direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×C8.D4, C40.51D4, C8.1(C5×D4), C4.Q8⋊5C10, (C2×Q16)⋊7C10, C4.62(D4×C10), (C10×Q16)⋊21C2, (C2×C20).329D4, C20.469(C2×D4), C23.17(C5×D4), C22⋊Q8.4C10, Q8⋊C4⋊18C10, (C22×C10).35D4, C22.94(D4×C10), C20.267(C4○D4), (C2×C40).334C22, (C2×C20).929C23, (C2×M4(2)).3C10, (C10×M4(2)).8C2, C10.153(C4⋊D4), (Q8×C10).166C22, C10.138(C8.C22), (C22×C20).427C22, (C5×C4.Q8)⋊14C2, C4.12(C5×C4○D4), (C2×C4).34(C5×D4), C4⋊C4.10(C2×C10), (C2×C8).23(C2×C10), C2.22(C5×C4⋊D4), (C5×Q8⋊C4)⋊41C2, (C2×C10).650(C2×D4), (C2×Q8).10(C2×C10), C2.13(C5×C8.C22), (C5×C22⋊Q8).14C2, (C5×C4⋊C4).232C22, (C22×C4).45(C2×C10), (C2×C4).104(C22×C10), SmallGroup(320,971)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C8.D4
G = < a,b,c,d | a5=b8=c4=1, d2=b4, ab=ba, ac=ca, ad=da, cbc-1=b3, dbd-1=b-1, dcd-1=b4c-1 >
Subgroups: 186 in 110 conjugacy classes, 54 normal (30 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, Q8, C23, C10, C10, C10, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), Q16, C22×C4, C2×Q8, C20, C20, C2×C10, C2×C10, Q8⋊C4, C4.Q8, C22⋊Q8, C2×M4(2), C2×Q16, C40, C40, C2×C20, C2×C20, C5×Q8, C22×C10, C8.D4, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C2×C40, C5×M4(2), C5×Q16, C22×C20, Q8×C10, C5×Q8⋊C4, C5×C4.Q8, C5×C22⋊Q8, C10×M4(2), C10×Q16, C5×C8.D4
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C4○D4, C2×C10, C4⋊D4, C8.C22, C5×D4, C22×C10, C8.D4, D4×C10, C5×C4○D4, C5×C4⋊D4, C5×C8.C22, C5×C8.D4
►On 160 points
Generators in S160
(1 16 49 57 17)(2 9 50 58 18)(3 10 51 59 19)(4 11 52 60 20)(5 12 53 61 21)(6 13 54 62 22)(7 14 55 63 23)(8 15 56 64 24)(25 85 73 33 65)(26 86 74 34 66)(27 87 75 35 67)(28 88 76 36 68)(29 81 77 37 69)(30 82 78 38 70)(31 83 79 39 71)(32 84 80 40 72)(41 156 120 148 112)(42 157 113 149 105)(43 158 114 150 106)(44 159 115 151 107)(45 160 116 152 108)(46 153 117 145 109)(47 154 118 146 110)(48 155 119 147 111)(89 126 141 97 133)(90 127 142 98 134)(91 128 143 99 135)(92 121 144 100 136)(93 122 137 101 129)(94 123 138 102 130)(95 124 139 103 131)(96 125 140 104 132)
(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)
(1 109 31 89)(2 112 32 92)(3 107 25 95)(4 110 26 90)(5 105 27 93)(6 108 28 96)(7 111 29 91)(8 106 30 94)(9 41 84 121)(10 44 85 124)(11 47 86 127)(12 42 87 122)(13 45 88 125)(14 48 81 128)(15 43 82 123)(16 46 83 126)(17 145 71 133)(18 148 72 136)(19 151 65 131)(20 146 66 134)(21 149 67 129)(22 152 68 132)(23 147 69 135)(24 150 70 130)(33 103 59 115)(34 98 60 118)(35 101 61 113)(36 104 62 116)(37 99 63 119)(38 102 64 114)(39 97 57 117)(40 100 58 120)(49 153 79 141)(50 156 80 144)(51 159 73 139)(52 154 74 142)(53 157 75 137)(54 160 76 140)(55 155 77 143)(56 158 78 138)
(1 93 5 89)(2 92 6 96)(3 91 7 95)(4 90 8 94)(9 121 13 125)(10 128 14 124)(11 127 15 123)(12 126 16 122)(17 129 21 133)(18 136 22 132)(19 135 23 131)(20 134 24 130)(25 111 29 107)(26 110 30 106)(27 109 31 105)(28 108 32 112)(33 119 37 115)(34 118 38 114)(35 117 39 113)(36 116 40 120)(41 88 45 84)(42 87 46 83)(43 86 47 82)(44 85 48 81)(49 137 53 141)(50 144 54 140)(51 143 55 139)(52 142 56 138)(57 101 61 97)(58 100 62 104)(59 99 63 103)(60 98 64 102)(65 147 69 151)(66 146 70 150)(67 145 71 149)(68 152 72 148)(73 155 77 159)(74 154 78 158)(75 153 79 157)(76 160 80 156)
G:=sub<Sym(160)| (1,16,49,57,17)(2,9,50,58,18)(3,10,51,59,19)(4,11,52,60,20)(5,12,53,61,21)(6,13,54,62,22)(7,14,55,63,23)(8,15,56,64,24)(25,85,73,33,65)(26,86,74,34,66)(27,87,75,35,67)(28,88,76,36,68)(29,81,77,37,69)(30,82,78,38,70)(31,83,79,39,71)(32,84,80,40,72)(41,156,120,148,112)(42,157,113,149,105)(43,158,114,150,106)(44,159,115,151,107)(45,160,116,152,108)(46,153,117,145,109)(47,154,118,146,110)(48,155,119,147,111)(89,126,141,97,133)(90,127,142,98,134)(91,128,143,99,135)(92,121,144,100,136)(93,122,137,101,129)(94,123,138,102,130)(95,124,139,103,131)(96,125,140,104,132), (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), (1,109,31,89)(2,112,32,92)(3,107,25,95)(4,110,26,90)(5,105,27,93)(6,108,28,96)(7,111,29,91)(8,106,30,94)(9,41,84,121)(10,44,85,124)(11,47,86,127)(12,42,87,122)(13,45,88,125)(14,48,81,128)(15,43,82,123)(16,46,83,126)(17,145,71,133)(18,148,72,136)(19,151,65,131)(20,146,66,134)(21,149,67,129)(22,152,68,132)(23,147,69,135)(24,150,70,130)(33,103,59,115)(34,98,60,118)(35,101,61,113)(36,104,62,116)(37,99,63,119)(38,102,64,114)(39,97,57,117)(40,100,58,120)(49,153,79,141)(50,156,80,144)(51,159,73,139)(52,154,74,142)(53,157,75,137)(54,160,76,140)(55,155,77,143)(56,158,78,138), (1,93,5,89)(2,92,6,96)(3,91,7,95)(4,90,8,94)(9,121,13,125)(10,128,14,124)(11,127,15,123)(12,126,16,122)(17,129,21,133)(18,136,22,132)(19,135,23,131)(20,134,24,130)(25,111,29,107)(26,110,30,106)(27,109,31,105)(28,108,32,112)(33,119,37,115)(34,118,38,114)(35,117,39,113)(36,116,40,120)(41,88,45,84)(42,87,46,83)(43,86,47,82)(44,85,48,81)(49,137,53,141)(50,144,54,140)(51,143,55,139)(52,142,56,138)(57,101,61,97)(58,100,62,104)(59,99,63,103)(60,98,64,102)(65,147,69,151)(66,146,70,150)(67,145,71,149)(68,152,72,148)(73,155,77,159)(74,154,78,158)(75,153,79,157)(76,160,80,156)>;
G:=Group( (1,16,49,57,17)(2,9,50,58,18)(3,10,51,59,19)(4,11,52,60,20)(5,12,53,61,21)(6,13,54,62,22)(7,14,55,63,23)(8,15,56,64,24)(25,85,73,33,65)(26,86,74,34,66)(27,87,75,35,67)(28,88,76,36,68)(29,81,77,37,69)(30,82,78,38,70)(31,83,79,39,71)(32,84,80,40,72)(41,156,120,148,112)(42,157,113,149,105)(43,158,114,150,106)(44,159,115,151,107)(45,160,116,152,108)(46,153,117,145,109)(47,154,118,146,110)(48,155,119,147,111)(89,126,141,97,133)(90,127,142,98,134)(91,128,143,99,135)(92,121,144,100,136)(93,122,137,101,129)(94,123,138,102,130)(95,124,139,103,131)(96,125,140,104,132), (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), (1,109,31,89)(2,112,32,92)(3,107,25,95)(4,110,26,90)(5,105,27,93)(6,108,28,96)(7,111,29,91)(8,106,30,94)(9,41,84,121)(10,44,85,124)(11,47,86,127)(12,42,87,122)(13,45,88,125)(14,48,81,128)(15,43,82,123)(16,46,83,126)(17,145,71,133)(18,148,72,136)(19,151,65,131)(20,146,66,134)(21,149,67,129)(22,152,68,132)(23,147,69,135)(24,150,70,130)(33,103,59,115)(34,98,60,118)(35,101,61,113)(36,104,62,116)(37,99,63,119)(38,102,64,114)(39,97,57,117)(40,100,58,120)(49,153,79,141)(50,156,80,144)(51,159,73,139)(52,154,74,142)(53,157,75,137)(54,160,76,140)(55,155,77,143)(56,158,78,138), (1,93,5,89)(2,92,6,96)(3,91,7,95)(4,90,8,94)(9,121,13,125)(10,128,14,124)(11,127,15,123)(12,126,16,122)(17,129,21,133)(18,136,22,132)(19,135,23,131)(20,134,24,130)(25,111,29,107)(26,110,30,106)(27,109,31,105)(28,108,32,112)(33,119,37,115)(34,118,38,114)(35,117,39,113)(36,116,40,120)(41,88,45,84)(42,87,46,83)(43,86,47,82)(44,85,48,81)(49,137,53,141)(50,144,54,140)(51,143,55,139)(52,142,56,138)(57,101,61,97)(58,100,62,104)(59,99,63,103)(60,98,64,102)(65,147,69,151)(66,146,70,150)(67,145,71,149)(68,152,72,148)(73,155,77,159)(74,154,78,158)(75,153,79,157)(76,160,80,156) );
G=PermutationGroup([[(1,16,49,57,17),(2,9,50,58,18),(3,10,51,59,19),(4,11,52,60,20),(5,12,53,61,21),(6,13,54,62,22),(7,14,55,63,23),(8,15,56,64,24),(25,85,73,33,65),(26,86,74,34,66),(27,87,75,35,67),(28,88,76,36,68),(29,81,77,37,69),(30,82,78,38,70),(31,83,79,39,71),(32,84,80,40,72),(41,156,120,148,112),(42,157,113,149,105),(43,158,114,150,106),(44,159,115,151,107),(45,160,116,152,108),(46,153,117,145,109),(47,154,118,146,110),(48,155,119,147,111),(89,126,141,97,133),(90,127,142,98,134),(91,128,143,99,135),(92,121,144,100,136),(93,122,137,101,129),(94,123,138,102,130),(95,124,139,103,131),(96,125,140,104,132)], [(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)], [(1,109,31,89),(2,112,32,92),(3,107,25,95),(4,110,26,90),(5,105,27,93),(6,108,28,96),(7,111,29,91),(8,106,30,94),(9,41,84,121),(10,44,85,124),(11,47,86,127),(12,42,87,122),(13,45,88,125),(14,48,81,128),(15,43,82,123),(16,46,83,126),(17,145,71,133),(18,148,72,136),(19,151,65,131),(20,146,66,134),(21,149,67,129),(22,152,68,132),(23,147,69,135),(24,150,70,130),(33,103,59,115),(34,98,60,118),(35,101,61,113),(36,104,62,116),(37,99,63,119),(38,102,64,114),(39,97,57,117),(40,100,58,120),(49,153,79,141),(50,156,80,144),(51,159,73,139),(52,154,74,142),(53,157,75,137),(54,160,76,140),(55,155,77,143),(56,158,78,138)], [(1,93,5,89),(2,92,6,96),(3,91,7,95),(4,90,8,94),(9,121,13,125),(10,128,14,124),(11,127,15,123),(12,126,16,122),(17,129,21,133),(18,136,22,132),(19,135,23,131),(20,134,24,130),(25,111,29,107),(26,110,30,106),(27,109,31,105),(28,108,32,112),(33,119,37,115),(34,118,38,114),(35,117,39,113),(36,116,40,120),(41,88,45,84),(42,87,46,83),(43,86,47,82),(44,85,48,81),(49,137,53,141),(50,144,54,140),(51,143,55,139),(52,142,56,138),(57,101,61,97),(58,100,62,104),(59,99,63,103),(60,98,64,102),(65,147,69,151),(66,146,70,150),(67,145,71,149),(68,152,72,148),(73,155,77,159),(74,154,78,158),(75,153,79,157),(76,160,80,156)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 10A | ··· | 10L | 10M | 10N | 10O | 10P | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 20M | ··· | 20AB | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 4 | 8 | 8 | 8 | 8 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 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.D4 | C5×Q8⋊C4 | C5×C4.Q8 | C5×C22⋊Q8 | C10×M4(2) | C10×Q16 | C8.D4 | Q8⋊C4 | C4.Q8 | C22⋊Q8 | C2×M4(2) | C2×Q16 | 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.D4 ►in GL8(𝔽41)
| 37 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 37 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 9 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 37 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 39 |
| 0 | 0 | 0 | 0 | 9 | 4 | 0 | 33 |
| 0 | 0 | 0 | 0 | 17 | 5 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 9 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 39 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 33 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 33 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 4 | 31 | 13 |
| 0 | 0 | 0 | 0 | 37 | 8 | 8 | 10 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 32 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 39 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 33 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 33 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 19 | 10 | 28 |
| 0 | 0 | 0 | 0 | 37 | 0 | 33 | 31 |
G:=sub<GL(8,GF(41))| [37,0,0,0,0,0,0,0,0,37,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[9,0,0,0,0,0,0,0,2,32,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,9,17,0,0,0,0,37,1,4,5,0,0,0,0,1,0,0,0,0,0,0,0,0,39,33,40],[40,9,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,1,0,0,0,0,0,0,39,1,0,0,0,0,0,0,0,0,33,33,9,37,0,0,0,0,3,8,4,8,0,0,0,0,0,0,31,8,0,0,0,0,0,0,13,10],[1,32,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,39,1,0,0,0,0,0,0,0,0,33,33,9,37,0,0,0,0,3,8,19,0,0,0,0,0,0,0,10,33,0,0,0,0,0,0,28,31] >;
C5×C8.D4 in GAP, Magma, Sage, TeX
C_5\times C_8.D_4
% in TeX
G:=Group("C5xC8.D4"); // GroupNames label
G:=SmallGroup(320,971);
// by ID
G=gap.SmallGroup(320,971);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,1120,589,1408,1766,1731,7004,172]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^8=c^4=1,d^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^3,d*b*d^-1=b^-1,d*c*d^-1=b^4*c^-1>;
// generators/relations