metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.174D10, C10.372- 1+4, C10.822+ 1+4, C4⋊Q8⋊12D5, (C4×D5)⋊2Q8, C20⋊Q8⋊45C2, C4.41(Q8×D5), D10.6(C2×Q8), C20.55(C2×Q8), C4⋊C4.124D10, (C2×Q8).86D10, Dic5.7(C2×Q8), D10⋊Q8.4C2, C4.Dic10⋊43C2, C20.6Q8⋊24C2, C42⋊D5.8C2, Dic5⋊Q8⋊27C2, C10.49(C22×Q8), (C4×C20).214C22, (C2×C20).106C23, (C2×C10).273C24, D10⋊3Q8.13C2, C2.86(D4⋊6D10), Dic5.Q8⋊41C2, C4⋊Dic5.252C22, (Q8×C10).140C22, C22.294(C23×D5), D10⋊C4.52C22, C5⋊5(C23.41C23), (C4×Dic5).170C22, (C2×Dic5).144C23, (C22×D5).244C23, C2.38(Q8.10D10), (C2×Dic10).197C22, C10.D4.167C22, C2.32(C2×Q8×D5), (C5×C4⋊Q8)⋊15C2, (D5×C4⋊C4).13C2, C4⋊C4⋊7D5.15C2, (C2×C4×D5).155C22, (C5×C4⋊C4).216C22, (C2×C4).219(C22×D5), SmallGroup(320,1401)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.174D10
G = < a,b,c,d | a4=b4=1, c10=d2=a2b2, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c9 >
Subgroups: 654 in 206 conjugacy classes, 103 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, Q8, C23, D5, C10, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, C2×Q8, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C4⋊C4, C42⋊C2, C22⋊Q8, C42.C2, C4⋊Q8, C4⋊Q8, Dic10, C4×D5, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, C23.41C23, C4×Dic5, C4×Dic5, C10.D4, C10.D4, C4⋊Dic5, C4⋊Dic5, D10⋊C4, D10⋊C4, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×C4×D5, Q8×C10, C20.6Q8, C42⋊D5, C20⋊Q8, Dic5.Q8, C4.Dic10, D5×C4⋊C4, C4⋊C4⋊7D5, D10⋊Q8, Dic5⋊Q8, D10⋊3Q8, C5×C4⋊Q8, C42.174D10
Quotients: C1, C2, C22, Q8, C23, D5, C2×Q8, C24, D10, C22×Q8, 2+ 1+4, 2- 1+4, C22×D5, C23.41C23, Q8×D5, C23×D5, D4⋊6D10, C2×Q8×D5, Q8.10D10, C42.174D10
►On 160 points
Generators in S160
(1 43 107 122)(2 123 108 44)(3 45 109 124)(4 125 110 46)(5 47 111 126)(6 127 112 48)(7 49 113 128)(8 129 114 50)(9 51 115 130)(10 131 116 52)(11 53 117 132)(12 133 118 54)(13 55 119 134)(14 135 120 56)(15 57 101 136)(16 137 102 58)(17 59 103 138)(18 139 104 60)(19 41 105 140)(20 121 106 42)(21 81 66 157)(22 158 67 82)(23 83 68 159)(24 160 69 84)(25 85 70 141)(26 142 71 86)(27 87 72 143)(28 144 73 88)(29 89 74 145)(30 146 75 90)(31 91 76 147)(32 148 77 92)(33 93 78 149)(34 150 79 94)(35 95 80 151)(36 152 61 96)(37 97 62 153)(38 154 63 98)(39 99 64 155)(40 156 65 100)
(1 94 117 160)(2 141 118 95)(3 96 119 142)(4 143 120 97)(5 98 101 144)(6 145 102 99)(7 100 103 146)(8 147 104 81)(9 82 105 148)(10 149 106 83)(11 84 107 150)(12 151 108 85)(13 86 109 152)(14 153 110 87)(15 88 111 154)(16 155 112 89)(17 90 113 156)(18 157 114 91)(19 92 115 158)(20 159 116 93)(21 50 76 139)(22 140 77 51)(23 52 78 121)(24 122 79 53)(25 54 80 123)(26 124 61 55)(27 56 62 125)(28 126 63 57)(29 58 64 127)(30 128 65 59)(31 60 66 129)(32 130 67 41)(33 42 68 131)(34 132 69 43)(35 44 70 133)(36 134 71 45)(37 46 72 135)(38 136 73 47)(39 48 74 137)(40 138 75 49)
(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 10 11 20)(2 19 12 9)(3 8 13 18)(4 17 14 7)(5 6 15 16)(21 71 31 61)(22 80 32 70)(23 69 33 79)(24 78 34 68)(25 67 35 77)(26 76 36 66)(27 65 37 75)(28 74 38 64)(29 63 39 73)(30 72 40 62)(41 54 51 44)(42 43 52 53)(45 50 55 60)(46 59 56 49)(47 48 57 58)(81 142 91 152)(82 151 92 141)(83 160 93 150)(84 149 94 159)(85 158 95 148)(86 147 96 157)(87 156 97 146)(88 145 98 155)(89 154 99 144)(90 143 100 153)(101 102 111 112)(103 120 113 110)(104 109 114 119)(105 118 115 108)(106 107 116 117)(121 122 131 132)(123 140 133 130)(124 129 134 139)(125 138 135 128)(126 127 136 137)
G:=sub<Sym(160)| (1,43,107,122)(2,123,108,44)(3,45,109,124)(4,125,110,46)(5,47,111,126)(6,127,112,48)(7,49,113,128)(8,129,114,50)(9,51,115,130)(10,131,116,52)(11,53,117,132)(12,133,118,54)(13,55,119,134)(14,135,120,56)(15,57,101,136)(16,137,102,58)(17,59,103,138)(18,139,104,60)(19,41,105,140)(20,121,106,42)(21,81,66,157)(22,158,67,82)(23,83,68,159)(24,160,69,84)(25,85,70,141)(26,142,71,86)(27,87,72,143)(28,144,73,88)(29,89,74,145)(30,146,75,90)(31,91,76,147)(32,148,77,92)(33,93,78,149)(34,150,79,94)(35,95,80,151)(36,152,61,96)(37,97,62,153)(38,154,63,98)(39,99,64,155)(40,156,65,100), (1,94,117,160)(2,141,118,95)(3,96,119,142)(4,143,120,97)(5,98,101,144)(6,145,102,99)(7,100,103,146)(8,147,104,81)(9,82,105,148)(10,149,106,83)(11,84,107,150)(12,151,108,85)(13,86,109,152)(14,153,110,87)(15,88,111,154)(16,155,112,89)(17,90,113,156)(18,157,114,91)(19,92,115,158)(20,159,116,93)(21,50,76,139)(22,140,77,51)(23,52,78,121)(24,122,79,53)(25,54,80,123)(26,124,61,55)(27,56,62,125)(28,126,63,57)(29,58,64,127)(30,128,65,59)(31,60,66,129)(32,130,67,41)(33,42,68,131)(34,132,69,43)(35,44,70,133)(36,134,71,45)(37,46,72,135)(38,136,73,47)(39,48,74,137)(40,138,75,49), (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), (1,10,11,20)(2,19,12,9)(3,8,13,18)(4,17,14,7)(5,6,15,16)(21,71,31,61)(22,80,32,70)(23,69,33,79)(24,78,34,68)(25,67,35,77)(26,76,36,66)(27,65,37,75)(28,74,38,64)(29,63,39,73)(30,72,40,62)(41,54,51,44)(42,43,52,53)(45,50,55,60)(46,59,56,49)(47,48,57,58)(81,142,91,152)(82,151,92,141)(83,160,93,150)(84,149,94,159)(85,158,95,148)(86,147,96,157)(87,156,97,146)(88,145,98,155)(89,154,99,144)(90,143,100,153)(101,102,111,112)(103,120,113,110)(104,109,114,119)(105,118,115,108)(106,107,116,117)(121,122,131,132)(123,140,133,130)(124,129,134,139)(125,138,135,128)(126,127,136,137)>;
G:=Group( (1,43,107,122)(2,123,108,44)(3,45,109,124)(4,125,110,46)(5,47,111,126)(6,127,112,48)(7,49,113,128)(8,129,114,50)(9,51,115,130)(10,131,116,52)(11,53,117,132)(12,133,118,54)(13,55,119,134)(14,135,120,56)(15,57,101,136)(16,137,102,58)(17,59,103,138)(18,139,104,60)(19,41,105,140)(20,121,106,42)(21,81,66,157)(22,158,67,82)(23,83,68,159)(24,160,69,84)(25,85,70,141)(26,142,71,86)(27,87,72,143)(28,144,73,88)(29,89,74,145)(30,146,75,90)(31,91,76,147)(32,148,77,92)(33,93,78,149)(34,150,79,94)(35,95,80,151)(36,152,61,96)(37,97,62,153)(38,154,63,98)(39,99,64,155)(40,156,65,100), (1,94,117,160)(2,141,118,95)(3,96,119,142)(4,143,120,97)(5,98,101,144)(6,145,102,99)(7,100,103,146)(8,147,104,81)(9,82,105,148)(10,149,106,83)(11,84,107,150)(12,151,108,85)(13,86,109,152)(14,153,110,87)(15,88,111,154)(16,155,112,89)(17,90,113,156)(18,157,114,91)(19,92,115,158)(20,159,116,93)(21,50,76,139)(22,140,77,51)(23,52,78,121)(24,122,79,53)(25,54,80,123)(26,124,61,55)(27,56,62,125)(28,126,63,57)(29,58,64,127)(30,128,65,59)(31,60,66,129)(32,130,67,41)(33,42,68,131)(34,132,69,43)(35,44,70,133)(36,134,71,45)(37,46,72,135)(38,136,73,47)(39,48,74,137)(40,138,75,49), (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), (1,10,11,20)(2,19,12,9)(3,8,13,18)(4,17,14,7)(5,6,15,16)(21,71,31,61)(22,80,32,70)(23,69,33,79)(24,78,34,68)(25,67,35,77)(26,76,36,66)(27,65,37,75)(28,74,38,64)(29,63,39,73)(30,72,40,62)(41,54,51,44)(42,43,52,53)(45,50,55,60)(46,59,56,49)(47,48,57,58)(81,142,91,152)(82,151,92,141)(83,160,93,150)(84,149,94,159)(85,158,95,148)(86,147,96,157)(87,156,97,146)(88,145,98,155)(89,154,99,144)(90,143,100,153)(101,102,111,112)(103,120,113,110)(104,109,114,119)(105,118,115,108)(106,107,116,117)(121,122,131,132)(123,140,133,130)(124,129,134,139)(125,138,135,128)(126,127,136,137) );
G=PermutationGroup([[(1,43,107,122),(2,123,108,44),(3,45,109,124),(4,125,110,46),(5,47,111,126),(6,127,112,48),(7,49,113,128),(8,129,114,50),(9,51,115,130),(10,131,116,52),(11,53,117,132),(12,133,118,54),(13,55,119,134),(14,135,120,56),(15,57,101,136),(16,137,102,58),(17,59,103,138),(18,139,104,60),(19,41,105,140),(20,121,106,42),(21,81,66,157),(22,158,67,82),(23,83,68,159),(24,160,69,84),(25,85,70,141),(26,142,71,86),(27,87,72,143),(28,144,73,88),(29,89,74,145),(30,146,75,90),(31,91,76,147),(32,148,77,92),(33,93,78,149),(34,150,79,94),(35,95,80,151),(36,152,61,96),(37,97,62,153),(38,154,63,98),(39,99,64,155),(40,156,65,100)], [(1,94,117,160),(2,141,118,95),(3,96,119,142),(4,143,120,97),(5,98,101,144),(6,145,102,99),(7,100,103,146),(8,147,104,81),(9,82,105,148),(10,149,106,83),(11,84,107,150),(12,151,108,85),(13,86,109,152),(14,153,110,87),(15,88,111,154),(16,155,112,89),(17,90,113,156),(18,157,114,91),(19,92,115,158),(20,159,116,93),(21,50,76,139),(22,140,77,51),(23,52,78,121),(24,122,79,53),(25,54,80,123),(26,124,61,55),(27,56,62,125),(28,126,63,57),(29,58,64,127),(30,128,65,59),(31,60,66,129),(32,130,67,41),(33,42,68,131),(34,132,69,43),(35,44,70,133),(36,134,71,45),(37,46,72,135),(38,136,73,47),(39,48,74,137),(40,138,75,49)], [(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)], [(1,10,11,20),(2,19,12,9),(3,8,13,18),(4,17,14,7),(5,6,15,16),(21,71,31,61),(22,80,32,70),(23,69,33,79),(24,78,34,68),(25,67,35,77),(26,76,36,66),(27,65,37,75),(28,74,38,64),(29,63,39,73),(30,72,40,62),(41,54,51,44),(42,43,52,53),(45,50,55,60),(46,59,56,49),(47,48,57,58),(81,142,91,152),(82,151,92,141),(83,160,93,150),(84,149,94,159),(85,158,95,148),(86,147,96,157),(87,156,97,146),(88,145,98,155),(89,154,99,144),(90,143,100,153),(101,102,111,112),(103,120,113,110),(104,109,114,119),(105,118,115,108),(106,107,116,117),(121,122,131,132),(123,140,133,130),(124,129,134,139),(125,138,135,128),(126,127,136,137)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | ··· | 4H | 4I | 4J | 4K | ··· | 4P | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 2 | 2 | 4 | ··· | 4 | 10 | 10 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | D5 | D10 | D10 | D10 | 2+ 1+4 | 2- 1+4 | Q8×D5 | D4⋊6D10 | Q8.10D10 |
| kernel | C42.174D10 | C20.6Q8 | C42⋊D5 | C20⋊Q8 | Dic5.Q8 | C4.Dic10 | D5×C4⋊C4 | C4⋊C4⋊7D5 | D10⋊Q8 | Dic5⋊Q8 | D10⋊3Q8 | C5×C4⋊Q8 | C4×D5 | C4⋊Q8 | C42 | C4⋊C4 | C2×Q8 | C10 | C10 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 4 | 2 | 2 | 8 | 4 | 1 | 1 | 4 | 4 | 4 |
Matrix representation of C42.174D10 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 0 | 0 | 11 |
| 0 | 0 | 26 | 0 | 0 | 0 |
| 0 | 0 | 0 | 26 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 1 | 0 | 0 |
| 0 | 0 | 40 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 24 | 1 |
| 0 | 0 | 0 | 0 | 40 | 17 |
| 27 | 7 | 0 | 0 | 0 | 0 |
| 7 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 34 | 27 | 27 |
| 0 | 0 | 7 | 1 | 14 | 2 |
| 0 | 0 | 7 | 7 | 7 | 7 |
| 0 | 0 | 34 | 40 | 34 | 40 |
| 14 | 34 | 0 | 0 | 0 | 0 |
| 34 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 34 | 27 | 27 |
| 0 | 0 | 1 | 7 | 2 | 14 |
| 0 | 0 | 7 | 7 | 7 | 7 |
| 0 | 0 | 40 | 34 | 40 | 34 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,26,0,0,0,0,0,0,26,0,0,11,0,0,0,0,0,0,11,0,0],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,24,40,0,0,0,0,1,17,0,0,0,0,0,0,24,40,0,0,0,0,1,17],[27,7,0,0,0,0,7,14,0,0,0,0,0,0,34,7,7,34,0,0,34,1,7,40,0,0,27,14,7,34,0,0,27,2,7,40],[14,34,0,0,0,0,34,27,0,0,0,0,0,0,34,1,7,40,0,0,34,7,7,34,0,0,27,2,7,40,0,0,27,14,7,34] >;
C42.174D10 in GAP, Magma, Sage, TeX
C_4^2._{174}D_{10} % in TeX
G:=Group("C4^2.174D10"); // GroupNames label
G:=SmallGroup(320,1401);
// by ID
G=gap.SmallGroup(320,1401);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,268,675,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=d^2=a^2*b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=b^-1,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^9>;
// generators/relations