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, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C4.Q8, C2.D8, 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, C20.Q8, C40⋊5C4, C5×C2.D8, C4⋊C4⋊7D5, D5×C2×C8, C8.27(C4×D5)
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, 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 55 108 155)(2 54 109 154)(3 53 110 153)(4 52 111 160)(5 51 112 159)(6 50 105 158)(7 49 106 157)(8 56 107 156)(9 66 116 24)(10 65 117 23)(11 72 118 22)(12 71 119 21)(13 70 120 20)(14 69 113 19)(15 68 114 18)(16 67 115 17)(25 78 132 128)(26 77 133 127)(27 76 134 126)(28 75 135 125)(29 74 136 124)(30 73 129 123)(31 80 130 122)(32 79 131 121)(33 82 143 61)(34 81 144 60)(35 88 137 59)(36 87 138 58)(37 86 139 57)(38 85 140 64)(39 84 141 63)(40 83 142 62)(41 101 95 145)(42 100 96 152)(43 99 89 151)(44 98 90 150)(45 97 91 149)(46 104 92 148)(47 103 93 147)(48 102 94 146)
(1 23 133 33 97)(2 24 134 34 98)(3 17 135 35 99)(4 18 136 36 100)(5 19 129 37 101)(6 20 130 38 102)(7 21 131 39 103)(8 22 132 40 104)(9 126 81 90 54)(10 127 82 91 55)(11 128 83 92 56)(12 121 84 93 49)(13 122 85 94 50)(14 123 86 95 51)(15 124 87 96 52)(16 125 88 89 53)(25 142 148 107 72)(26 143 149 108 65)(27 144 150 109 66)(28 137 151 110 67)(29 138 152 111 68)(30 139 145 112 69)(31 140 146 105 70)(32 141 147 106 71)(41 159 113 73 57)(42 160 114 74 58)(43 153 115 75 59)(44 154 116 76 60)(45 155 117 77 61)(46 156 118 78 62)(47 157 119 79 63)(48 158 120 80 64)
(1 149)(2 150)(3 151)(4 152)(5 145)(6 146)(7 147)(8 148)(9 64)(10 57)(11 58)(12 59)(13 60)(14 61)(15 62)(16 63)(17 137)(18 138)(19 139)(20 140)(21 141)(22 142)(23 143)(24 144)(25 132)(26 133)(27 134)(28 135)(29 136)(30 129)(31 130)(32 131)(33 65)(34 66)(35 67)(36 68)(37 69)(38 70)(39 71)(40 72)(41 55)(42 56)(43 49)(44 50)(45 51)(46 52)(47 53)(48 54)(73 127)(74 128)(75 121)(76 122)(77 123)(78 124)(79 125)(80 126)(81 120)(82 113)(83 114)(84 115)(85 116)(86 117)(87 118)(88 119)(89 157)(90 158)(91 159)(92 160)(93 153)(94 154)(95 155)(96 156)(97 108)(98 109)(99 110)(100 111)(101 112)(102 105)(103 106)(104 107)
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), (1,55,108,155)(2,54,109,154)(3,53,110,153)(4,52,111,160)(5,51,112,159)(6,50,105,158)(7,49,106,157)(8,56,107,156)(9,66,116,24)(10,65,117,23)(11,72,118,22)(12,71,119,21)(13,70,120,20)(14,69,113,19)(15,68,114,18)(16,67,115,17)(25,78,132,128)(26,77,133,127)(27,76,134,126)(28,75,135,125)(29,74,136,124)(30,73,129,123)(31,80,130,122)(32,79,131,121)(33,82,143,61)(34,81,144,60)(35,88,137,59)(36,87,138,58)(37,86,139,57)(38,85,140,64)(39,84,141,63)(40,83,142,62)(41,101,95,145)(42,100,96,152)(43,99,89,151)(44,98,90,150)(45,97,91,149)(46,104,92,148)(47,103,93,147)(48,102,94,146), (1,23,133,33,97)(2,24,134,34,98)(3,17,135,35,99)(4,18,136,36,100)(5,19,129,37,101)(6,20,130,38,102)(7,21,131,39,103)(8,22,132,40,104)(9,126,81,90,54)(10,127,82,91,55)(11,128,83,92,56)(12,121,84,93,49)(13,122,85,94,50)(14,123,86,95,51)(15,124,87,96,52)(16,125,88,89,53)(25,142,148,107,72)(26,143,149,108,65)(27,144,150,109,66)(28,137,151,110,67)(29,138,152,111,68)(30,139,145,112,69)(31,140,146,105,70)(32,141,147,106,71)(41,159,113,73,57)(42,160,114,74,58)(43,153,115,75,59)(44,154,116,76,60)(45,155,117,77,61)(46,156,118,78,62)(47,157,119,79,63)(48,158,120,80,64), (1,149)(2,150)(3,151)(4,152)(5,145)(6,146)(7,147)(8,148)(9,64)(10,57)(11,58)(12,59)(13,60)(14,61)(15,62)(16,63)(17,137)(18,138)(19,139)(20,140)(21,141)(22,142)(23,143)(24,144)(25,132)(26,133)(27,134)(28,135)(29,136)(30,129)(31,130)(32,131)(33,65)(34,66)(35,67)(36,68)(37,69)(38,70)(39,71)(40,72)(41,55)(42,56)(43,49)(44,50)(45,51)(46,52)(47,53)(48,54)(73,127)(74,128)(75,121)(76,122)(77,123)(78,124)(79,125)(80,126)(81,120)(82,113)(83,114)(84,115)(85,116)(86,117)(87,118)(88,119)(89,157)(90,158)(91,159)(92,160)(93,153)(94,154)(95,155)(96,156)(97,108)(98,109)(99,110)(100,111)(101,112)(102,105)(103,106)(104,107)>;
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), (1,55,108,155)(2,54,109,154)(3,53,110,153)(4,52,111,160)(5,51,112,159)(6,50,105,158)(7,49,106,157)(8,56,107,156)(9,66,116,24)(10,65,117,23)(11,72,118,22)(12,71,119,21)(13,70,120,20)(14,69,113,19)(15,68,114,18)(16,67,115,17)(25,78,132,128)(26,77,133,127)(27,76,134,126)(28,75,135,125)(29,74,136,124)(30,73,129,123)(31,80,130,122)(32,79,131,121)(33,82,143,61)(34,81,144,60)(35,88,137,59)(36,87,138,58)(37,86,139,57)(38,85,140,64)(39,84,141,63)(40,83,142,62)(41,101,95,145)(42,100,96,152)(43,99,89,151)(44,98,90,150)(45,97,91,149)(46,104,92,148)(47,103,93,147)(48,102,94,146), (1,23,133,33,97)(2,24,134,34,98)(3,17,135,35,99)(4,18,136,36,100)(5,19,129,37,101)(6,20,130,38,102)(7,21,131,39,103)(8,22,132,40,104)(9,126,81,90,54)(10,127,82,91,55)(11,128,83,92,56)(12,121,84,93,49)(13,122,85,94,50)(14,123,86,95,51)(15,124,87,96,52)(16,125,88,89,53)(25,142,148,107,72)(26,143,149,108,65)(27,144,150,109,66)(28,137,151,110,67)(29,138,152,111,68)(30,139,145,112,69)(31,140,146,105,70)(32,141,147,106,71)(41,159,113,73,57)(42,160,114,74,58)(43,153,115,75,59)(44,154,116,76,60)(45,155,117,77,61)(46,156,118,78,62)(47,157,119,79,63)(48,158,120,80,64), (1,149)(2,150)(3,151)(4,152)(5,145)(6,146)(7,147)(8,148)(9,64)(10,57)(11,58)(12,59)(13,60)(14,61)(15,62)(16,63)(17,137)(18,138)(19,139)(20,140)(21,141)(22,142)(23,143)(24,144)(25,132)(26,133)(27,134)(28,135)(29,136)(30,129)(31,130)(32,131)(33,65)(34,66)(35,67)(36,68)(37,69)(38,70)(39,71)(40,72)(41,55)(42,56)(43,49)(44,50)(45,51)(46,52)(47,53)(48,54)(73,127)(74,128)(75,121)(76,122)(77,123)(78,124)(79,125)(80,126)(81,120)(82,113)(83,114)(84,115)(85,116)(86,117)(87,118)(88,119)(89,157)(90,158)(91,159)(92,160)(93,153)(94,154)(95,155)(96,156)(97,108)(98,109)(99,110)(100,111)(101,112)(102,105)(103,106)(104,107) );
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)], [(1,55,108,155),(2,54,109,154),(3,53,110,153),(4,52,111,160),(5,51,112,159),(6,50,105,158),(7,49,106,157),(8,56,107,156),(9,66,116,24),(10,65,117,23),(11,72,118,22),(12,71,119,21),(13,70,120,20),(14,69,113,19),(15,68,114,18),(16,67,115,17),(25,78,132,128),(26,77,133,127),(27,76,134,126),(28,75,135,125),(29,74,136,124),(30,73,129,123),(31,80,130,122),(32,79,131,121),(33,82,143,61),(34,81,144,60),(35,88,137,59),(36,87,138,58),(37,86,139,57),(38,85,140,64),(39,84,141,63),(40,83,142,62),(41,101,95,145),(42,100,96,152),(43,99,89,151),(44,98,90,150),(45,97,91,149),(46,104,92,148),(47,103,93,147),(48,102,94,146)], [(1,23,133,33,97),(2,24,134,34,98),(3,17,135,35,99),(4,18,136,36,100),(5,19,129,37,101),(6,20,130,38,102),(7,21,131,39,103),(8,22,132,40,104),(9,126,81,90,54),(10,127,82,91,55),(11,128,83,92,56),(12,121,84,93,49),(13,122,85,94,50),(14,123,86,95,51),(15,124,87,96,52),(16,125,88,89,53),(25,142,148,107,72),(26,143,149,108,65),(27,144,150,109,66),(28,137,151,110,67),(29,138,152,111,68),(30,139,145,112,69),(31,140,146,105,70),(32,141,147,106,71),(41,159,113,73,57),(42,160,114,74,58),(43,153,115,75,59),(44,154,116,76,60),(45,155,117,77,61),(46,156,118,78,62),(47,157,119,79,63),(48,158,120,80,64)], [(1,149),(2,150),(3,151),(4,152),(5,145),(6,146),(7,147),(8,148),(9,64),(10,57),(11,58),(12,59),(13,60),(14,61),(15,62),(16,63),(17,137),(18,138),(19,139),(20,140),(21,141),(22,142),(23,143),(24,144),(25,132),(26,133),(27,134),(28,135),(29,136),(30,129),(31,130),(32,131),(33,65),(34,66),(35,67),(36,68),(37,69),(38,70),(39,71),(40,72),(41,55),(42,56),(43,49),(44,50),(45,51),(46,52),(47,53),(48,54),(73,127),(74,128),(75,121),(76,122),(77,123),(78,124),(79,125),(80,126),(81,120),(82,113),(83,114),(84,115),(85,116),(86,117),(87,118),(88,119),(89,157),(90,158),(91,159),(92,160),(93,153),(94,154),(95,155),(96,156),(97,108),(98,109),(99,110),(100,111),(101,112),(102,105),(103,106),(104,107)]])
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