metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C8×D5)⋊2C4, C8.27(C4×D5), C4.30(Q8×D5), C2.D8⋊14D5, C40.58(C2×C4), C40⋊5C4⋊21C2, (C4×D5).16Q8, C20.21(C2×Q8), C4⋊C4.170D10, (C2×C8).230D10, C22.90(D4×D5), D10.16(C4⋊C4), C10.28(C4○D8), C2.4(D8⋊3D5), (C2×C40).82C22, C20.Q8⋊19C2, (C22×D5).85D4, C2.4(Q8.D10), Dic5.39(C4⋊C4), (C2×C20).296C23, C20.108(C22×C4), (C2×Dic5).277D4, C5⋊3(C23.25D4), C4⋊Dic5.122C22, (D5×C2×C8).3C2, C4.80(C2×C4×D5), C2.15(D5×C4⋊C4), (C5×C2.D8)⋊4C2, C10.37(C2×C4⋊C4), C5⋊2C8.39(C2×C4), C4⋊C4⋊7D5.7C2, (C4×D5).76(C2×C4), (C2×C10).301(C2×D4), (C5×C4⋊C4).89C22, (C2×C4×D5).305C22, (C2×C4).399(C22×D5), (C2×C5⋊2C8).242C22, SmallGroup(320,507)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8.27(C4×D5)
G = < a,b,c,d | a8=b4=c5=d2=1, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd=a4b, dcd=c-1 >
Subgroups: 382 in 114 conjugacy classes, 55 normal (27 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×6], C22, C22 [×4], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×9], C23, D5 [×2], C10 [×3], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×2], C2×C8, C2×C8 [×5], C22×C4, Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C4.Q8 [×2], C2.D8, C2.D8, C42⋊C2 [×2], C22×C8, C5⋊2C8 [×2], C40 [×2], C4×D5 [×4], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5, C23.25D4, C8×D5 [×4], C2×C5⋊2C8, C4×Dic5 [×2], C4⋊Dic5 [×2], D10⋊C4 [×2], C5×C4⋊C4 [×2], C2×C40, C2×C4×D5, C20.Q8 [×2], C40⋊5C4, C5×C2.D8, C4⋊C4⋊7D5 [×2], D5×C2×C8, C8.27(C4×D5)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, D5, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, D10 [×3], C2×C4⋊C4, C4○D8 [×2], C4×D5 [×2], C22×D5, C23.25D4, C2×C4×D5, D4×D5, Q8×D5, D5×C4⋊C4, D8⋊3D5, Q8.D10, C8.27(C4×D5)
►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 55 25 155)(2 54 26 154)(3 53 27 153)(4 52 28 160)(5 51 29 159)(6 50 30 158)(7 49 31 157)(8 56 32 156)(9 143 63 21)(10 142 64 20)(11 141 57 19)(12 140 58 18)(13 139 59 17)(14 138 60 24)(15 137 61 23)(16 144 62 22)(33 89 66 126)(34 96 67 125)(35 95 68 124)(36 94 69 123)(37 93 70 122)(38 92 71 121)(39 91 72 128)(40 90 65 127)(41 112 76 134)(42 111 77 133)(43 110 78 132)(44 109 79 131)(45 108 80 130)(46 107 73 129)(47 106 74 136)(48 105 75 135)(81 152 113 99)(82 151 114 98)(83 150 115 97)(84 149 116 104)(85 148 117 103)(86 147 118 102)(87 146 119 101)(88 145 120 100)
(1 103 133 21 39)(2 104 134 22 40)(3 97 135 23 33)(4 98 136 24 34)(5 99 129 17 35)(6 100 130 18 36)(7 101 131 19 37)(8 102 132 20 38)(9 91 55 85 42)(10 92 56 86 43)(11 93 49 87 44)(12 94 50 88 45)(13 95 51 81 46)(14 96 52 82 47)(15 89 53 83 48)(16 90 54 84 41)(25 148 111 143 72)(26 149 112 144 65)(27 150 105 137 66)(28 151 106 138 67)(29 152 107 139 68)(30 145 108 140 69)(31 146 109 141 70)(32 147 110 142 71)(57 122 157 119 79)(58 123 158 120 80)(59 124 159 113 73)(60 125 160 114 74)(61 126 153 115 75)(62 127 154 116 76)(63 128 155 117 77)(64 121 156 118 78)
(1 72)(2 65)(3 66)(4 67)(5 68)(6 69)(7 70)(8 71)(9 113)(10 114)(11 115)(12 116)(13 117)(14 118)(15 119)(16 120)(17 152)(18 145)(19 146)(20 147)(21 148)(22 149)(23 150)(24 151)(25 39)(26 40)(27 33)(28 34)(29 35)(30 36)(31 37)(32 38)(41 80)(42 73)(43 74)(44 75)(45 76)(46 77)(47 78)(48 79)(49 126)(50 127)(51 128)(52 121)(53 122)(54 123)(55 124)(56 125)(57 83)(58 84)(59 85)(60 86)(61 87)(62 88)(63 81)(64 82)(89 157)(90 158)(91 159)(92 160)(93 153)(94 154)(95 155)(96 156)(97 137)(98 138)(99 139)(100 140)(101 141)(102 142)(103 143)(104 144)(105 135)(106 136)(107 129)(108 130)(109 131)(110 132)(111 133)(112 134)
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), (1,55,25,155)(2,54,26,154)(3,53,27,153)(4,52,28,160)(5,51,29,159)(6,50,30,158)(7,49,31,157)(8,56,32,156)(9,143,63,21)(10,142,64,20)(11,141,57,19)(12,140,58,18)(13,139,59,17)(14,138,60,24)(15,137,61,23)(16,144,62,22)(33,89,66,126)(34,96,67,125)(35,95,68,124)(36,94,69,123)(37,93,70,122)(38,92,71,121)(39,91,72,128)(40,90,65,127)(41,112,76,134)(42,111,77,133)(43,110,78,132)(44,109,79,131)(45,108,80,130)(46,107,73,129)(47,106,74,136)(48,105,75,135)(81,152,113,99)(82,151,114,98)(83,150,115,97)(84,149,116,104)(85,148,117,103)(86,147,118,102)(87,146,119,101)(88,145,120,100), (1,103,133,21,39)(2,104,134,22,40)(3,97,135,23,33)(4,98,136,24,34)(5,99,129,17,35)(6,100,130,18,36)(7,101,131,19,37)(8,102,132,20,38)(9,91,55,85,42)(10,92,56,86,43)(11,93,49,87,44)(12,94,50,88,45)(13,95,51,81,46)(14,96,52,82,47)(15,89,53,83,48)(16,90,54,84,41)(25,148,111,143,72)(26,149,112,144,65)(27,150,105,137,66)(28,151,106,138,67)(29,152,107,139,68)(30,145,108,140,69)(31,146,109,141,70)(32,147,110,142,71)(57,122,157,119,79)(58,123,158,120,80)(59,124,159,113,73)(60,125,160,114,74)(61,126,153,115,75)(62,127,154,116,76)(63,128,155,117,77)(64,121,156,118,78), (1,72)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,71)(9,113)(10,114)(11,115)(12,116)(13,117)(14,118)(15,119)(16,120)(17,152)(18,145)(19,146)(20,147)(21,148)(22,149)(23,150)(24,151)(25,39)(26,40)(27,33)(28,34)(29,35)(30,36)(31,37)(32,38)(41,80)(42,73)(43,74)(44,75)(45,76)(46,77)(47,78)(48,79)(49,126)(50,127)(51,128)(52,121)(53,122)(54,123)(55,124)(56,125)(57,83)(58,84)(59,85)(60,86)(61,87)(62,88)(63,81)(64,82)(89,157)(90,158)(91,159)(92,160)(93,153)(94,154)(95,155)(96,156)(97,137)(98,138)(99,139)(100,140)(101,141)(102,142)(103,143)(104,144)(105,135)(106,136)(107,129)(108,130)(109,131)(110,132)(111,133)(112,134)>;
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), (1,55,25,155)(2,54,26,154)(3,53,27,153)(4,52,28,160)(5,51,29,159)(6,50,30,158)(7,49,31,157)(8,56,32,156)(9,143,63,21)(10,142,64,20)(11,141,57,19)(12,140,58,18)(13,139,59,17)(14,138,60,24)(15,137,61,23)(16,144,62,22)(33,89,66,126)(34,96,67,125)(35,95,68,124)(36,94,69,123)(37,93,70,122)(38,92,71,121)(39,91,72,128)(40,90,65,127)(41,112,76,134)(42,111,77,133)(43,110,78,132)(44,109,79,131)(45,108,80,130)(46,107,73,129)(47,106,74,136)(48,105,75,135)(81,152,113,99)(82,151,114,98)(83,150,115,97)(84,149,116,104)(85,148,117,103)(86,147,118,102)(87,146,119,101)(88,145,120,100), (1,103,133,21,39)(2,104,134,22,40)(3,97,135,23,33)(4,98,136,24,34)(5,99,129,17,35)(6,100,130,18,36)(7,101,131,19,37)(8,102,132,20,38)(9,91,55,85,42)(10,92,56,86,43)(11,93,49,87,44)(12,94,50,88,45)(13,95,51,81,46)(14,96,52,82,47)(15,89,53,83,48)(16,90,54,84,41)(25,148,111,143,72)(26,149,112,144,65)(27,150,105,137,66)(28,151,106,138,67)(29,152,107,139,68)(30,145,108,140,69)(31,146,109,141,70)(32,147,110,142,71)(57,122,157,119,79)(58,123,158,120,80)(59,124,159,113,73)(60,125,160,114,74)(61,126,153,115,75)(62,127,154,116,76)(63,128,155,117,77)(64,121,156,118,78), (1,72)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,71)(9,113)(10,114)(11,115)(12,116)(13,117)(14,118)(15,119)(16,120)(17,152)(18,145)(19,146)(20,147)(21,148)(22,149)(23,150)(24,151)(25,39)(26,40)(27,33)(28,34)(29,35)(30,36)(31,37)(32,38)(41,80)(42,73)(43,74)(44,75)(45,76)(46,77)(47,78)(48,79)(49,126)(50,127)(51,128)(52,121)(53,122)(54,123)(55,124)(56,125)(57,83)(58,84)(59,85)(60,86)(61,87)(62,88)(63,81)(64,82)(89,157)(90,158)(91,159)(92,160)(93,153)(94,154)(95,155)(96,156)(97,137)(98,138)(99,139)(100,140)(101,141)(102,142)(103,143)(104,144)(105,135)(106,136)(107,129)(108,130)(109,131)(110,132)(111,133)(112,134) );
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)], [(1,55,25,155),(2,54,26,154),(3,53,27,153),(4,52,28,160),(5,51,29,159),(6,50,30,158),(7,49,31,157),(8,56,32,156),(9,143,63,21),(10,142,64,20),(11,141,57,19),(12,140,58,18),(13,139,59,17),(14,138,60,24),(15,137,61,23),(16,144,62,22),(33,89,66,126),(34,96,67,125),(35,95,68,124),(36,94,69,123),(37,93,70,122),(38,92,71,121),(39,91,72,128),(40,90,65,127),(41,112,76,134),(42,111,77,133),(43,110,78,132),(44,109,79,131),(45,108,80,130),(46,107,73,129),(47,106,74,136),(48,105,75,135),(81,152,113,99),(82,151,114,98),(83,150,115,97),(84,149,116,104),(85,148,117,103),(86,147,118,102),(87,146,119,101),(88,145,120,100)], [(1,103,133,21,39),(2,104,134,22,40),(3,97,135,23,33),(4,98,136,24,34),(5,99,129,17,35),(6,100,130,18,36),(7,101,131,19,37),(8,102,132,20,38),(9,91,55,85,42),(10,92,56,86,43),(11,93,49,87,44),(12,94,50,88,45),(13,95,51,81,46),(14,96,52,82,47),(15,89,53,83,48),(16,90,54,84,41),(25,148,111,143,72),(26,149,112,144,65),(27,150,105,137,66),(28,151,106,138,67),(29,152,107,139,68),(30,145,108,140,69),(31,146,109,141,70),(32,147,110,142,71),(57,122,157,119,79),(58,123,158,120,80),(59,124,159,113,73),(60,125,160,114,74),(61,126,153,115,75),(62,127,154,116,76),(63,128,155,117,77),(64,121,156,118,78)], [(1,72),(2,65),(3,66),(4,67),(5,68),(6,69),(7,70),(8,71),(9,113),(10,114),(11,115),(12,116),(13,117),(14,118),(15,119),(16,120),(17,152),(18,145),(19,146),(20,147),(21,148),(22,149),(23,150),(24,151),(25,39),(26,40),(27,33),(28,34),(29,35),(30,36),(31,37),(32,38),(41,80),(42,73),(43,74),(44,75),(45,76),(46,77),(47,78),(48,79),(49,126),(50,127),(51,128),(52,121),(53,122),(54,123),(55,124),(56,125),(57,83),(58,84),(59,85),(60,86),(61,87),(62,88),(63,81),(64,82),(89,157),(90,158),(91,159),(92,160),(93,153),(94,154),(95,155),(96,156),(97,137),(98,138),(99,139),(100,140),(101,141),(102,142),(103,143),(104,144),(105,135),(106,136),(107,129),(108,130),(109,131),(110,132),(111,133),(112,134)])
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 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | + | + | - | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | Q8 | D4 | D4 | D5 | D10 | D10 | C4○D8 | C4×D5 | Q8×D5 | D4×D5 | D8⋊3D5 | Q8.D10 |
| kernel | C8.27(C4×D5) | C20.Q8 | C40⋊5C4 | C5×C2.D8 | C4⋊C4⋊7D5 | D5×C2×C8 | C8×D5 | C4×D5 | C2×Dic5 | C22×D5 | C2.D8 | C4⋊C4 | C2×C8 | C10 | C8 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 8 | 2 | 1 | 1 | 2 | 4 | 2 | 8 | 8 | 2 | 2 | 4 | 4 |
Matrix representation of C8.27(C4×D5) ►in GL4(𝔽41) generated by
| 3 | 20 | 0 | 0 |
| 0 | 14 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 30 | 30 | 0 | 0 |
| 37 | 11 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 0 | 1 | 34 |
| 1 | 15 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 34 | 1 |
| 0 | 0 | 34 | 7 |
G:=sub<GL(4,GF(41))| [3,0,0,0,20,14,0,0,0,0,40,0,0,0,0,40],[30,37,0,0,30,11,0,0,0,0,9,0,0,0,0,9],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,40,34],[1,0,0,0,15,40,0,0,0,0,34,34,0,0,1,7] >;
C8.27(C4×D5) in GAP, Magma, Sage, TeX
C_8._{27}(C_4\times D_5) % in TeX
G:=Group("C8.27(C4xD5)"); // GroupNames label
G:=SmallGroup(320,507);
// by ID
G=gap.SmallGroup(320,507);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,758,219,58,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^4=c^5=d^2=1,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations