direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×C8⋊3D4, C40⋊21D4, C8⋊3(C5×D4), (C2×D8)⋊9C10, C8⋊C4⋊4C10, C4.5(D4×C10), (C10×D8)⋊23C2, C4⋊1D4⋊4C10, (C2×SD16)⋊3C10, C20.312(C2×D4), C4.4D4⋊5C10, (C2×C20).343D4, (C10×SD16)⋊14C2, C42.29(C2×C10), C10.46(C4⋊1D4), (C2×C40).275C22, (C2×C20).952C23, (C4×C20).271C22, C22.117(D4×C10), C10.147(C8⋊C22), (D4×C10).205C22, (Q8×C10).179C22, (C5×C8⋊C4)⋊13C2, (C2×C4).44(C5×D4), C2.9(C5×C4⋊1D4), (C5×C4⋊1D4)⋊14C2, (C2×C8).27(C2×C10), C2.22(C5×C8⋊C22), (C2×D4).28(C2×C10), (C5×C4.4D4)⋊25C2, (C2×C10).673(C2×D4), (C2×Q8).23(C2×C10), (C2×C4).127(C22×C10), SmallGroup(320,997)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C8⋊3D4
G = < a,b,c,d | a5=b8=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd=b-1, dcd=c-1 >
Subgroups: 322 in 144 conjugacy classes, 58 normal (22 characteristic)
C1, C2, C2 [×2], C2 [×3], C4 [×2], C4 [×3], C22, C22 [×9], C5, C8 [×4], C2×C4, C2×C4 [×2], C2×C4, D4 [×10], Q8 [×2], C23 [×3], C10, C10 [×2], C10 [×3], C42, C22⋊C4 [×2], C2×C8 [×2], D8 [×4], SD16 [×4], C2×D4, C2×D4 [×2], C2×D4 [×2], C2×Q8, C20 [×2], C20 [×3], C2×C10, C2×C10 [×9], C8⋊C4, C4.4D4, C4⋊1D4, C2×D8 [×2], C2×SD16 [×2], C40 [×4], C2×C20, C2×C20 [×2], C2×C20, C5×D4 [×10], C5×Q8 [×2], C22×C10 [×3], C8⋊3D4, C4×C20, C5×C22⋊C4 [×2], C2×C40 [×2], C5×D8 [×4], C5×SD16 [×4], D4×C10, D4×C10 [×2], D4×C10 [×2], Q8×C10, C5×C8⋊C4, C5×C4.4D4, C5×C4⋊1D4, C10×D8 [×2], C10×SD16 [×2], C5×C8⋊3D4
Quotients: C1, C2 [×7], C22 [×7], C5, D4 [×6], C23, C10 [×7], C2×D4 [×3], C2×C10 [×7], C4⋊1D4, C8⋊C22 [×2], C5×D4 [×6], C22×C10, C8⋊3D4, D4×C10 [×3], C5×C4⋊1D4, C5×C8⋊C22 [×2], C5×C8⋊3D4
►On 160 points
Generators in S160
(1 39 111 31 103)(2 40 112 32 104)(3 33 105 25 97)(4 34 106 26 98)(5 35 107 27 99)(6 36 108 28 100)(7 37 109 29 101)(8 38 110 30 102)(9 22 46 118 93)(10 23 47 119 94)(11 24 48 120 95)(12 17 41 113 96)(13 18 42 114 89)(14 19 43 115 90)(15 20 44 116 91)(16 21 45 117 92)(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 87 153 73 145)(66 88 154 74 146)(67 81 155 75 147)(68 82 156 76 148)(69 83 157 77 149)(70 84 158 78 150)(71 85 159 79 151)(72 86 160 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 71 127 119)(2 68 128 116)(3 65 121 113)(4 70 122 118)(5 67 123 115)(6 72 124 120)(7 69 125 117)(8 66 126 114)(9 106 158 58)(10 111 159 63)(11 108 160 60)(12 105 153 57)(13 110 154 62)(14 107 155 59)(15 112 156 64)(16 109 157 61)(17 25 73 129)(18 30 74 134)(19 27 75 131)(20 32 76 136)(21 29 77 133)(22 26 78 130)(23 31 79 135)(24 28 80 132)(33 87 137 96)(34 84 138 93)(35 81 139 90)(36 86 140 95)(37 83 141 92)(38 88 142 89)(39 85 143 94)(40 82 144 91)(41 97 145 49)(42 102 146 54)(43 99 147 51)(44 104 148 56)(45 101 149 53)(46 98 150 50)(47 103 151 55)(48 100 152 52)
(1 122)(2 121)(3 128)(4 127)(5 126)(6 125)(7 124)(8 123)(9 10)(11 16)(12 15)(13 14)(17 20)(18 19)(21 24)(22 23)(25 136)(26 135)(27 134)(28 133)(29 132)(30 131)(31 130)(32 129)(33 144)(34 143)(35 142)(36 141)(37 140)(38 139)(39 138)(40 137)(41 44)(42 43)(45 48)(46 47)(49 104)(50 103)(51 102)(52 101)(53 100)(54 99)(55 98)(56 97)(57 112)(58 111)(59 110)(60 109)(61 108)(62 107)(63 106)(64 105)(65 68)(66 67)(69 72)(70 71)(73 76)(74 75)(77 80)(78 79)(81 88)(82 87)(83 86)(84 85)(89 90)(91 96)(92 95)(93 94)(113 116)(114 115)(117 120)(118 119)(145 148)(146 147)(149 152)(150 151)(153 156)(154 155)(157 160)(158 159)
G:=sub<Sym(160)| (1,39,111,31,103)(2,40,112,32,104)(3,33,105,25,97)(4,34,106,26,98)(5,35,107,27,99)(6,36,108,28,100)(7,37,109,29,101)(8,38,110,30,102)(9,22,46,118,93)(10,23,47,119,94)(11,24,48,120,95)(12,17,41,113,96)(13,18,42,114,89)(14,19,43,115,90)(15,20,44,116,91)(16,21,45,117,92)(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)(65,87,153,73,145)(66,88,154,74,146)(67,81,155,75,147)(68,82,156,76,148)(69,83,157,77,149)(70,84,158,78,150)(71,85,159,79,151)(72,86,160,80,152), (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), (1,71,127,119)(2,68,128,116)(3,65,121,113)(4,70,122,118)(5,67,123,115)(6,72,124,120)(7,69,125,117)(8,66,126,114)(9,106,158,58)(10,111,159,63)(11,108,160,60)(12,105,153,57)(13,110,154,62)(14,107,155,59)(15,112,156,64)(16,109,157,61)(17,25,73,129)(18,30,74,134)(19,27,75,131)(20,32,76,136)(21,29,77,133)(22,26,78,130)(23,31,79,135)(24,28,80,132)(33,87,137,96)(34,84,138,93)(35,81,139,90)(36,86,140,95)(37,83,141,92)(38,88,142,89)(39,85,143,94)(40,82,144,91)(41,97,145,49)(42,102,146,54)(43,99,147,51)(44,104,148,56)(45,101,149,53)(46,98,150,50)(47,103,151,55)(48,100,152,52), (1,122)(2,121)(3,128)(4,127)(5,126)(6,125)(7,124)(8,123)(9,10)(11,16)(12,15)(13,14)(17,20)(18,19)(21,24)(22,23)(25,136)(26,135)(27,134)(28,133)(29,132)(30,131)(31,130)(32,129)(33,144)(34,143)(35,142)(36,141)(37,140)(38,139)(39,138)(40,137)(41,44)(42,43)(45,48)(46,47)(49,104)(50,103)(51,102)(52,101)(53,100)(54,99)(55,98)(56,97)(57,112)(58,111)(59,110)(60,109)(61,108)(62,107)(63,106)(64,105)(65,68)(66,67)(69,72)(70,71)(73,76)(74,75)(77,80)(78,79)(81,88)(82,87)(83,86)(84,85)(89,90)(91,96)(92,95)(93,94)(113,116)(114,115)(117,120)(118,119)(145,148)(146,147)(149,152)(150,151)(153,156)(154,155)(157,160)(158,159)>;
G:=Group( (1,39,111,31,103)(2,40,112,32,104)(3,33,105,25,97)(4,34,106,26,98)(5,35,107,27,99)(6,36,108,28,100)(7,37,109,29,101)(8,38,110,30,102)(9,22,46,118,93)(10,23,47,119,94)(11,24,48,120,95)(12,17,41,113,96)(13,18,42,114,89)(14,19,43,115,90)(15,20,44,116,91)(16,21,45,117,92)(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)(65,87,153,73,145)(66,88,154,74,146)(67,81,155,75,147)(68,82,156,76,148)(69,83,157,77,149)(70,84,158,78,150)(71,85,159,79,151)(72,86,160,80,152), (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), (1,71,127,119)(2,68,128,116)(3,65,121,113)(4,70,122,118)(5,67,123,115)(6,72,124,120)(7,69,125,117)(8,66,126,114)(9,106,158,58)(10,111,159,63)(11,108,160,60)(12,105,153,57)(13,110,154,62)(14,107,155,59)(15,112,156,64)(16,109,157,61)(17,25,73,129)(18,30,74,134)(19,27,75,131)(20,32,76,136)(21,29,77,133)(22,26,78,130)(23,31,79,135)(24,28,80,132)(33,87,137,96)(34,84,138,93)(35,81,139,90)(36,86,140,95)(37,83,141,92)(38,88,142,89)(39,85,143,94)(40,82,144,91)(41,97,145,49)(42,102,146,54)(43,99,147,51)(44,104,148,56)(45,101,149,53)(46,98,150,50)(47,103,151,55)(48,100,152,52), (1,122)(2,121)(3,128)(4,127)(5,126)(6,125)(7,124)(8,123)(9,10)(11,16)(12,15)(13,14)(17,20)(18,19)(21,24)(22,23)(25,136)(26,135)(27,134)(28,133)(29,132)(30,131)(31,130)(32,129)(33,144)(34,143)(35,142)(36,141)(37,140)(38,139)(39,138)(40,137)(41,44)(42,43)(45,48)(46,47)(49,104)(50,103)(51,102)(52,101)(53,100)(54,99)(55,98)(56,97)(57,112)(58,111)(59,110)(60,109)(61,108)(62,107)(63,106)(64,105)(65,68)(66,67)(69,72)(70,71)(73,76)(74,75)(77,80)(78,79)(81,88)(82,87)(83,86)(84,85)(89,90)(91,96)(92,95)(93,94)(113,116)(114,115)(117,120)(118,119)(145,148)(146,147)(149,152)(150,151)(153,156)(154,155)(157,160)(158,159) );
G=PermutationGroup([(1,39,111,31,103),(2,40,112,32,104),(3,33,105,25,97),(4,34,106,26,98),(5,35,107,27,99),(6,36,108,28,100),(7,37,109,29,101),(8,38,110,30,102),(9,22,46,118,93),(10,23,47,119,94),(11,24,48,120,95),(12,17,41,113,96),(13,18,42,114,89),(14,19,43,115,90),(15,20,44,116,91),(16,21,45,117,92),(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),(65,87,153,73,145),(66,88,154,74,146),(67,81,155,75,147),(68,82,156,76,148),(69,83,157,77,149),(70,84,158,78,150),(71,85,159,79,151),(72,86,160,80,152)], [(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)], [(1,71,127,119),(2,68,128,116),(3,65,121,113),(4,70,122,118),(5,67,123,115),(6,72,124,120),(7,69,125,117),(8,66,126,114),(9,106,158,58),(10,111,159,63),(11,108,160,60),(12,105,153,57),(13,110,154,62),(14,107,155,59),(15,112,156,64),(16,109,157,61),(17,25,73,129),(18,30,74,134),(19,27,75,131),(20,32,76,136),(21,29,77,133),(22,26,78,130),(23,31,79,135),(24,28,80,132),(33,87,137,96),(34,84,138,93),(35,81,139,90),(36,86,140,95),(37,83,141,92),(38,88,142,89),(39,85,143,94),(40,82,144,91),(41,97,145,49),(42,102,146,54),(43,99,147,51),(44,104,148,56),(45,101,149,53),(46,98,150,50),(47,103,151,55),(48,100,152,52)], [(1,122),(2,121),(3,128),(4,127),(5,126),(6,125),(7,124),(8,123),(9,10),(11,16),(12,15),(13,14),(17,20),(18,19),(21,24),(22,23),(25,136),(26,135),(27,134),(28,133),(29,132),(30,131),(31,130),(32,129),(33,144),(34,143),(35,142),(36,141),(37,140),(38,139),(39,138),(40,137),(41,44),(42,43),(45,48),(46,47),(49,104),(50,103),(51,102),(52,101),(53,100),(54,99),(55,98),(56,97),(57,112),(58,111),(59,110),(60,109),(61,108),(62,107),(63,106),(64,105),(65,68),(66,67),(69,72),(70,71),(73,76),(74,75),(77,80),(78,79),(81,88),(82,87),(83,86),(84,85),(89,90),(91,96),(92,95),(93,94),(113,116),(114,115),(117,120),(118,119),(145,148),(146,147),(149,152),(150,151),(153,156),(154,155),(157,160),(158,159)])
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 | ··· | 10X | 20A | ··· | 20H | 20I | ··· | 20P | 20Q | 20R | 20S | 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 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 8 | 2 | 2 | 4 | 4 | 8 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 8 | ··· | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | D4 | D4 | C5×D4 | C5×D4 | C8⋊C22 | C5×C8⋊C22 |
| kernel | C5×C8⋊3D4 | C5×C8⋊C4 | C5×C4.4D4 | C5×C4⋊1D4 | C10×D8 | C10×SD16 | C8⋊3D4 | C8⋊C4 | C4.4D4 | C4⋊1D4 | C2×D8 | C2×SD16 | C40 | C2×C20 | C8 | C2×C4 | C10 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 4 | 2 | 16 | 8 | 2 | 8 |
Matrix representation of C5×C8⋊3D4 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 0 | 0 | 0 |
| 0 | 0 | 0 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 0 | 0 | 10 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 19 | 38 | 0 | 19 |
| 0 | 0 | 22 | 19 | 19 | 22 |
| 0 | 0 | 3 | 38 | 25 | 16 |
| 0 | 0 | 19 | 19 | 3 | 19 |
| 22 | 40 | 0 | 0 | 0 | 0 |
| 34 | 19 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 40 | 1 | 39 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 3 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 3 | 0 | 22 |
| 0 | 0 | 22 | 19 | 19 | 22 |
| 0 | 0 | 3 | 38 | 25 | 16 |
| 0 | 0 | 3 | 0 | 3 | 16 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,19,22,3,19,0,0,38,19,38,19,0,0,0,19,25,3,0,0,19,22,16,19],[22,34,0,0,0,0,40,19,0,0,0,0,0,0,0,1,1,0,0,0,0,40,0,0,0,0,1,1,0,0,0,0,0,39,0,1],[1,3,0,0,0,0,0,40,0,0,0,0,0,0,22,22,3,3,0,0,3,19,38,0,0,0,0,19,25,3,0,0,22,22,16,16] >;
C5×C8⋊3D4 in GAP, Magma, Sage, TeX
C_5\times C_8\rtimes_3D_4
% in TeX
G:=Group("C5xC8:3D4"); // GroupNames label
G:=SmallGroup(320,997);
// by ID
G=gap.SmallGroup(320,997);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,589,288,1766,1731,436,7004,172]);
// 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^5,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations