direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×C8⋊2D4, C40⋊20D4, C8⋊2(C5×D4), (C2×D8)⋊7C10, C4.Q8⋊4C10, (C10×D8)⋊21C2, C4⋊D4⋊4C10, C4.61(D4×C10), C20.468(C2×D4), (C2×C20).328D4, D4⋊C4⋊18C10, C23.16(C5×D4), (C2×M4(2))⋊2C10, (C22×C10).34D4, C22.93(D4×C10), C20.266(C4○D4), (C10×M4(2))⋊12C2, (C2×C40).333C22, (C2×C20).928C23, C10.152(C4⋊D4), C10.138(C8⋊C22), (D4×C10).192C22, (C22×C20).426C22, C4⋊C4.9(C2×C10), (C5×C4.Q8)⋊13C2, C4.11(C5×C4○D4), (C2×C4).33(C5×D4), (C5×C4⋊D4)⋊31C2, (C2×C8).22(C2×C10), C2.21(C5×C4⋊D4), C2.13(C5×C8⋊C22), (C5×D4⋊C4)⋊41C2, (C2×D4).15(C2×C10), (C2×C10).649(C2×D4), (C5×C4⋊C4).231C22, (C22×C4).44(C2×C10), (C2×C4).103(C22×C10), SmallGroup(320,970)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C8⋊2D4
G = < a,b,c,d | a5=b8=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b3, dbd=b-1, dcd=c-1 >
Subgroups: 282 in 130 conjugacy classes, 54 normal (30 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, C23, C23, C10, C10, C10, C22⋊C4, C4⋊C4, C2×C8, M4(2), D8, C22×C4, C2×D4, C2×D4, C20, C20, C2×C10, C2×C10, D4⋊C4, C4.Q8, C4⋊D4, C2×M4(2), C2×D8, C40, C40, C2×C20, C2×C20, C5×D4, C22×C10, C22×C10, C8⋊2D4, C5×C22⋊C4, C5×C4⋊C4, C2×C40, C5×M4(2), C5×D8, C22×C20, D4×C10, D4×C10, C5×D4⋊C4, C5×C4.Q8, C5×C4⋊D4, C10×M4(2), C10×D8, C5×C8⋊2D4
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C4○D4, C2×C10, C4⋊D4, C8⋊C22, C5×D4, C22×C10, C8⋊2D4, D4×C10, C5×C4○D4, C5×C4⋊D4, C5×C8⋊C22, C5×C8⋊2D4
►On 160 points
Generators in S160
(1 105 25 97 17)(2 106 26 98 18)(3 107 27 99 19)(4 108 28 100 20)(5 109 29 101 21)(6 110 30 102 22)(7 111 31 103 23)(8 112 32 104 24)(9 46 118 38 96)(10 47 119 39 89)(11 48 120 40 90)(12 41 113 33 91)(13 42 114 34 92)(14 43 115 35 93)(15 44 116 36 94)(16 45 117 37 95)(49 121 137 57 129)(50 122 138 58 130)(51 123 139 59 131)(52 124 140 60 132)(53 125 141 61 133)(54 126 142 62 134)(55 127 143 63 135)(56 128 144 64 136)(65 86 153 73 145)(66 87 154 74 146)(67 88 155 75 147)(68 81 156 76 148)(69 82 157 77 149)(70 83 158 78 150)(71 84 159 79 151)(72 85 160 80 152)
(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 33 123 67)(2 36 124 70)(3 39 125 65)(4 34 126 68)(5 37 127 71)(6 40 128 66)(7 35 121 69)(8 38 122 72)(9 58 160 32)(10 61 153 27)(11 64 154 30)(12 59 155 25)(13 62 156 28)(14 57 157 31)(15 60 158 26)(16 63 159 29)(17 113 51 147)(18 116 52 150)(19 119 53 145)(20 114 54 148)(21 117 55 151)(22 120 56 146)(23 115 49 149)(24 118 50 152)(41 131 75 97)(42 134 76 100)(43 129 77 103)(44 132 78 98)(45 135 79 101)(46 130 80 104)(47 133 73 99)(48 136 74 102)(81 108 92 142)(82 111 93 137)(83 106 94 140)(84 109 95 143)(85 112 96 138)(86 107 89 141)(87 110 90 144)(88 105 91 139)
(2 8)(3 7)(4 6)(9 158)(10 157)(11 156)(12 155)(13 154)(14 153)(15 160)(16 159)(18 24)(19 23)(20 22)(26 32)(27 31)(28 30)(33 67)(34 66)(35 65)(36 72)(37 71)(38 70)(39 69)(40 68)(41 75)(42 74)(43 73)(44 80)(45 79)(46 78)(47 77)(48 76)(49 53)(50 52)(54 56)(57 61)(58 60)(62 64)(81 90)(82 89)(83 96)(84 95)(85 94)(86 93)(87 92)(88 91)(98 104)(99 103)(100 102)(106 112)(107 111)(108 110)(113 147)(114 146)(115 145)(116 152)(117 151)(118 150)(119 149)(120 148)(121 125)(122 124)(126 128)(129 133)(130 132)(134 136)(137 141)(138 140)(142 144)
G:=sub<Sym(160)| (1,105,25,97,17)(2,106,26,98,18)(3,107,27,99,19)(4,108,28,100,20)(5,109,29,101,21)(6,110,30,102,22)(7,111,31,103,23)(8,112,32,104,24)(9,46,118,38,96)(10,47,119,39,89)(11,48,120,40,90)(12,41,113,33,91)(13,42,114,34,92)(14,43,115,35,93)(15,44,116,36,94)(16,45,117,37,95)(49,121,137,57,129)(50,122,138,58,130)(51,123,139,59,131)(52,124,140,60,132)(53,125,141,61,133)(54,126,142,62,134)(55,127,143,63,135)(56,128,144,64,136)(65,86,153,73,145)(66,87,154,74,146)(67,88,155,75,147)(68,81,156,76,148)(69,82,157,77,149)(70,83,158,78,150)(71,84,159,79,151)(72,85,160,80,152), (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), (1,33,123,67)(2,36,124,70)(3,39,125,65)(4,34,126,68)(5,37,127,71)(6,40,128,66)(7,35,121,69)(8,38,122,72)(9,58,160,32)(10,61,153,27)(11,64,154,30)(12,59,155,25)(13,62,156,28)(14,57,157,31)(15,60,158,26)(16,63,159,29)(17,113,51,147)(18,116,52,150)(19,119,53,145)(20,114,54,148)(21,117,55,151)(22,120,56,146)(23,115,49,149)(24,118,50,152)(41,131,75,97)(42,134,76,100)(43,129,77,103)(44,132,78,98)(45,135,79,101)(46,130,80,104)(47,133,73,99)(48,136,74,102)(81,108,92,142)(82,111,93,137)(83,106,94,140)(84,109,95,143)(85,112,96,138)(86,107,89,141)(87,110,90,144)(88,105,91,139), (2,8)(3,7)(4,6)(9,158)(10,157)(11,156)(12,155)(13,154)(14,153)(15,160)(16,159)(18,24)(19,23)(20,22)(26,32)(27,31)(28,30)(33,67)(34,66)(35,65)(36,72)(37,71)(38,70)(39,69)(40,68)(41,75)(42,74)(43,73)(44,80)(45,79)(46,78)(47,77)(48,76)(49,53)(50,52)(54,56)(57,61)(58,60)(62,64)(81,90)(82,89)(83,96)(84,95)(85,94)(86,93)(87,92)(88,91)(98,104)(99,103)(100,102)(106,112)(107,111)(108,110)(113,147)(114,146)(115,145)(116,152)(117,151)(118,150)(119,149)(120,148)(121,125)(122,124)(126,128)(129,133)(130,132)(134,136)(137,141)(138,140)(142,144)>;
G:=Group( (1,105,25,97,17)(2,106,26,98,18)(3,107,27,99,19)(4,108,28,100,20)(5,109,29,101,21)(6,110,30,102,22)(7,111,31,103,23)(8,112,32,104,24)(9,46,118,38,96)(10,47,119,39,89)(11,48,120,40,90)(12,41,113,33,91)(13,42,114,34,92)(14,43,115,35,93)(15,44,116,36,94)(16,45,117,37,95)(49,121,137,57,129)(50,122,138,58,130)(51,123,139,59,131)(52,124,140,60,132)(53,125,141,61,133)(54,126,142,62,134)(55,127,143,63,135)(56,128,144,64,136)(65,86,153,73,145)(66,87,154,74,146)(67,88,155,75,147)(68,81,156,76,148)(69,82,157,77,149)(70,83,158,78,150)(71,84,159,79,151)(72,85,160,80,152), (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), (1,33,123,67)(2,36,124,70)(3,39,125,65)(4,34,126,68)(5,37,127,71)(6,40,128,66)(7,35,121,69)(8,38,122,72)(9,58,160,32)(10,61,153,27)(11,64,154,30)(12,59,155,25)(13,62,156,28)(14,57,157,31)(15,60,158,26)(16,63,159,29)(17,113,51,147)(18,116,52,150)(19,119,53,145)(20,114,54,148)(21,117,55,151)(22,120,56,146)(23,115,49,149)(24,118,50,152)(41,131,75,97)(42,134,76,100)(43,129,77,103)(44,132,78,98)(45,135,79,101)(46,130,80,104)(47,133,73,99)(48,136,74,102)(81,108,92,142)(82,111,93,137)(83,106,94,140)(84,109,95,143)(85,112,96,138)(86,107,89,141)(87,110,90,144)(88,105,91,139), (2,8)(3,7)(4,6)(9,158)(10,157)(11,156)(12,155)(13,154)(14,153)(15,160)(16,159)(18,24)(19,23)(20,22)(26,32)(27,31)(28,30)(33,67)(34,66)(35,65)(36,72)(37,71)(38,70)(39,69)(40,68)(41,75)(42,74)(43,73)(44,80)(45,79)(46,78)(47,77)(48,76)(49,53)(50,52)(54,56)(57,61)(58,60)(62,64)(81,90)(82,89)(83,96)(84,95)(85,94)(86,93)(87,92)(88,91)(98,104)(99,103)(100,102)(106,112)(107,111)(108,110)(113,147)(114,146)(115,145)(116,152)(117,151)(118,150)(119,149)(120,148)(121,125)(122,124)(126,128)(129,133)(130,132)(134,136)(137,141)(138,140)(142,144) );
G=PermutationGroup([[(1,105,25,97,17),(2,106,26,98,18),(3,107,27,99,19),(4,108,28,100,20),(5,109,29,101,21),(6,110,30,102,22),(7,111,31,103,23),(8,112,32,104,24),(9,46,118,38,96),(10,47,119,39,89),(11,48,120,40,90),(12,41,113,33,91),(13,42,114,34,92),(14,43,115,35,93),(15,44,116,36,94),(16,45,117,37,95),(49,121,137,57,129),(50,122,138,58,130),(51,123,139,59,131),(52,124,140,60,132),(53,125,141,61,133),(54,126,142,62,134),(55,127,143,63,135),(56,128,144,64,136),(65,86,153,73,145),(66,87,154,74,146),(67,88,155,75,147),(68,81,156,76,148),(69,82,157,77,149),(70,83,158,78,150),(71,84,159,79,151),(72,85,160,80,152)], [(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)], [(1,33,123,67),(2,36,124,70),(3,39,125,65),(4,34,126,68),(5,37,127,71),(6,40,128,66),(7,35,121,69),(8,38,122,72),(9,58,160,32),(10,61,153,27),(11,64,154,30),(12,59,155,25),(13,62,156,28),(14,57,157,31),(15,60,158,26),(16,63,159,29),(17,113,51,147),(18,116,52,150),(19,119,53,145),(20,114,54,148),(21,117,55,151),(22,120,56,146),(23,115,49,149),(24,118,50,152),(41,131,75,97),(42,134,76,100),(43,129,77,103),(44,132,78,98),(45,135,79,101),(46,130,80,104),(47,133,73,99),(48,136,74,102),(81,108,92,142),(82,111,93,137),(83,106,94,140),(84,109,95,143),(85,112,96,138),(86,107,89,141),(87,110,90,144),(88,105,91,139)], [(2,8),(3,7),(4,6),(9,158),(10,157),(11,156),(12,155),(13,154),(14,153),(15,160),(16,159),(18,24),(19,23),(20,22),(26,32),(27,31),(28,30),(33,67),(34,66),(35,65),(36,72),(37,71),(38,70),(39,69),(40,68),(41,75),(42,74),(43,73),(44,80),(45,79),(46,78),(47,77),(48,76),(49,53),(50,52),(54,56),(57,61),(58,60),(62,64),(81,90),(82,89),(83,96),(84,95),(85,94),(86,93),(87,92),(88,91),(98,104),(99,103),(100,102),(106,112),(107,111),(108,110),(113,147),(114,146),(115,145),(116,152),(117,151),(118,150),(119,149),(120,148),(121,125),(122,124),(126,128),(129,133),(130,132),(134,136),(137,141),(138,140),(142,144)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 10A | ··· | 10L | 10M | 10N | 10O | 10P | 10Q | ··· | 10X | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 20M | ··· | 20T | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 8 | 2 | 2 | 4 | 8 | 8 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
80 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 | D4 | C4○D4 | C5×D4 | C5×D4 | C5×D4 | C5×C4○D4 | C8⋊C22 | C5×C8⋊C22 |
| kernel | C5×C8⋊2D4 | C5×D4⋊C4 | C5×C4.Q8 | C5×C4⋊D4 | C10×M4(2) | C10×D8 | C8⋊2D4 | D4⋊C4 | C4.Q8 | C4⋊D4 | C2×M4(2) | C2×D8 | C40 | C2×C20 | C22×C10 | C20 | C8 | C2×C4 | C23 | C4 | C10 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 4 | 8 | 4 | 8 | 4 | 4 | 2 | 1 | 1 | 2 | 8 | 4 | 4 | 8 | 2 | 8 |
Matrix representation of C5×C8⋊2D4 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 28 | 1 | 1 |
| 0 | 0 | 40 | 2 | 40 | 1 |
| 0 | 0 | 8 | 31 | 7 | 22 |
| 0 | 0 | 22 | 14 | 10 | 21 |
| 40 | 2 | 0 | 0 | 0 | 0 |
| 40 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 27 | 2 | 0 |
| 0 | 0 | 32 | 11 | 0 | 39 |
| 0 | 0 | 8 | 38 | 29 | 27 |
| 0 | 0 | 40 | 1 | 32 | 30 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 29 | 0 | 40 | 0 |
| 0 | 0 | 0 | 30 | 0 | 1 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,40,8,22,0,0,28,2,31,14,0,0,1,40,7,10,0,0,1,1,22,21],[40,40,0,0,0,0,2,1,0,0,0,0,0,0,12,32,8,40,0,0,27,11,38,1,0,0,2,0,29,32,0,0,0,39,27,30],[1,1,0,0,0,0,0,40,0,0,0,0,0,0,1,0,29,0,0,0,0,40,0,30,0,0,0,0,40,0,0,0,0,0,0,1] >;
C5×C8⋊2D4 in GAP, Magma, Sage, TeX
C_5\times C_8\rtimes_2D_4
% in TeX
G:=Group("C5xC8:2D4"); // GroupNames label
G:=SmallGroup(320,970);
// by ID
G=gap.SmallGroup(320,970);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,589,1408,1766,1731,7004,172]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^8=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^3,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations