direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×C8⋊D4, C40⋊19D4, C8⋊1(C5×D4), C2.D8⋊12C10, C4.60(D4×C10), C22⋊Q8⋊4C10, (C2×SD16)⋊1C10, (C2×C20).327D4, C4⋊D4.4C10, C20.467(C2×D4), D4⋊C4⋊17C10, C23.15(C5×D4), Q8⋊C4⋊17C10, (C10×SD16)⋊12C2, (C2×M4(2))⋊1C10, C22.92(D4×C10), (C22×C10).33D4, C20.265(C4○D4), (C10×M4(2))⋊11C2, (C2×C40).332C22, (C2×C20).927C23, C10.151(C4⋊D4), C10.137(C8⋊C22), (D4×C10).191C22, (Q8×C10).165C22, C10.137(C8.C22), (C22×C20).425C22, C4⋊C4.8(C2×C10), (C5×C2.D8)⋊27C2, C4.10(C5×C4○D4), (C2×C4).32(C5×D4), (C2×C8).21(C2×C10), C2.20(C5×C4⋊D4), C2.12(C5×C8⋊C22), (C5×C22⋊Q8)⋊31C2, (C2×Q8).9(C2×C10), (C5×D4⋊C4)⋊40C2, (C5×Q8⋊C4)⋊40C2, (C2×D4).14(C2×C10), (C2×C10).648(C2×D4), (C5×C4⋊D4).14C2, C2.12(C5×C8.C22), (C5×C4⋊C4).230C22, (C22×C4).43(C2×C10), (C2×C4).102(C22×C10), SmallGroup(320,969)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C8⋊D4
G = < a,b,c,d | a5=b8=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd=b3, dcd=c-1 >
Subgroups: 234 in 120 conjugacy classes, 54 normal (50 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C10, C10, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C20, C20, C2×C10, C2×C10, D4⋊C4, Q8⋊C4, C2.D8, C4⋊D4, C22⋊Q8, C2×M4(2), C2×SD16, C40, C40, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C22×C10, C8⋊D4, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C2×C40, C5×M4(2), C5×SD16, C22×C20, D4×C10, D4×C10, Q8×C10, C5×D4⋊C4, C5×Q8⋊C4, C5×C2.D8, C5×C4⋊D4, C5×C22⋊Q8, C10×M4(2), C10×SD16, C5×C8⋊D4
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C4○D4, C2×C10, C4⋊D4, C8⋊C22, C8.C22, C5×D4, C22×C10, C8⋊D4, D4×C10, C5×C4○D4, C5×C4⋊D4, C5×C8⋊C22, C5×C8.C22, C5×C8⋊D4
►On 160 points
Generators in S160
(1 143 61 135 53)(2 144 62 136 54)(3 137 63 129 55)(4 138 64 130 56)(5 139 57 131 49)(6 140 58 132 50)(7 141 59 133 51)(8 142 60 134 52)(9 33 20 94 99)(10 34 21 95 100)(11 35 22 96 101)(12 36 23 89 102)(13 37 24 90 103)(14 38 17 91 104)(15 39 18 92 97)(16 40 19 93 98)(25 41 126 118 107)(26 42 127 119 108)(27 43 128 120 109)(28 44 121 113 110)(29 45 122 114 111)(30 46 123 115 112)(31 47 124 116 105)(32 48 125 117 106)(65 83 155 73 147)(66 84 156 74 148)(67 85 157 75 149)(68 86 158 76 150)(69 87 159 77 151)(70 88 160 78 152)(71 81 153 79 145)(72 82 154 80 146)
(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 69 9 28)(2 68 10 27)(3 67 11 26)(4 66 12 25)(5 65 13 32)(6 72 14 31)(7 71 15 30)(8 70 16 29)(17 124 58 154)(18 123 59 153)(19 122 60 160)(20 121 61 159)(21 128 62 158)(22 127 63 157)(23 126 64 156)(24 125 57 155)(33 44 143 87)(34 43 144 86)(35 42 137 85)(36 41 138 84)(37 48 139 83)(38 47 140 82)(39 46 141 81)(40 45 142 88)(49 147 103 106)(50 146 104 105)(51 145 97 112)(52 152 98 111)(53 151 99 110)(54 150 100 109)(55 149 101 108)(56 148 102 107)(73 90 117 131)(74 89 118 130)(75 96 119 129)(76 95 120 136)(77 94 113 135)(78 93 114 134)(79 92 115 133)(80 91 116 132)
(2 4)(3 7)(6 8)(10 12)(11 15)(14 16)(17 19)(18 22)(21 23)(25 68)(26 71)(27 66)(28 69)(29 72)(30 67)(31 70)(32 65)(34 36)(35 39)(38 40)(41 86)(42 81)(43 84)(44 87)(45 82)(46 85)(47 88)(48 83)(50 52)(51 55)(54 56)(58 60)(59 63)(62 64)(73 117)(74 120)(75 115)(76 118)(77 113)(78 116)(79 119)(80 114)(89 95)(91 93)(92 96)(97 101)(98 104)(100 102)(105 152)(106 147)(107 150)(108 145)(109 148)(110 151)(111 146)(112 149)(121 159)(122 154)(123 157)(124 160)(125 155)(126 158)(127 153)(128 156)(129 133)(130 136)(132 134)(137 141)(138 144)(140 142)
G:=sub<Sym(160)| (1,143,61,135,53)(2,144,62,136,54)(3,137,63,129,55)(4,138,64,130,56)(5,139,57,131,49)(6,140,58,132,50)(7,141,59,133,51)(8,142,60,134,52)(9,33,20,94,99)(10,34,21,95,100)(11,35,22,96,101)(12,36,23,89,102)(13,37,24,90,103)(14,38,17,91,104)(15,39,18,92,97)(16,40,19,93,98)(25,41,126,118,107)(26,42,127,119,108)(27,43,128,120,109)(28,44,121,113,110)(29,45,122,114,111)(30,46,123,115,112)(31,47,124,116,105)(32,48,125,117,106)(65,83,155,73,147)(66,84,156,74,148)(67,85,157,75,149)(68,86,158,76,150)(69,87,159,77,151)(70,88,160,78,152)(71,81,153,79,145)(72,82,154,80,146), (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), (1,69,9,28)(2,68,10,27)(3,67,11,26)(4,66,12,25)(5,65,13,32)(6,72,14,31)(7,71,15,30)(8,70,16,29)(17,124,58,154)(18,123,59,153)(19,122,60,160)(20,121,61,159)(21,128,62,158)(22,127,63,157)(23,126,64,156)(24,125,57,155)(33,44,143,87)(34,43,144,86)(35,42,137,85)(36,41,138,84)(37,48,139,83)(38,47,140,82)(39,46,141,81)(40,45,142,88)(49,147,103,106)(50,146,104,105)(51,145,97,112)(52,152,98,111)(53,151,99,110)(54,150,100,109)(55,149,101,108)(56,148,102,107)(73,90,117,131)(74,89,118,130)(75,96,119,129)(76,95,120,136)(77,94,113,135)(78,93,114,134)(79,92,115,133)(80,91,116,132), (2,4)(3,7)(6,8)(10,12)(11,15)(14,16)(17,19)(18,22)(21,23)(25,68)(26,71)(27,66)(28,69)(29,72)(30,67)(31,70)(32,65)(34,36)(35,39)(38,40)(41,86)(42,81)(43,84)(44,87)(45,82)(46,85)(47,88)(48,83)(50,52)(51,55)(54,56)(58,60)(59,63)(62,64)(73,117)(74,120)(75,115)(76,118)(77,113)(78,116)(79,119)(80,114)(89,95)(91,93)(92,96)(97,101)(98,104)(100,102)(105,152)(106,147)(107,150)(108,145)(109,148)(110,151)(111,146)(112,149)(121,159)(122,154)(123,157)(124,160)(125,155)(126,158)(127,153)(128,156)(129,133)(130,136)(132,134)(137,141)(138,144)(140,142)>;
G:=Group( (1,143,61,135,53)(2,144,62,136,54)(3,137,63,129,55)(4,138,64,130,56)(5,139,57,131,49)(6,140,58,132,50)(7,141,59,133,51)(8,142,60,134,52)(9,33,20,94,99)(10,34,21,95,100)(11,35,22,96,101)(12,36,23,89,102)(13,37,24,90,103)(14,38,17,91,104)(15,39,18,92,97)(16,40,19,93,98)(25,41,126,118,107)(26,42,127,119,108)(27,43,128,120,109)(28,44,121,113,110)(29,45,122,114,111)(30,46,123,115,112)(31,47,124,116,105)(32,48,125,117,106)(65,83,155,73,147)(66,84,156,74,148)(67,85,157,75,149)(68,86,158,76,150)(69,87,159,77,151)(70,88,160,78,152)(71,81,153,79,145)(72,82,154,80,146), (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), (1,69,9,28)(2,68,10,27)(3,67,11,26)(4,66,12,25)(5,65,13,32)(6,72,14,31)(7,71,15,30)(8,70,16,29)(17,124,58,154)(18,123,59,153)(19,122,60,160)(20,121,61,159)(21,128,62,158)(22,127,63,157)(23,126,64,156)(24,125,57,155)(33,44,143,87)(34,43,144,86)(35,42,137,85)(36,41,138,84)(37,48,139,83)(38,47,140,82)(39,46,141,81)(40,45,142,88)(49,147,103,106)(50,146,104,105)(51,145,97,112)(52,152,98,111)(53,151,99,110)(54,150,100,109)(55,149,101,108)(56,148,102,107)(73,90,117,131)(74,89,118,130)(75,96,119,129)(76,95,120,136)(77,94,113,135)(78,93,114,134)(79,92,115,133)(80,91,116,132), (2,4)(3,7)(6,8)(10,12)(11,15)(14,16)(17,19)(18,22)(21,23)(25,68)(26,71)(27,66)(28,69)(29,72)(30,67)(31,70)(32,65)(34,36)(35,39)(38,40)(41,86)(42,81)(43,84)(44,87)(45,82)(46,85)(47,88)(48,83)(50,52)(51,55)(54,56)(58,60)(59,63)(62,64)(73,117)(74,120)(75,115)(76,118)(77,113)(78,116)(79,119)(80,114)(89,95)(91,93)(92,96)(97,101)(98,104)(100,102)(105,152)(106,147)(107,150)(108,145)(109,148)(110,151)(111,146)(112,149)(121,159)(122,154)(123,157)(124,160)(125,155)(126,158)(127,153)(128,156)(129,133)(130,136)(132,134)(137,141)(138,144)(140,142) );
G=PermutationGroup([[(1,143,61,135,53),(2,144,62,136,54),(3,137,63,129,55),(4,138,64,130,56),(5,139,57,131,49),(6,140,58,132,50),(7,141,59,133,51),(8,142,60,134,52),(9,33,20,94,99),(10,34,21,95,100),(11,35,22,96,101),(12,36,23,89,102),(13,37,24,90,103),(14,38,17,91,104),(15,39,18,92,97),(16,40,19,93,98),(25,41,126,118,107),(26,42,127,119,108),(27,43,128,120,109),(28,44,121,113,110),(29,45,122,114,111),(30,46,123,115,112),(31,47,124,116,105),(32,48,125,117,106),(65,83,155,73,147),(66,84,156,74,148),(67,85,157,75,149),(68,86,158,76,150),(69,87,159,77,151),(70,88,160,78,152),(71,81,153,79,145),(72,82,154,80,146)], [(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)], [(1,69,9,28),(2,68,10,27),(3,67,11,26),(4,66,12,25),(5,65,13,32),(6,72,14,31),(7,71,15,30),(8,70,16,29),(17,124,58,154),(18,123,59,153),(19,122,60,160),(20,121,61,159),(21,128,62,158),(22,127,63,157),(23,126,64,156),(24,125,57,155),(33,44,143,87),(34,43,144,86),(35,42,137,85),(36,41,138,84),(37,48,139,83),(38,47,140,82),(39,46,141,81),(40,45,142,88),(49,147,103,106),(50,146,104,105),(51,145,97,112),(52,152,98,111),(53,151,99,110),(54,150,100,109),(55,149,101,108),(56,148,102,107),(73,90,117,131),(74,89,118,130),(75,96,119,129),(76,95,120,136),(77,94,113,135),(78,93,114,134),(79,92,115,133),(80,91,116,132)], [(2,4),(3,7),(6,8),(10,12),(11,15),(14,16),(17,19),(18,22),(21,23),(25,68),(26,71),(27,66),(28,69),(29,72),(30,67),(31,70),(32,65),(34,36),(35,39),(38,40),(41,86),(42,81),(43,84),(44,87),(45,82),(46,85),(47,88),(48,83),(50,52),(51,55),(54,56),(58,60),(59,63),(62,64),(73,117),(74,120),(75,115),(76,118),(77,113),(78,116),(79,119),(80,114),(89,95),(91,93),(92,96),(97,101),(98,104),(100,102),(105,152),(106,147),(107,150),(108,145),(109,148),(110,151),(111,146),(112,149),(121,159),(122,154),(123,157),(124,160),(125,155),(126,158),(127,153),(128,156),(129,133),(130,136),(132,134),(137,141),(138,144),(140,142)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 10A | ··· | 10L | 10M | 10N | 10O | 10P | 10Q | 10R | 10S | 10T | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 20M | ··· | 20X | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 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 | 2 | 2 | 4 | 8 | 8 | 8 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 8 | 8 | 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 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | |||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | C10 | C10 | D4 | D4 | D4 | C4○D4 | C5×D4 | C5×D4 | C5×D4 | C5×C4○D4 | C8⋊C22 | C8.C22 | C5×C8⋊C22 | C5×C8.C22 |
| kernel | C5×C8⋊D4 | C5×D4⋊C4 | C5×Q8⋊C4 | C5×C2.D8 | C5×C4⋊D4 | C5×C22⋊Q8 | C10×M4(2) | C10×SD16 | C8⋊D4 | D4⋊C4 | Q8⋊C4 | C2.D8 | C4⋊D4 | C22⋊Q8 | C2×M4(2) | C2×SD16 | C40 | C2×C20 | C22×C10 | C20 | C8 | C2×C4 | C23 | C4 | C10 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 1 | 1 | 2 | 8 | 4 | 4 | 8 | 1 | 1 | 4 | 4 |
Matrix representation of C5×C8⋊D4 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 37 | 0 | 0 | 0 |
| 0 | 0 | 0 | 37 | 0 | 0 |
| 0 | 0 | 0 | 0 | 37 | 0 |
| 0 | 0 | 0 | 0 | 0 | 37 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 29 | 40 | 1 |
| 0 | 0 | 12 | 12 | 40 | 40 |
| 0 | 0 | 1 | 40 | 29 | 12 |
| 0 | 0 | 1 | 1 | 29 | 29 |
| 40 | 2 | 0 | 0 | 0 | 0 |
| 40 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 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 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,37,0,0,0,0,0,0,37,0,0,0,0,0,0,37,0,0,0,0,0,0,37],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,12,12,1,1,0,0,29,12,40,1,0,0,40,40,29,29,0,0,1,40,12,29],[40,40,0,0,0,0,2,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,1,0,0,0,0,0,0,40,0,0],[1,1,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40] >;
C5×C8⋊D4 in GAP, Magma, Sage, TeX
C_5\times C_8\rtimes D_4
% in TeX
G:=Group("C5xC8:D4"); // GroupNames label
G:=SmallGroup(320,969);
// by ID
G=gap.SmallGroup(320,969);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,589,288,1766,1731,7004,172]);
// 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^-1,d*b*d=b^3,d*c*d=c^-1>;
// generators/relations