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, C2, C4, C4, C22, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C10, C4×C8, C22⋊C8, C4⋊C8, C4×D4, C22×C8, C22×C8, C5⋊2C8, C40, C40, C4×D5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C8×D4, C8×D5, C2×C5⋊2C8, C4×Dic5, C10.D4, D10⋊C4, C23.D5, C2×C40, C2×C40, 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, C4, C22, C8, C2×C4, D4, C23, D5, C2×C8, C22×C4, C2×D4, C4○D4, D10, C4×D4, C22×C8, C8○D4, C4×D5, C5⋊D4, C22×D5, C8×D4, C8×D5, 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 12 81 107 54)(2 13 82 108 55)(3 14 83 109 56)(4 15 84 110 49)(5 16 85 111 50)(6 9 86 112 51)(7 10 87 105 52)(8 11 88 106 53)(17 48 141 156 104)(18 41 142 157 97)(19 42 143 158 98)(20 43 144 159 99)(21 44 137 160 100)(22 45 138 153 101)(23 46 139 154 102)(24 47 140 155 103)(25 114 129 69 62)(26 115 130 70 63)(27 116 131 71 64)(28 117 132 72 57)(29 118 133 65 58)(30 119 134 66 59)(31 120 135 67 60)(32 113 136 68 61)(33 78 89 124 150)(34 79 90 125 151)(35 80 91 126 152)(36 73 92 127 145)(37 74 93 128 146)(38 75 94 121 147)(39 76 95 122 148)(40 77 96 123 149)
(1 153 121 62)(2 154 122 63)(3 155 123 64)(4 156 124 57)(5 157 125 58)(6 158 126 59)(7 159 127 60)(8 160 128 61)(9 143 152 66)(10 144 145 67)(11 137 146 68)(12 138 147 69)(13 139 148 70)(14 140 149 71)(15 141 150 72)(16 142 151 65)(17 78 117 110)(18 79 118 111)(19 80 119 112)(20 73 120 105)(21 74 113 106)(22 75 114 107)(23 76 115 108)(24 77 116 109)(25 54 101 94)(26 55 102 95)(27 56 103 96)(28 49 104 89)(29 50 97 90)(30 51 98 91)(31 52 99 92)(32 53 100 93)(33 132 84 48)(34 133 85 41)(35 134 86 42)(36 135 87 43)(37 136 88 44)(38 129 81 45)(39 130 82 46)(40 131 83 47)
(9 51)(10 52)(11 53)(12 54)(13 55)(14 56)(15 49)(16 50)(17 132)(18 133)(19 134)(20 135)(21 136)(22 129)(23 130)(24 131)(25 138)(26 139)(27 140)(28 141)(29 142)(30 143)(31 144)(32 137)(33 78)(34 79)(35 80)(36 73)(37 74)(38 75)(39 76)(40 77)(41 118)(42 119)(43 120)(44 113)(45 114)(46 115)(47 116)(48 117)(57 156)(58 157)(59 158)(60 159)(61 160)(62 153)(63 154)(64 155)(65 97)(66 98)(67 99)(68 100)(69 101)(70 102)(71 103)(72 104)(81 107)(82 108)(83 109)(84 110)(85 111)(86 112)(87 105)(88 106)(89 150)(90 151)(91 152)(92 145)(93 146)(94 147)(95 148)(96 149)
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), (1,12,81,107,54)(2,13,82,108,55)(3,14,83,109,56)(4,15,84,110,49)(5,16,85,111,50)(6,9,86,112,51)(7,10,87,105,52)(8,11,88,106,53)(17,48,141,156,104)(18,41,142,157,97)(19,42,143,158,98)(20,43,144,159,99)(21,44,137,160,100)(22,45,138,153,101)(23,46,139,154,102)(24,47,140,155,103)(25,114,129,69,62)(26,115,130,70,63)(27,116,131,71,64)(28,117,132,72,57)(29,118,133,65,58)(30,119,134,66,59)(31,120,135,67,60)(32,113,136,68,61)(33,78,89,124,150)(34,79,90,125,151)(35,80,91,126,152)(36,73,92,127,145)(37,74,93,128,146)(38,75,94,121,147)(39,76,95,122,148)(40,77,96,123,149), (1,153,121,62)(2,154,122,63)(3,155,123,64)(4,156,124,57)(5,157,125,58)(6,158,126,59)(7,159,127,60)(8,160,128,61)(9,143,152,66)(10,144,145,67)(11,137,146,68)(12,138,147,69)(13,139,148,70)(14,140,149,71)(15,141,150,72)(16,142,151,65)(17,78,117,110)(18,79,118,111)(19,80,119,112)(20,73,120,105)(21,74,113,106)(22,75,114,107)(23,76,115,108)(24,77,116,109)(25,54,101,94)(26,55,102,95)(27,56,103,96)(28,49,104,89)(29,50,97,90)(30,51,98,91)(31,52,99,92)(32,53,100,93)(33,132,84,48)(34,133,85,41)(35,134,86,42)(36,135,87,43)(37,136,88,44)(38,129,81,45)(39,130,82,46)(40,131,83,47), (9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,49)(16,50)(17,132)(18,133)(19,134)(20,135)(21,136)(22,129)(23,130)(24,131)(25,138)(26,139)(27,140)(28,141)(29,142)(30,143)(31,144)(32,137)(33,78)(34,79)(35,80)(36,73)(37,74)(38,75)(39,76)(40,77)(41,118)(42,119)(43,120)(44,113)(45,114)(46,115)(47,116)(48,117)(57,156)(58,157)(59,158)(60,159)(61,160)(62,153)(63,154)(64,155)(65,97)(66,98)(67,99)(68,100)(69,101)(70,102)(71,103)(72,104)(81,107)(82,108)(83,109)(84,110)(85,111)(86,112)(87,105)(88,106)(89,150)(90,151)(91,152)(92,145)(93,146)(94,147)(95,148)(96,149)>;
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), (1,12,81,107,54)(2,13,82,108,55)(3,14,83,109,56)(4,15,84,110,49)(5,16,85,111,50)(6,9,86,112,51)(7,10,87,105,52)(8,11,88,106,53)(17,48,141,156,104)(18,41,142,157,97)(19,42,143,158,98)(20,43,144,159,99)(21,44,137,160,100)(22,45,138,153,101)(23,46,139,154,102)(24,47,140,155,103)(25,114,129,69,62)(26,115,130,70,63)(27,116,131,71,64)(28,117,132,72,57)(29,118,133,65,58)(30,119,134,66,59)(31,120,135,67,60)(32,113,136,68,61)(33,78,89,124,150)(34,79,90,125,151)(35,80,91,126,152)(36,73,92,127,145)(37,74,93,128,146)(38,75,94,121,147)(39,76,95,122,148)(40,77,96,123,149), (1,153,121,62)(2,154,122,63)(3,155,123,64)(4,156,124,57)(5,157,125,58)(6,158,126,59)(7,159,127,60)(8,160,128,61)(9,143,152,66)(10,144,145,67)(11,137,146,68)(12,138,147,69)(13,139,148,70)(14,140,149,71)(15,141,150,72)(16,142,151,65)(17,78,117,110)(18,79,118,111)(19,80,119,112)(20,73,120,105)(21,74,113,106)(22,75,114,107)(23,76,115,108)(24,77,116,109)(25,54,101,94)(26,55,102,95)(27,56,103,96)(28,49,104,89)(29,50,97,90)(30,51,98,91)(31,52,99,92)(32,53,100,93)(33,132,84,48)(34,133,85,41)(35,134,86,42)(36,135,87,43)(37,136,88,44)(38,129,81,45)(39,130,82,46)(40,131,83,47), (9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,49)(16,50)(17,132)(18,133)(19,134)(20,135)(21,136)(22,129)(23,130)(24,131)(25,138)(26,139)(27,140)(28,141)(29,142)(30,143)(31,144)(32,137)(33,78)(34,79)(35,80)(36,73)(37,74)(38,75)(39,76)(40,77)(41,118)(42,119)(43,120)(44,113)(45,114)(46,115)(47,116)(48,117)(57,156)(58,157)(59,158)(60,159)(61,160)(62,153)(63,154)(64,155)(65,97)(66,98)(67,99)(68,100)(69,101)(70,102)(71,103)(72,104)(81,107)(82,108)(83,109)(84,110)(85,111)(86,112)(87,105)(88,106)(89,150)(90,151)(91,152)(92,145)(93,146)(94,147)(95,148)(96,149) );
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)], [(1,12,81,107,54),(2,13,82,108,55),(3,14,83,109,56),(4,15,84,110,49),(5,16,85,111,50),(6,9,86,112,51),(7,10,87,105,52),(8,11,88,106,53),(17,48,141,156,104),(18,41,142,157,97),(19,42,143,158,98),(20,43,144,159,99),(21,44,137,160,100),(22,45,138,153,101),(23,46,139,154,102),(24,47,140,155,103),(25,114,129,69,62),(26,115,130,70,63),(27,116,131,71,64),(28,117,132,72,57),(29,118,133,65,58),(30,119,134,66,59),(31,120,135,67,60),(32,113,136,68,61),(33,78,89,124,150),(34,79,90,125,151),(35,80,91,126,152),(36,73,92,127,145),(37,74,93,128,146),(38,75,94,121,147),(39,76,95,122,148),(40,77,96,123,149)], [(1,153,121,62),(2,154,122,63),(3,155,123,64),(4,156,124,57),(5,157,125,58),(6,158,126,59),(7,159,127,60),(8,160,128,61),(9,143,152,66),(10,144,145,67),(11,137,146,68),(12,138,147,69),(13,139,148,70),(14,140,149,71),(15,141,150,72),(16,142,151,65),(17,78,117,110),(18,79,118,111),(19,80,119,112),(20,73,120,105),(21,74,113,106),(22,75,114,107),(23,76,115,108),(24,77,116,109),(25,54,101,94),(26,55,102,95),(27,56,103,96),(28,49,104,89),(29,50,97,90),(30,51,98,91),(31,52,99,92),(32,53,100,93),(33,132,84,48),(34,133,85,41),(35,134,86,42),(36,135,87,43),(37,136,88,44),(38,129,81,45),(39,130,82,46),(40,131,83,47)], [(9,51),(10,52),(11,53),(12,54),(13,55),(14,56),(15,49),(16,50),(17,132),(18,133),(19,134),(20,135),(21,136),(22,129),(23,130),(24,131),(25,138),(26,139),(27,140),(28,141),(29,142),(30,143),(31,144),(32,137),(33,78),(34,79),(35,80),(36,73),(37,74),(38,75),(39,76),(40,77),(41,118),(42,119),(43,120),(44,113),(45,114),(46,115),(47,116),(48,117),(57,156),(58,157),(59,158),(60,159),(61,160),(62,153),(63,154),(64,155),(65,97),(66,98),(67,99),(68,100),(69,101),(70,102),(71,103),(72,104),(81,107),(82,108),(83,109),(84,110),(85,111),(86,112),(87,105),(88,106),(89,150),(90,151),(91,152),(92,145),(93,146),(94,147),(95,148),(96,149)]])
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