metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.166D10, C10.752+ 1+4, C4⋊1D4.7D5, (D4×Dic5)⋊33C2, (C4×Dic10)⋊50C2, (C2×D4).114D10, C20.133(C4○D4), C20.17D4⋊25C2, C4.17(D4⋊2D5), (C4×C20).202C22, (C2×C20).634C23, (C2×C10).257C24, C2.79(D4⋊6D10), C23.63(C22×D5), (D4×C10).160C22, C4⋊Dic5.380C22, (C22×C10).71C23, C22.278(C23×D5), C23.D5.71C22, C23.18D10⋊26C2, C5⋊5(C22.53C24), (C4×Dic5).162C22, (C2×Dic5).133C23, (C2×Dic10).308C22, C10.D4.163C22, (C22×Dic5).156C22, C10.95(C2×C4○D4), (C5×C4⋊1D4).6C2, C2.59(C2×D4⋊2D5), (C2×C4).595(C22×D5), SmallGroup(320,1385)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.166D10
G = < a,b,c,d | a4=b4=c10=1, d2=b2, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, dbd-1=a2b, dcd-1=c-1 >
Subgroups: 726 in 236 conjugacy classes, 99 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C10, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic5, C20, C20, C2×C10, C2×C10, C4×D4, C4×Q8, C22.D4, C4.4D4, C4⋊1D4, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×D4, C22×C10, C22.53C24, C4×Dic5, C10.D4, C4⋊Dic5, C23.D5, C4×C20, C2×Dic10, C22×Dic5, D4×C10, C4×Dic10, D4×Dic5, C23.18D10, C20.17D4, C5×C4⋊1D4, C42.166D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2+ 1+4, C22×D5, C22.53C24, D4⋊2D5, C23×D5, C2×D4⋊2D5, D4⋊6D10, C42.166D10
►On 160 points
Generators in S160
(1 58 53 6)(2 7 54 59)(3 60 55 8)(4 9 56 51)(5 52 57 10)(11 85 29 77)(12 78 30 86)(13 87 21 79)(14 80 22 88)(15 89 23 71)(16 72 24 90)(17 81 25 73)(18 74 26 82)(19 83 27 75)(20 76 28 84)(31 36 64 69)(32 70 65 37)(33 38 66 61)(34 62 67 39)(35 40 68 63)(41 128 158 101)(42 102 159 129)(43 130 160 103)(44 104 151 121)(45 122 152 105)(46 106 153 123)(47 124 154 107)(48 108 155 125)(49 126 156 109)(50 110 157 127)(91 138 133 96)(92 97 134 139)(93 140 135 98)(94 99 136 131)(95 132 137 100)(111 148 143 116)(112 117 144 149)(113 150 145 118)(114 119 146 141)(115 142 147 120)
(1 73 63 30)(2 21 64 74)(3 75 65 22)(4 23 66 76)(5 77 67 24)(6 25 68 78)(7 79 69 26)(8 27 70 80)(9 71 61 28)(10 29 62 72)(11 39 90 52)(12 53 81 40)(13 31 82 54)(14 55 83 32)(15 33 84 56)(16 57 85 34)(17 35 86 58)(18 59 87 36)(19 37 88 60)(20 51 89 38)(41 148 123 96)(42 97 124 149)(43 150 125 98)(44 99 126 141)(45 142 127 100)(46 91 128 143)(47 144 129 92)(48 93 130 145)(49 146 121 94)(50 95 122 147)(101 111 153 133)(102 134 154 112)(103 113 155 135)(104 136 156 114)(105 115 157 137)(106 138 158 116)(107 117 159 139)(108 140 160 118)(109 119 151 131)(110 132 152 120)
(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)
(1 146 63 94)(2 145 64 93)(3 144 65 92)(4 143 66 91)(5 142 67 100)(6 141 68 99)(7 150 69 98)(8 149 70 97)(9 148 61 96)(10 147 62 95)(11 50 90 122)(12 49 81 121)(13 48 82 130)(14 47 83 129)(15 46 84 128)(16 45 85 127)(17 44 86 126)(18 43 87 125)(19 42 88 124)(20 41 89 123)(21 155 74 103)(22 154 75 102)(23 153 76 101)(24 152 77 110)(25 151 78 109)(26 160 79 108)(27 159 80 107)(28 158 71 106)(29 157 72 105)(30 156 73 104)(31 135 54 113)(32 134 55 112)(33 133 56 111)(34 132 57 120)(35 131 58 119)(36 140 59 118)(37 139 60 117)(38 138 51 116)(39 137 52 115)(40 136 53 114)
G:=sub<Sym(160)| (1,58,53,6)(2,7,54,59)(3,60,55,8)(4,9,56,51)(5,52,57,10)(11,85,29,77)(12,78,30,86)(13,87,21,79)(14,80,22,88)(15,89,23,71)(16,72,24,90)(17,81,25,73)(18,74,26,82)(19,83,27,75)(20,76,28,84)(31,36,64,69)(32,70,65,37)(33,38,66,61)(34,62,67,39)(35,40,68,63)(41,128,158,101)(42,102,159,129)(43,130,160,103)(44,104,151,121)(45,122,152,105)(46,106,153,123)(47,124,154,107)(48,108,155,125)(49,126,156,109)(50,110,157,127)(91,138,133,96)(92,97,134,139)(93,140,135,98)(94,99,136,131)(95,132,137,100)(111,148,143,116)(112,117,144,149)(113,150,145,118)(114,119,146,141)(115,142,147,120), (1,73,63,30)(2,21,64,74)(3,75,65,22)(4,23,66,76)(5,77,67,24)(6,25,68,78)(7,79,69,26)(8,27,70,80)(9,71,61,28)(10,29,62,72)(11,39,90,52)(12,53,81,40)(13,31,82,54)(14,55,83,32)(15,33,84,56)(16,57,85,34)(17,35,86,58)(18,59,87,36)(19,37,88,60)(20,51,89,38)(41,148,123,96)(42,97,124,149)(43,150,125,98)(44,99,126,141)(45,142,127,100)(46,91,128,143)(47,144,129,92)(48,93,130,145)(49,146,121,94)(50,95,122,147)(101,111,153,133)(102,134,154,112)(103,113,155,135)(104,136,156,114)(105,115,157,137)(106,138,158,116)(107,117,159,139)(108,140,160,118)(109,119,151,131)(110,132,152,120), (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), (1,146,63,94)(2,145,64,93)(3,144,65,92)(4,143,66,91)(5,142,67,100)(6,141,68,99)(7,150,69,98)(8,149,70,97)(9,148,61,96)(10,147,62,95)(11,50,90,122)(12,49,81,121)(13,48,82,130)(14,47,83,129)(15,46,84,128)(16,45,85,127)(17,44,86,126)(18,43,87,125)(19,42,88,124)(20,41,89,123)(21,155,74,103)(22,154,75,102)(23,153,76,101)(24,152,77,110)(25,151,78,109)(26,160,79,108)(27,159,80,107)(28,158,71,106)(29,157,72,105)(30,156,73,104)(31,135,54,113)(32,134,55,112)(33,133,56,111)(34,132,57,120)(35,131,58,119)(36,140,59,118)(37,139,60,117)(38,138,51,116)(39,137,52,115)(40,136,53,114)>;
G:=Group( (1,58,53,6)(2,7,54,59)(3,60,55,8)(4,9,56,51)(5,52,57,10)(11,85,29,77)(12,78,30,86)(13,87,21,79)(14,80,22,88)(15,89,23,71)(16,72,24,90)(17,81,25,73)(18,74,26,82)(19,83,27,75)(20,76,28,84)(31,36,64,69)(32,70,65,37)(33,38,66,61)(34,62,67,39)(35,40,68,63)(41,128,158,101)(42,102,159,129)(43,130,160,103)(44,104,151,121)(45,122,152,105)(46,106,153,123)(47,124,154,107)(48,108,155,125)(49,126,156,109)(50,110,157,127)(91,138,133,96)(92,97,134,139)(93,140,135,98)(94,99,136,131)(95,132,137,100)(111,148,143,116)(112,117,144,149)(113,150,145,118)(114,119,146,141)(115,142,147,120), (1,73,63,30)(2,21,64,74)(3,75,65,22)(4,23,66,76)(5,77,67,24)(6,25,68,78)(7,79,69,26)(8,27,70,80)(9,71,61,28)(10,29,62,72)(11,39,90,52)(12,53,81,40)(13,31,82,54)(14,55,83,32)(15,33,84,56)(16,57,85,34)(17,35,86,58)(18,59,87,36)(19,37,88,60)(20,51,89,38)(41,148,123,96)(42,97,124,149)(43,150,125,98)(44,99,126,141)(45,142,127,100)(46,91,128,143)(47,144,129,92)(48,93,130,145)(49,146,121,94)(50,95,122,147)(101,111,153,133)(102,134,154,112)(103,113,155,135)(104,136,156,114)(105,115,157,137)(106,138,158,116)(107,117,159,139)(108,140,160,118)(109,119,151,131)(110,132,152,120), (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), (1,146,63,94)(2,145,64,93)(3,144,65,92)(4,143,66,91)(5,142,67,100)(6,141,68,99)(7,150,69,98)(8,149,70,97)(9,148,61,96)(10,147,62,95)(11,50,90,122)(12,49,81,121)(13,48,82,130)(14,47,83,129)(15,46,84,128)(16,45,85,127)(17,44,86,126)(18,43,87,125)(19,42,88,124)(20,41,89,123)(21,155,74,103)(22,154,75,102)(23,153,76,101)(24,152,77,110)(25,151,78,109)(26,160,79,108)(27,159,80,107)(28,158,71,106)(29,157,72,105)(30,156,73,104)(31,135,54,113)(32,134,55,112)(33,133,56,111)(34,132,57,120)(35,131,58,119)(36,140,59,118)(37,139,60,117)(38,138,51,116)(39,137,52,115)(40,136,53,114) );
G=PermutationGroup([[(1,58,53,6),(2,7,54,59),(3,60,55,8),(4,9,56,51),(5,52,57,10),(11,85,29,77),(12,78,30,86),(13,87,21,79),(14,80,22,88),(15,89,23,71),(16,72,24,90),(17,81,25,73),(18,74,26,82),(19,83,27,75),(20,76,28,84),(31,36,64,69),(32,70,65,37),(33,38,66,61),(34,62,67,39),(35,40,68,63),(41,128,158,101),(42,102,159,129),(43,130,160,103),(44,104,151,121),(45,122,152,105),(46,106,153,123),(47,124,154,107),(48,108,155,125),(49,126,156,109),(50,110,157,127),(91,138,133,96),(92,97,134,139),(93,140,135,98),(94,99,136,131),(95,132,137,100),(111,148,143,116),(112,117,144,149),(113,150,145,118),(114,119,146,141),(115,142,147,120)], [(1,73,63,30),(2,21,64,74),(3,75,65,22),(4,23,66,76),(5,77,67,24),(6,25,68,78),(7,79,69,26),(8,27,70,80),(9,71,61,28),(10,29,62,72),(11,39,90,52),(12,53,81,40),(13,31,82,54),(14,55,83,32),(15,33,84,56),(16,57,85,34),(17,35,86,58),(18,59,87,36),(19,37,88,60),(20,51,89,38),(41,148,123,96),(42,97,124,149),(43,150,125,98),(44,99,126,141),(45,142,127,100),(46,91,128,143),(47,144,129,92),(48,93,130,145),(49,146,121,94),(50,95,122,147),(101,111,153,133),(102,134,154,112),(103,113,155,135),(104,136,156,114),(105,115,157,137),(106,138,158,116),(107,117,159,139),(108,140,160,118),(109,119,151,131),(110,132,152,120)], [(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)], [(1,146,63,94),(2,145,64,93),(3,144,65,92),(4,143,66,91),(5,142,67,100),(6,141,68,99),(7,150,69,98),(8,149,70,97),(9,148,61,96),(10,147,62,95),(11,50,90,122),(12,49,81,121),(13,48,82,130),(14,47,83,129),(15,46,84,128),(16,45,85,127),(17,44,86,126),(18,43,87,125),(19,42,88,124),(20,41,89,123),(21,155,74,103),(22,154,75,102),(23,153,76,101),(24,152,77,110),(25,151,78,109),(26,160,79,108),(27,159,80,107),(28,158,71,106),(29,157,72,105),(30,156,73,104),(31,135,54,113),(32,134,55,112),(33,133,56,111),(34,132,57,120),(35,131,58,119),(36,140,59,118),(37,139,60,117),(38,138,51,116),(39,137,52,115),(40,136,53,114)]])
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | ··· | 4M | 4N | 4O | 4P | 4Q | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | 2+ 1+4 | D4⋊2D5 | D4⋊6D10 |
| kernel | C42.166D10 | C4×Dic10 | D4×Dic5 | C23.18D10 | C20.17D4 | C5×C4⋊1D4 | C4⋊1D4 | C20 | C42 | C2×D4 | C10 | C4 | C2 |
| # reps | 1 | 2 | 4 | 4 | 4 | 1 | 2 | 8 | 2 | 12 | 1 | 8 | 4 |
Matrix representation of C42.166D10 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 35 | 0 | 0 |
| 0 | 0 | 6 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 25 | 0 | 0 |
| 0 | 0 | 13 | 39 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 0 | 0 | 0 | 0 | 9 | 0 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,6,0,0,0,0,35,35,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,2,13,0,0,0,0,25,39,0,0,0,0,0,0,0,9,0,0,0,0,9,0] >;
C42.166D10 in GAP, Magma, Sage, TeX
C_4^2._{166}D_{10} % in TeX
G:=Group("C4^2.166D10"); // GroupNames label
G:=SmallGroup(320,1385);
// by ID
G=gap.SmallGroup(320,1385);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,219,1571,570,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^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,d*c*d^-1=c^-1>;
// generators/relations