direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8×C5⋊D4, C40⋊31D4, C5⋊8(C8×D4), D10⋊6(C2×C8), (C22×C8)⋊2D5, C22⋊2(C8×D5), Dic5⋊3(C2×C8), (C22×C40)⋊15C2, (C8×Dic5)⋊26C2, C10.107(C4×D4), (C2×C8).345D10, C20.435(C2×D4), D10⋊1C8⋊42C2, C23.33(C4×D5), C10.41(C8○D4), C10.44(C22×C8), C20.8Q8⋊43C2, C23.D5.22C4, D10⋊C4.30C4, C20.251(C4○D4), C4.135(C4○D20), C20.55D4⋊32C2, (C2×C20).859C23, (C2×C40).353C22, C10.D4.30C4, (C22×C4).401D10, C2.5(D20.3C4), (C22×C20).560C22, (C4×Dic5).313C22, (D5×C2×C8)⋊24C2, C2.20(D5×C2×C8), C2.3(C4×C5⋊D4), (C2×C10)⋊11(C2×C8), (C2×C4).93(C4×D5), C22.61(C2×C4×D5), (C2×C5⋊D4).27C4, (C4×C5⋊D4).19C2, C4.125(C2×C5⋊D4), (C2×C20).382(C2×C4), (C2×C4×D5).353C22, (C22×D5).80(C2×C4), (C2×C4).801(C22×D5), (C22×C10).162(C2×C4), (C2×C10).230(C22×C4), (C2×C5⋊2C8).328C22, (C2×Dic5).111(C2×C4), SmallGroup(320,736)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8×C5⋊D4
G = < a,b,c,d | a8=b5=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 382 in 134 conjugacy classes, 63 normal (47 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×5], C22, C22 [×2], C22 [×6], C5, C8 [×2], C8 [×3], C2×C4 [×2], C2×C4 [×7], D4 [×4], C23, C23, D5 [×2], C10 [×3], C10 [×2], C42, C22⋊C4 [×2], C4⋊C4, C2×C8 [×2], C2×C8 [×6], C22×C4, C22×C4, C2×D4, Dic5 [×2], Dic5 [×2], C20 [×2], C20, D10 [×2], D10 [×2], C2×C10, C2×C10 [×2], C2×C10 [×2], C4×C8, C22⋊C8 [×2], C4⋊C8, C4×D4, C22×C8, C22×C8, C5⋊2C8 [×2], C40 [×2], C40, C4×D5 [×2], C2×Dic5 [×3], C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×2], C22×D5, C22×C10, C8×D4, C8×D5 [×2], C2×C5⋊2C8 [×2], C4×Dic5, C10.D4, D10⋊C4, C23.D5, C2×C40 [×2], C2×C40 [×2], C2×C4×D5, C2×C5⋊D4, C22×C20, C8×Dic5, C20.8Q8, D10⋊1C8, C20.55D4, D5×C2×C8, C4×C5⋊D4, C22×C40, C8×C5⋊D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×2], C23, D5, C2×C8 [×6], C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C22×C8, C8○D4, C4×D5 [×2], C5⋊D4 [×2], C22×D5, C8×D4, C8×D5 [×2], C2×C4×D5, C4○D20, C2×C5⋊D4, D5×C2×C8, D20.3C4, C4×C5⋊D4, C8×C5⋊D4
►On 160 points
Generators in S160
(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 54 121 107 73)(2 55 122 108 74)(3 56 123 109 75)(4 49 124 110 76)(5 50 125 111 77)(6 51 126 112 78)(7 52 127 105 79)(8 53 128 106 80)(9 91 86 152 34)(10 92 87 145 35)(11 93 88 146 36)(12 94 81 147 37)(13 95 82 148 38)(14 96 83 149 39)(15 89 84 150 40)(16 90 85 151 33)(17 48 116 144 72)(18 41 117 137 65)(19 42 118 138 66)(20 43 119 139 67)(21 44 120 140 68)(22 45 113 141 69)(23 46 114 142 70)(24 47 115 143 71)(25 157 131 103 62)(26 158 132 104 63)(27 159 133 97 64)(28 160 134 98 57)(29 153 135 99 58)(30 154 136 100 59)(31 155 129 101 60)(32 156 130 102 61)
(1 131 81 22)(2 132 82 23)(3 133 83 24)(4 134 84 17)(5 135 85 18)(6 136 86 19)(7 129 87 20)(8 130 88 21)(9 118 112 59)(10 119 105 60)(11 120 106 61)(12 113 107 62)(13 114 108 63)(14 115 109 64)(15 116 110 57)(16 117 111 58)(25 37 141 121)(26 38 142 122)(27 39 143 123)(28 40 144 124)(29 33 137 125)(30 34 138 126)(31 35 139 127)(32 36 140 128)(41 77 99 90)(42 78 100 91)(43 79 101 92)(44 80 102 93)(45 73 103 94)(46 74 104 95)(47 75 97 96)(48 76 98 89)(49 160 150 72)(50 153 151 65)(51 154 152 66)(52 155 145 67)(53 156 146 68)(54 157 147 69)(55 158 148 70)(56 159 149 71)
(9 34)(10 35)(11 36)(12 37)(13 38)(14 39)(15 40)(16 33)(17 134)(18 135)(19 136)(20 129)(21 130)(22 131)(23 132)(24 133)(25 113)(26 114)(27 115)(28 116)(29 117)(30 118)(31 119)(32 120)(41 153)(42 154)(43 155)(44 156)(45 157)(46 158)(47 159)(48 160)(49 76)(50 77)(51 78)(52 79)(53 80)(54 73)(55 74)(56 75)(57 144)(58 137)(59 138)(60 139)(61 140)(62 141)(63 142)(64 143)(65 99)(66 100)(67 101)(68 102)(69 103)(70 104)(71 97)(72 98)(89 150)(90 151)(91 152)(92 145)(93 146)(94 147)(95 148)(96 149)(105 127)(106 128)(107 121)(108 122)(109 123)(110 124)(111 125)(112 126)
G:=sub<Sym(160)| (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), (1,54,121,107,73)(2,55,122,108,74)(3,56,123,109,75)(4,49,124,110,76)(5,50,125,111,77)(6,51,126,112,78)(7,52,127,105,79)(8,53,128,106,80)(9,91,86,152,34)(10,92,87,145,35)(11,93,88,146,36)(12,94,81,147,37)(13,95,82,148,38)(14,96,83,149,39)(15,89,84,150,40)(16,90,85,151,33)(17,48,116,144,72)(18,41,117,137,65)(19,42,118,138,66)(20,43,119,139,67)(21,44,120,140,68)(22,45,113,141,69)(23,46,114,142,70)(24,47,115,143,71)(25,157,131,103,62)(26,158,132,104,63)(27,159,133,97,64)(28,160,134,98,57)(29,153,135,99,58)(30,154,136,100,59)(31,155,129,101,60)(32,156,130,102,61), (1,131,81,22)(2,132,82,23)(3,133,83,24)(4,134,84,17)(5,135,85,18)(6,136,86,19)(7,129,87,20)(8,130,88,21)(9,118,112,59)(10,119,105,60)(11,120,106,61)(12,113,107,62)(13,114,108,63)(14,115,109,64)(15,116,110,57)(16,117,111,58)(25,37,141,121)(26,38,142,122)(27,39,143,123)(28,40,144,124)(29,33,137,125)(30,34,138,126)(31,35,139,127)(32,36,140,128)(41,77,99,90)(42,78,100,91)(43,79,101,92)(44,80,102,93)(45,73,103,94)(46,74,104,95)(47,75,97,96)(48,76,98,89)(49,160,150,72)(50,153,151,65)(51,154,152,66)(52,155,145,67)(53,156,146,68)(54,157,147,69)(55,158,148,70)(56,159,149,71), (9,34)(10,35)(11,36)(12,37)(13,38)(14,39)(15,40)(16,33)(17,134)(18,135)(19,136)(20,129)(21,130)(22,131)(23,132)(24,133)(25,113)(26,114)(27,115)(28,116)(29,117)(30,118)(31,119)(32,120)(41,153)(42,154)(43,155)(44,156)(45,157)(46,158)(47,159)(48,160)(49,76)(50,77)(51,78)(52,79)(53,80)(54,73)(55,74)(56,75)(57,144)(58,137)(59,138)(60,139)(61,140)(62,141)(63,142)(64,143)(65,99)(66,100)(67,101)(68,102)(69,103)(70,104)(71,97)(72,98)(89,150)(90,151)(91,152)(92,145)(93,146)(94,147)(95,148)(96,149)(105,127)(106,128)(107,121)(108,122)(109,123)(110,124)(111,125)(112,126)>;
G:=Group( (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), (1,54,121,107,73)(2,55,122,108,74)(3,56,123,109,75)(4,49,124,110,76)(5,50,125,111,77)(6,51,126,112,78)(7,52,127,105,79)(8,53,128,106,80)(9,91,86,152,34)(10,92,87,145,35)(11,93,88,146,36)(12,94,81,147,37)(13,95,82,148,38)(14,96,83,149,39)(15,89,84,150,40)(16,90,85,151,33)(17,48,116,144,72)(18,41,117,137,65)(19,42,118,138,66)(20,43,119,139,67)(21,44,120,140,68)(22,45,113,141,69)(23,46,114,142,70)(24,47,115,143,71)(25,157,131,103,62)(26,158,132,104,63)(27,159,133,97,64)(28,160,134,98,57)(29,153,135,99,58)(30,154,136,100,59)(31,155,129,101,60)(32,156,130,102,61), (1,131,81,22)(2,132,82,23)(3,133,83,24)(4,134,84,17)(5,135,85,18)(6,136,86,19)(7,129,87,20)(8,130,88,21)(9,118,112,59)(10,119,105,60)(11,120,106,61)(12,113,107,62)(13,114,108,63)(14,115,109,64)(15,116,110,57)(16,117,111,58)(25,37,141,121)(26,38,142,122)(27,39,143,123)(28,40,144,124)(29,33,137,125)(30,34,138,126)(31,35,139,127)(32,36,140,128)(41,77,99,90)(42,78,100,91)(43,79,101,92)(44,80,102,93)(45,73,103,94)(46,74,104,95)(47,75,97,96)(48,76,98,89)(49,160,150,72)(50,153,151,65)(51,154,152,66)(52,155,145,67)(53,156,146,68)(54,157,147,69)(55,158,148,70)(56,159,149,71), (9,34)(10,35)(11,36)(12,37)(13,38)(14,39)(15,40)(16,33)(17,134)(18,135)(19,136)(20,129)(21,130)(22,131)(23,132)(24,133)(25,113)(26,114)(27,115)(28,116)(29,117)(30,118)(31,119)(32,120)(41,153)(42,154)(43,155)(44,156)(45,157)(46,158)(47,159)(48,160)(49,76)(50,77)(51,78)(52,79)(53,80)(54,73)(55,74)(56,75)(57,144)(58,137)(59,138)(60,139)(61,140)(62,141)(63,142)(64,143)(65,99)(66,100)(67,101)(68,102)(69,103)(70,104)(71,97)(72,98)(89,150)(90,151)(91,152)(92,145)(93,146)(94,147)(95,148)(96,149)(105,127)(106,128)(107,121)(108,122)(109,123)(110,124)(111,125)(112,126) );
G=PermutationGroup([(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)], [(1,54,121,107,73),(2,55,122,108,74),(3,56,123,109,75),(4,49,124,110,76),(5,50,125,111,77),(6,51,126,112,78),(7,52,127,105,79),(8,53,128,106,80),(9,91,86,152,34),(10,92,87,145,35),(11,93,88,146,36),(12,94,81,147,37),(13,95,82,148,38),(14,96,83,149,39),(15,89,84,150,40),(16,90,85,151,33),(17,48,116,144,72),(18,41,117,137,65),(19,42,118,138,66),(20,43,119,139,67),(21,44,120,140,68),(22,45,113,141,69),(23,46,114,142,70),(24,47,115,143,71),(25,157,131,103,62),(26,158,132,104,63),(27,159,133,97,64),(28,160,134,98,57),(29,153,135,99,58),(30,154,136,100,59),(31,155,129,101,60),(32,156,130,102,61)], [(1,131,81,22),(2,132,82,23),(3,133,83,24),(4,134,84,17),(5,135,85,18),(6,136,86,19),(7,129,87,20),(8,130,88,21),(9,118,112,59),(10,119,105,60),(11,120,106,61),(12,113,107,62),(13,114,108,63),(14,115,109,64),(15,116,110,57),(16,117,111,58),(25,37,141,121),(26,38,142,122),(27,39,143,123),(28,40,144,124),(29,33,137,125),(30,34,138,126),(31,35,139,127),(32,36,140,128),(41,77,99,90),(42,78,100,91),(43,79,101,92),(44,80,102,93),(45,73,103,94),(46,74,104,95),(47,75,97,96),(48,76,98,89),(49,160,150,72),(50,153,151,65),(51,154,152,66),(52,155,145,67),(53,156,146,68),(54,157,147,69),(55,158,148,70),(56,159,149,71)], [(9,34),(10,35),(11,36),(12,37),(13,38),(14,39),(15,40),(16,33),(17,134),(18,135),(19,136),(20,129),(21,130),(22,131),(23,132),(24,133),(25,113),(26,114),(27,115),(28,116),(29,117),(30,118),(31,119),(32,120),(41,153),(42,154),(43,155),(44,156),(45,157),(46,158),(47,159),(48,160),(49,76),(50,77),(51,78),(52,79),(53,80),(54,73),(55,74),(56,75),(57,144),(58,137),(59,138),(60,139),(61,140),(62,141),(63,142),(64,143),(65,99),(66,100),(67,101),(68,102),(69,103),(70,104),(71,97),(72,98),(89,150),(90,151),(91,152),(92,145),(93,146),(94,147),(95,148),(96,149),(105,127),(106,128),(107,121),(108,122),(109,123),(110,124),(111,125),(112,126)])
104 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4L | 5A | 5B | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 8M | ··· | 8T | 10A | ··· | 10N | 20A | ··· | 20P | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 10 | 10 | 1 | 1 | 1 | 1 | 2 | 2 | 10 | ··· | 10 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
104 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C8 | D4 | D5 | C4○D4 | D10 | D10 | C8○D4 | C5⋊D4 | C4×D5 | C4×D5 | C4○D20 | C8×D5 | D20.3C4 |
| kernel | C8×C5⋊D4 | C8×Dic5 | C20.8Q8 | D10⋊1C8 | C20.55D4 | D5×C2×C8 | C4×C5⋊D4 | C22×C40 | C10.D4 | D10⋊C4 | C23.D5 | C2×C5⋊D4 | C5⋊D4 | C40 | C22×C8 | C20 | C2×C8 | C22×C4 | C10 | C8 | C2×C4 | C23 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 16 | 2 | 2 | 2 | 4 | 2 | 4 | 8 | 4 | 4 | 8 | 16 | 16 |
Matrix representation of C8×C5⋊D4 ►in GL3(𝔽41) generated by
| 3 | 0 | 0 |
| 0 | 38 | 0 |
| 0 | 0 | 38 |
| 1 | 0 | 0 |
| 0 | 7 | 40 |
| 0 | 8 | 40 |
| 1 | 0 | 0 |
| 0 | 3 | 18 |
| 0 | 4 | 38 |
| 40 | 0 | 0 |
| 0 | 34 | 1 |
| 0 | 34 | 7 |
G:=sub<GL(3,GF(41))| [3,0,0,0,38,0,0,0,38],[1,0,0,0,7,8,0,40,40],[1,0,0,0,3,4,0,18,38],[40,0,0,0,34,34,0,1,7] >;
C8×C5⋊D4 in GAP, Magma, Sage, TeX
C_8\times C_5\rtimes D_4
% in TeX
G:=Group("C8xC5:D4"); // GroupNames label
G:=SmallGroup(320,736);
// by ID
G=gap.SmallGroup(320,736);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,58,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^5=c^4=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