metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C8×D5)⋊4C4, C8.30(C4×D5), C4.24(Q8×D5), C4.Q8⋊14D5, C40.75(C2×C4), C40⋊6C4⋊25C2, (C4×D5).14Q8, C20.13(C2×Q8), C4⋊C4.161D10, (C2×C8).259D10, C22.84(D4×D5), D10.14(C4⋊C4), C10.54(C4○D8), C10.D8⋊14C2, (C22×D5).83D4, Dic5.38(C4⋊C4), (C2×C20).276C23, C20.102(C22×C4), (C2×C40).160C22, (C2×Dic5).274D4, C5⋊2(C23.25D4), C2.6(SD16⋊3D5), C4⋊Dic5.108C22, (D5×C2×C8).8C2, C4.77(C2×C4×D5), C2.12(D5×C4⋊C4), (C5×C4.Q8)⋊8C2, C10.34(C2×C4⋊C4), C5⋊2C8.38(C2×C4), C4⋊C4⋊7D5.4C2, (C4×D5).74(C2×C4), (C2×C10).281(C2×D4), (C5×C4⋊C4).69C22, (C2×C4×D5).302C22, (C2×C4).379(C22×D5), (C2×C5⋊2C8).235C22, SmallGroup(320,487)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C8×D5)⋊C4
G = < a,b,c,d | a8=b5=c2=d4=1, ab=ba, ac=ca, dad-1=a3, cbc=b-1, bd=db, dcd-1=a4c >
Subgroups: 382 in 114 conjugacy classes, 55 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C4.Q8, C4.Q8, C2.D8, C42⋊C2, C22×C8, C5⋊2C8, C40, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C23.25D4, C8×D5, C2×C5⋊2C8, C4×Dic5, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C2×C40, C2×C4×D5, C10.D8, C40⋊6C4, C5×C4.Q8, C4⋊C4⋊7D5, D5×C2×C8, (C8×D5)⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D5, C4⋊C4, C22×C4, C2×D4, C2×Q8, D10, C2×C4⋊C4, C4○D8, C4×D5, C22×D5, C23.25D4, C2×C4×D5, D4×D5, Q8×D5, D5×C4⋊C4, SD16⋊3D5, (C8×D5)⋊C4
►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 42 133 35 97)(2 43 134 36 98)(3 44 135 37 99)(4 45 136 38 100)(5 46 129 39 101)(6 47 130 40 102)(7 48 131 33 103)(8 41 132 34 104)(9 124 81 93 49)(10 125 82 94 50)(11 126 83 95 51)(12 127 84 96 52)(13 128 85 89 53)(14 121 86 90 54)(15 122 87 91 55)(16 123 88 92 56)(17 58 154 116 76)(18 59 155 117 77)(19 60 156 118 78)(20 61 157 119 79)(21 62 158 120 80)(22 63 159 113 73)(23 64 160 114 74)(24 57 153 115 75)(25 137 146 107 70)(26 138 147 108 71)(27 139 148 109 72)(28 140 149 110 65)(29 141 150 111 66)(30 142 151 112 67)(31 143 152 105 68)(32 144 145 106 69)
(1 148)(2 149)(3 150)(4 151)(5 152)(6 145)(7 146)(8 147)(9 20)(10 21)(11 22)(12 23)(13 24)(14 17)(15 18)(16 19)(25 131)(26 132)(27 133)(28 134)(29 135)(30 136)(31 129)(32 130)(33 70)(34 71)(35 72)(36 65)(37 66)(38 67)(39 68)(40 69)(41 138)(42 139)(43 140)(44 141)(45 142)(46 143)(47 144)(48 137)(49 61)(50 62)(51 63)(52 64)(53 57)(54 58)(55 59)(56 60)(73 126)(74 127)(75 128)(76 121)(77 122)(78 123)(79 124)(80 125)(81 119)(82 120)(83 113)(84 114)(85 115)(86 116)(87 117)(88 118)(89 153)(90 154)(91 155)(92 156)(93 157)(94 158)(95 159)(96 160)(97 109)(98 110)(99 111)(100 112)(101 105)(102 106)(103 107)(104 108)
(1 55 105 155)(2 50 106 158)(3 53 107 153)(4 56 108 156)(5 51 109 159)(6 54 110 154)(7 49 111 157)(8 52 112 160)(9 66 119 48)(10 69 120 43)(11 72 113 46)(12 67 114 41)(13 70 115 44)(14 65 116 47)(15 68 117 42)(16 71 118 45)(17 40 86 140)(18 35 87 143)(19 38 88 138)(20 33 81 141)(21 36 82 144)(22 39 83 139)(23 34 84 142)(24 37 85 137)(25 75 135 128)(26 78 136 123)(27 73 129 126)(28 76 130 121)(29 79 131 124)(30 74 132 127)(31 77 133 122)(32 80 134 125)(57 99 89 146)(58 102 90 149)(59 97 91 152)(60 100 92 147)(61 103 93 150)(62 98 94 145)(63 101 95 148)(64 104 96 151)
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), (1,42,133,35,97)(2,43,134,36,98)(3,44,135,37,99)(4,45,136,38,100)(5,46,129,39,101)(6,47,130,40,102)(7,48,131,33,103)(8,41,132,34,104)(9,124,81,93,49)(10,125,82,94,50)(11,126,83,95,51)(12,127,84,96,52)(13,128,85,89,53)(14,121,86,90,54)(15,122,87,91,55)(16,123,88,92,56)(17,58,154,116,76)(18,59,155,117,77)(19,60,156,118,78)(20,61,157,119,79)(21,62,158,120,80)(22,63,159,113,73)(23,64,160,114,74)(24,57,153,115,75)(25,137,146,107,70)(26,138,147,108,71)(27,139,148,109,72)(28,140,149,110,65)(29,141,150,111,66)(30,142,151,112,67)(31,143,152,105,68)(32,144,145,106,69), (1,148)(2,149)(3,150)(4,151)(5,152)(6,145)(7,146)(8,147)(9,20)(10,21)(11,22)(12,23)(13,24)(14,17)(15,18)(16,19)(25,131)(26,132)(27,133)(28,134)(29,135)(30,136)(31,129)(32,130)(33,70)(34,71)(35,72)(36,65)(37,66)(38,67)(39,68)(40,69)(41,138)(42,139)(43,140)(44,141)(45,142)(46,143)(47,144)(48,137)(49,61)(50,62)(51,63)(52,64)(53,57)(54,58)(55,59)(56,60)(73,126)(74,127)(75,128)(76,121)(77,122)(78,123)(79,124)(80,125)(81,119)(82,120)(83,113)(84,114)(85,115)(86,116)(87,117)(88,118)(89,153)(90,154)(91,155)(92,156)(93,157)(94,158)(95,159)(96,160)(97,109)(98,110)(99,111)(100,112)(101,105)(102,106)(103,107)(104,108), (1,55,105,155)(2,50,106,158)(3,53,107,153)(4,56,108,156)(5,51,109,159)(6,54,110,154)(7,49,111,157)(8,52,112,160)(9,66,119,48)(10,69,120,43)(11,72,113,46)(12,67,114,41)(13,70,115,44)(14,65,116,47)(15,68,117,42)(16,71,118,45)(17,40,86,140)(18,35,87,143)(19,38,88,138)(20,33,81,141)(21,36,82,144)(22,39,83,139)(23,34,84,142)(24,37,85,137)(25,75,135,128)(26,78,136,123)(27,73,129,126)(28,76,130,121)(29,79,131,124)(30,74,132,127)(31,77,133,122)(32,80,134,125)(57,99,89,146)(58,102,90,149)(59,97,91,152)(60,100,92,147)(61,103,93,150)(62,98,94,145)(63,101,95,148)(64,104,96,151)>;
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), (1,42,133,35,97)(2,43,134,36,98)(3,44,135,37,99)(4,45,136,38,100)(5,46,129,39,101)(6,47,130,40,102)(7,48,131,33,103)(8,41,132,34,104)(9,124,81,93,49)(10,125,82,94,50)(11,126,83,95,51)(12,127,84,96,52)(13,128,85,89,53)(14,121,86,90,54)(15,122,87,91,55)(16,123,88,92,56)(17,58,154,116,76)(18,59,155,117,77)(19,60,156,118,78)(20,61,157,119,79)(21,62,158,120,80)(22,63,159,113,73)(23,64,160,114,74)(24,57,153,115,75)(25,137,146,107,70)(26,138,147,108,71)(27,139,148,109,72)(28,140,149,110,65)(29,141,150,111,66)(30,142,151,112,67)(31,143,152,105,68)(32,144,145,106,69), (1,148)(2,149)(3,150)(4,151)(5,152)(6,145)(7,146)(8,147)(9,20)(10,21)(11,22)(12,23)(13,24)(14,17)(15,18)(16,19)(25,131)(26,132)(27,133)(28,134)(29,135)(30,136)(31,129)(32,130)(33,70)(34,71)(35,72)(36,65)(37,66)(38,67)(39,68)(40,69)(41,138)(42,139)(43,140)(44,141)(45,142)(46,143)(47,144)(48,137)(49,61)(50,62)(51,63)(52,64)(53,57)(54,58)(55,59)(56,60)(73,126)(74,127)(75,128)(76,121)(77,122)(78,123)(79,124)(80,125)(81,119)(82,120)(83,113)(84,114)(85,115)(86,116)(87,117)(88,118)(89,153)(90,154)(91,155)(92,156)(93,157)(94,158)(95,159)(96,160)(97,109)(98,110)(99,111)(100,112)(101,105)(102,106)(103,107)(104,108), (1,55,105,155)(2,50,106,158)(3,53,107,153)(4,56,108,156)(5,51,109,159)(6,54,110,154)(7,49,111,157)(8,52,112,160)(9,66,119,48)(10,69,120,43)(11,72,113,46)(12,67,114,41)(13,70,115,44)(14,65,116,47)(15,68,117,42)(16,71,118,45)(17,40,86,140)(18,35,87,143)(19,38,88,138)(20,33,81,141)(21,36,82,144)(22,39,83,139)(23,34,84,142)(24,37,85,137)(25,75,135,128)(26,78,136,123)(27,73,129,126)(28,76,130,121)(29,79,131,124)(30,74,132,127)(31,77,133,122)(32,80,134,125)(57,99,89,146)(58,102,90,149)(59,97,91,152)(60,100,92,147)(61,103,93,150)(62,98,94,145)(63,101,95,148)(64,104,96,151) );
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)], [(1,42,133,35,97),(2,43,134,36,98),(3,44,135,37,99),(4,45,136,38,100),(5,46,129,39,101),(6,47,130,40,102),(7,48,131,33,103),(8,41,132,34,104),(9,124,81,93,49),(10,125,82,94,50),(11,126,83,95,51),(12,127,84,96,52),(13,128,85,89,53),(14,121,86,90,54),(15,122,87,91,55),(16,123,88,92,56),(17,58,154,116,76),(18,59,155,117,77),(19,60,156,118,78),(20,61,157,119,79),(21,62,158,120,80),(22,63,159,113,73),(23,64,160,114,74),(24,57,153,115,75),(25,137,146,107,70),(26,138,147,108,71),(27,139,148,109,72),(28,140,149,110,65),(29,141,150,111,66),(30,142,151,112,67),(31,143,152,105,68),(32,144,145,106,69)], [(1,148),(2,149),(3,150),(4,151),(5,152),(6,145),(7,146),(8,147),(9,20),(10,21),(11,22),(12,23),(13,24),(14,17),(15,18),(16,19),(25,131),(26,132),(27,133),(28,134),(29,135),(30,136),(31,129),(32,130),(33,70),(34,71),(35,72),(36,65),(37,66),(38,67),(39,68),(40,69),(41,138),(42,139),(43,140),(44,141),(45,142),(46,143),(47,144),(48,137),(49,61),(50,62),(51,63),(52,64),(53,57),(54,58),(55,59),(56,60),(73,126),(74,127),(75,128),(76,121),(77,122),(78,123),(79,124),(80,125),(81,119),(82,120),(83,113),(84,114),(85,115),(86,116),(87,117),(88,118),(89,153),(90,154),(91,155),(92,156),(93,157),(94,158),(95,159),(96,160),(97,109),(98,110),(99,111),(100,112),(101,105),(102,106),(103,107),(104,108)], [(1,55,105,155),(2,50,106,158),(3,53,107,153),(4,56,108,156),(5,51,109,159),(6,54,110,154),(7,49,111,157),(8,52,112,160),(9,66,119,48),(10,69,120,43),(11,72,113,46),(12,67,114,41),(13,70,115,44),(14,65,116,47),(15,68,117,42),(16,71,118,45),(17,40,86,140),(18,35,87,143),(19,38,88,138),(20,33,81,141),(21,36,82,144),(22,39,83,139),(23,34,84,142),(24,37,85,137),(25,75,135,128),(26,78,136,123),(27,73,129,126),(28,76,130,121),(29,79,131,124),(30,74,132,127),(31,77,133,122),(32,80,134,125),(57,99,89,146),(58,102,90,149),(59,97,91,152),(60,100,92,147),(61,103,93,150),(62,98,94,145),(63,101,95,148),(64,104,96,151)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10F | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | + | + | - | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | Q8 | D4 | D4 | D5 | D10 | D10 | C4○D8 | C4×D5 | Q8×D5 | D4×D5 | SD16⋊3D5 |
| kernel | (C8×D5)⋊C4 | C10.D8 | C40⋊6C4 | C5×C4.Q8 | C4⋊C4⋊7D5 | D5×C2×C8 | C8×D5 | C4×D5 | C2×Dic5 | C22×D5 | C4.Q8 | C4⋊C4 | C2×C8 | C10 | C8 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 8 | 2 | 1 | 1 | 2 | 4 | 2 | 8 | 8 | 2 | 2 | 8 |
Matrix representation of (C8×D5)⋊C4 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 29 | 3 |
| 35 | 40 | 0 | 0 |
| 36 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 7 | 0 | 0 |
| 6 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 1 | 1 |
| 32 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 40 | 39 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,27,29,0,0,0,3],[35,36,0,0,40,40,0,0,0,0,1,0,0,0,0,1],[0,6,0,0,7,0,0,0,0,0,40,1,0,0,0,1],[32,0,0,0,0,32,0,0,0,0,40,0,0,0,39,1] >;
(C8×D5)⋊C4 in GAP, Magma, Sage, TeX
(C_8\times D_5)\rtimes C_4
% in TeX
G:=Group("(C8xD5):C4"); // GroupNames label
G:=SmallGroup(320,487);
// by ID
G=gap.SmallGroup(320,487);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,758,555,58,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^5=c^2=d^4=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^3,c*b*c=b^-1,b*d=d*b,d*c*d^-1=a^4*c>;
// generators/relations