60,63,7,131,20,23,136),(3,135,24,19,132,6,64,59),(4,58,57,5,133,18,17,134),(9,50,94,93,51,16,128,127),(10,126,121,15,52,92,95,49),(11,56,96,91,53,14,122,125),(12,124,123,13,54,90,89,55),(25,144,97,104,137,32,72,71),(26,70,65,31,138,103,98,143),(27,142,99,102,139,30,66,69),(28,68,67,29,140,101,100,141),(33,147,110,105,152,36,76,79),(34,78,77,35,145,112,111,146),(37,151,106,109,148,40,80,75),(38,74,73,39,149,108,107,150),(41,160,113,120,153,48,82,81),(42,88,83,47,154,119,114,159),(43,158,115,118,155,46,84,87),(44,86,85,45,156,117,116,157)]])
95 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 10A | ··· | 10L | 10M | ··· | 10T | 20A | ··· | 20P | 20Q | 20R | 20S | 20T | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 1 | ··· | 1 | 8 | ··· | 8 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
95 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C4 | C5 | C10 | C10 | C20 | D4 | D8 | SD16 | C5×D4 | C5×D8 | C5×SD16 | C4.D4 | C5×C4.D4 |
| kernel | C5×C4.D8 | C5×C4⋊C8 | C5×C4⋊1D4 | D4×C10 | C4.D8 | C4⋊C8 | C4⋊1D4 | C2×D4 | C2×C20 | C20 | C20 | C2×C4 | C4 | C4 | C10 | C2 |
| # reps | 1 | 2 | 1 | 4 | 4 | 8 | 4 | 16 | 2 | 4 | 4 | 8 | 16 | 16 | 1 | 4 |
Matrix representation of C5×C4.D8 ►in GL4(𝔽41) generated by
| 18 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 12 | 29 | 0 | 0 |
| 29 | 29 | 0 | 0 |
| 0 | 0 | 15 | 26 |
| 0 | 0 | 15 | 15 |
| 12 | 12 | 0 | 0 |
| 29 | 12 | 0 | 0 |
| 0 | 0 | 26 | 15 |
| 0 | 0 | 15 | 15 |
G:=sub<GL(4,GF(41))| [18,0,0,0,0,18,0,0,0,0,1,0,0,0,0,1],[0,40,0,0,1,0,0,0,0,0,40,0,0,0,0,40],[12,29,0,0,29,29,0,0,0,0,15,15,0,0,26,15],[12,29,0,0,12,12,0,0,0,0,26,15,0,0,15,15] >;
C5×C4.D8 in GAP, Magma, Sage, TeX
C_5\times C_4.D_8
% in TeX
G:=Group("C5xC4.D8"); // GroupNames label
G:=SmallGroup(320,136);
// by ID
G=gap.SmallGroup(320,136);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,2803,2530,248,4911,242]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^4=c^8=1,d^2=b,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b*c^-1>;
// generators/relations