metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.165D10, C10.1032- (1+4), C10.1432+ (1+4), C20⋊Q8⋊42C2, C20⋊2Q8⋊9C2, C4⋊C4.120D10, C42⋊2D5⋊2C2, C42⋊2C2⋊9D5, (C4×C20).9C22, D10⋊2Q8⋊43C2, D10⋊Q8⋊46C2, (C2×C20).97C23, C22⋊C4.42D10, C4.Dic10⋊41C2, (C2×C10).256C24, C4⋊Dic5.55C22, C2.68(D4⋊8D10), C23.62(C22×D5), Dic5.Q8⋊40C2, D10.12D4.5C2, C23.D10⋊47C2, (C22×C10).70C23, Dic5.5D4.5C2, C22.D20.4C2, C22.277(C23×D5), Dic5.14D4⋊48C2, C23.D5.70C22, D10⋊C4.48C22, C5⋊5(C22.57C24), (C2×Dic10).45C22, (C2×Dic5).132C23, (C4×Dic5).161C22, C10.D4.11C22, (C22×D5).115C23, C2.67(D4.10D10), (C22×Dic5).155C22, C4⋊C4⋊D5⋊45C2, (C2×C4×D5).146C22, (C5×C42⋊2C2)⋊11C2, (C5×C4⋊C4).207C22, (C2×C4).212(C22×D5), (C2×C5⋊D4).76C22, (C5×C22⋊C4).81C22, SmallGroup(320,1384)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 678 in 196 conjugacy classes, 91 normal (all characteristic)
C1, C2 [×3], C2 [×2], C4 [×13], C22, C22 [×6], C5, C2×C4 [×6], C2×C4 [×9], D4, Q8 [×3], C23, C23, D5, C10 [×3], C10, C42, C42 [×2], C22⋊C4 [×3], C22⋊C4 [×7], C4⋊C4 [×3], C4⋊C4 [×13], C22×C4 [×2], C2×D4, C2×Q8 [×3], Dic5 [×7], C20 [×6], D10 [×3], C2×C10, C2×C10 [×3], C22⋊Q8 [×4], C22.D4 [×2], C4.4D4, C42.C2 [×2], C42⋊2C2, C42⋊2C2 [×3], C4⋊Q8 [×2], Dic10 [×3], C4×D5, C2×Dic5 [×7], C2×Dic5, C5⋊D4, C2×C20 [×6], C22×D5, C22×C10, C22.57C24, C4×Dic5 [×2], C10.D4 [×7], C4⋊Dic5 [×6], D10⋊C4 [×5], C23.D5 [×2], C4×C20, C5×C22⋊C4 [×3], C5×C4⋊C4 [×3], C2×Dic10 [×3], C2×C4×D5, C22×Dic5, C2×C5⋊D4, C20⋊2Q8, C42⋊2D5, Dic5.14D4 [×2], C23.D10, D10.12D4, Dic5.5D4, C22.D20, C20⋊Q8, Dic5.Q8, C4.Dic10, D10⋊Q8, D10⋊2Q8, C4⋊C4⋊D5, C5×C42⋊2C2, C42.165D10
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C24, D10 [×7], 2+ (1+4), 2- (1+4) [×2], C22×D5 [×7], C22.57C24, C23×D5, D4⋊8D10, D4.10D10 [×2], C42.165D10
Generators and relations
G = < a,b,c,d | a4=b4=1, c10=d2=a2b2, ab=ba, cac-1=ab2, dad-1=a-1, cbc-1=a2b-1, dbd-1=b-1, dcd-1=c9 >
►On 160 points
Generators in S160
(1 89 24 149)(2 160 25 100)(3 91 26 151)(4 142 27 82)(5 93 28 153)(6 144 29 84)(7 95 30 155)(8 146 31 86)(9 97 32 157)(10 148 33 88)(11 99 34 159)(12 150 35 90)(13 81 36 141)(14 152 37 92)(15 83 38 143)(16 154 39 94)(17 85 40 145)(18 156 21 96)(19 87 22 147)(20 158 23 98)(41 72 120 130)(42 121 101 63)(43 74 102 132)(44 123 103 65)(45 76 104 134)(46 125 105 67)(47 78 106 136)(48 127 107 69)(49 80 108 138)(50 129 109 71)(51 62 110 140)(52 131 111 73)(53 64 112 122)(54 133 113 75)(55 66 114 124)(56 135 115 77)(57 68 116 126)(58 137 117 79)(59 70 118 128)(60 139 119 61)
(1 122 34 74)(2 133 35 65)(3 124 36 76)(4 135 37 67)(5 126 38 78)(6 137 39 69)(7 128 40 80)(8 139 21 71)(9 130 22 62)(10 121 23 73)(11 132 24 64)(12 123 25 75)(13 134 26 66)(14 125 27 77)(15 136 28 68)(16 127 29 79)(17 138 30 70)(18 129 31 61)(19 140 32 72)(20 131 33 63)(41 147 110 97)(42 158 111 88)(43 149 112 99)(44 160 113 90)(45 151 114 81)(46 142 115 92)(47 153 116 83)(48 144 117 94)(49 155 118 85)(50 146 119 96)(51 157 120 87)(52 148 101 98)(53 159 102 89)(54 150 103 100)(55 141 104 91)(56 152 105 82)(57 143 106 93)(58 154 107 84)(59 145 108 95)(60 156 109 86)
(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 26 31 36)(22 35 32 25)(23 24 33 34)(27 40 37 30)(28 29 38 39)(41 103 51 113)(42 112 52 102)(43 101 53 111)(44 110 54 120)(45 119 55 109)(46 108 56 118)(47 117 57 107)(48 106 58 116)(49 115 59 105)(50 104 60 114)(61 124 71 134)(62 133 72 123)(63 122 73 132)(64 131 74 121)(65 140 75 130)(66 129 76 139)(67 138 77 128)(68 127 78 137)(69 136 79 126)(70 125 80 135)(81 96 91 86)(82 85 92 95)(83 94 93 84)(87 90 97 100)(88 99 98 89)(141 156 151 146)(142 145 152 155)(143 154 153 144)(147 150 157 160)(148 159 158 149)
G:=sub<Sym(160)| (1,89,24,149)(2,160,25,100)(3,91,26,151)(4,142,27,82)(5,93,28,153)(6,144,29,84)(7,95,30,155)(8,146,31,86)(9,97,32,157)(10,148,33,88)(11,99,34,159)(12,150,35,90)(13,81,36,141)(14,152,37,92)(15,83,38,143)(16,154,39,94)(17,85,40,145)(18,156,21,96)(19,87,22,147)(20,158,23,98)(41,72,120,130)(42,121,101,63)(43,74,102,132)(44,123,103,65)(45,76,104,134)(46,125,105,67)(47,78,106,136)(48,127,107,69)(49,80,108,138)(50,129,109,71)(51,62,110,140)(52,131,111,73)(53,64,112,122)(54,133,113,75)(55,66,114,124)(56,135,115,77)(57,68,116,126)(58,137,117,79)(59,70,118,128)(60,139,119,61), (1,122,34,74)(2,133,35,65)(3,124,36,76)(4,135,37,67)(5,126,38,78)(6,137,39,69)(7,128,40,80)(8,139,21,71)(9,130,22,62)(10,121,23,73)(11,132,24,64)(12,123,25,75)(13,134,26,66)(14,125,27,77)(15,136,28,68)(16,127,29,79)(17,138,30,70)(18,129,31,61)(19,140,32,72)(20,131,33,63)(41,147,110,97)(42,158,111,88)(43,149,112,99)(44,160,113,90)(45,151,114,81)(46,142,115,92)(47,153,116,83)(48,144,117,94)(49,155,118,85)(50,146,119,96)(51,157,120,87)(52,148,101,98)(53,159,102,89)(54,150,103,100)(55,141,104,91)(56,152,105,82)(57,143,106,93)(58,154,107,84)(59,145,108,95)(60,156,109,86), (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)(21,26,31,36)(22,35,32,25)(23,24,33,34)(27,40,37,30)(28,29,38,39)(41,103,51,113)(42,112,52,102)(43,101,53,111)(44,110,54,120)(45,119,55,109)(46,108,56,118)(47,117,57,107)(48,106,58,116)(49,115,59,105)(50,104,60,114)(61,124,71,134)(62,133,72,123)(63,122,73,132)(64,131,74,121)(65,140,75,130)(66,129,76,139)(67,138,77,128)(68,127,78,137)(69,136,79,126)(70,125,80,135)(81,96,91,86)(82,85,92,95)(83,94,93,84)(87,90,97,100)(88,99,98,89)(141,156,151,146)(142,145,152,155)(143,154,153,144)(147,150,157,160)(148,159,158,149)>;
G:=Group( (1,89,24,149)(2,160,25,100)(3,91,26,151)(4,142,27,82)(5,93,28,153)(6,144,29,84)(7,95,30,155)(8,146,31,86)(9,97,32,157)(10,148,33,88)(11,99,34,159)(12,150,35,90)(13,81,36,141)(14,152,37,92)(15,83,38,143)(16,154,39,94)(17,85,40,145)(18,156,21,96)(19,87,22,147)(20,158,23,98)(41,72,120,130)(42,121,101,63)(43,74,102,132)(44,123,103,65)(45,76,104,134)(46,125,105,67)(47,78,106,136)(48,127,107,69)(49,80,108,138)(50,129,109,71)(51,62,110,140)(52,131,111,73)(53,64,112,122)(54,133,113,75)(55,66,114,124)(56,135,115,77)(57,68,116,126)(58,137,117,79)(59,70,118,128)(60,139,119,61), (1,122,34,74)(2,133,35,65)(3,124,36,76)(4,135,37,67)(5,126,38,78)(6,137,39,69)(7,128,40,80)(8,139,21,71)(9,130,22,62)(10,121,23,73)(11,132,24,64)(12,123,25,75)(13,134,26,66)(14,125,27,77)(15,136,28,68)(16,127,29,79)(17,138,30,70)(18,129,31,61)(19,140,32,72)(20,131,33,63)(41,147,110,97)(42,158,111,88)(43,149,112,99)(44,160,113,90)(45,151,114,81)(46,142,115,92)(47,153,116,83)(48,144,117,94)(49,155,118,85)(50,146,119,96)(51,157,120,87)(52,148,101,98)(53,159,102,89)(54,150,103,100)(55,141,104,91)(56,152,105,82)(57,143,106,93)(58,154,107,84)(59,145,108,95)(60,156,109,86), (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)(21,26,31,36)(22,35,32,25)(23,24,33,34)(27,40,37,30)(28,29,38,39)(41,103,51,113)(42,112,52,102)(43,101,53,111)(44,110,54,120)(45,119,55,109)(46,108,56,118)(47,117,57,107)(48,106,58,116)(49,115,59,105)(50,104,60,114)(61,124,71,134)(62,133,72,123)(63,122,73,132)(64,131,74,121)(65,140,75,130)(66,129,76,139)(67,138,77,128)(68,127,78,137)(69,136,79,126)(70,125,80,135)(81,96,91,86)(82,85,92,95)(83,94,93,84)(87,90,97,100)(88,99,98,89)(141,156,151,146)(142,145,152,155)(143,154,153,144)(147,150,157,160)(148,159,158,149) );
G=PermutationGroup([(1,89,24,149),(2,160,25,100),(3,91,26,151),(4,142,27,82),(5,93,28,153),(6,144,29,84),(7,95,30,155),(8,146,31,86),(9,97,32,157),(10,148,33,88),(11,99,34,159),(12,150,35,90),(13,81,36,141),(14,152,37,92),(15,83,38,143),(16,154,39,94),(17,85,40,145),(18,156,21,96),(19,87,22,147),(20,158,23,98),(41,72,120,130),(42,121,101,63),(43,74,102,132),(44,123,103,65),(45,76,104,134),(46,125,105,67),(47,78,106,136),(48,127,107,69),(49,80,108,138),(50,129,109,71),(51,62,110,140),(52,131,111,73),(53,64,112,122),(54,133,113,75),(55,66,114,124),(56,135,115,77),(57,68,116,126),(58,137,117,79),(59,70,118,128),(60,139,119,61)], [(1,122,34,74),(2,133,35,65),(3,124,36,76),(4,135,37,67),(5,126,38,78),(6,137,39,69),(7,128,40,80),(8,139,21,71),(9,130,22,62),(10,121,23,73),(11,132,24,64),(12,123,25,75),(13,134,26,66),(14,125,27,77),(15,136,28,68),(16,127,29,79),(17,138,30,70),(18,129,31,61),(19,140,32,72),(20,131,33,63),(41,147,110,97),(42,158,111,88),(43,149,112,99),(44,160,113,90),(45,151,114,81),(46,142,115,92),(47,153,116,83),(48,144,117,94),(49,155,118,85),(50,146,119,96),(51,157,120,87),(52,148,101,98),(53,159,102,89),(54,150,103,100),(55,141,104,91),(56,152,105,82),(57,143,106,93),(58,154,107,84),(59,145,108,95),(60,156,109,86)], [(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),(21,26,31,36),(22,35,32,25),(23,24,33,34),(27,40,37,30),(28,29,38,39),(41,103,51,113),(42,112,52,102),(43,101,53,111),(44,110,54,120),(45,119,55,109),(46,108,56,118),(47,117,57,107),(48,106,58,116),(49,115,59,105),(50,104,60,114),(61,124,71,134),(62,133,72,123),(63,122,73,132),(64,131,74,121),(65,140,75,130),(66,129,76,139),(67,138,77,128),(68,127,78,137),(69,136,79,126),(70,125,80,135),(81,96,91,86),(82,85,92,95),(83,94,93,84),(87,90,97,100),(88,99,98,89),(141,156,151,146),(142,145,152,155),(143,154,153,144),(147,150,157,160),(148,159,158,149)])
Matrix representation ►G ⊆ GL8(𝔽41)
| 2 | 13 | 37 | 15 | 0 | 0 | 0 | 0 |
| 28 | 39 | 26 | 4 | 0 | 0 | 0 | 0 |
| 2 | 13 | 39 | 28 | 0 | 0 | 0 | 0 |
| 28 | 39 | 13 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 39 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 39 | 28 |
| 0 | 0 | 0 | 0 | 0 | 0 | 13 | 2 |
| 1 | 0 | 39 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 39 | 0 | 0 | 0 | 0 |
| 1 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 3 | 3 | 39 | 39 | 0 | 0 | 0 | 0 |
| 38 | 21 | 2 | 27 | 0 | 0 | 0 | 0 |
| 2 | 2 | 38 | 38 | 0 | 0 | 0 | 0 |
| 39 | 14 | 3 | 20 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 28 | 28 |
| 0 | 0 | 0 | 0 | 0 | 0 | 13 | 32 |
| 3 | 3 | 39 | 39 | 0 | 0 | 0 | 0 |
| 21 | 38 | 27 | 2 | 0 | 0 | 0 | 0 |
| 2 | 2 | 38 | 38 | 0 | 0 | 0 | 0 |
| 14 | 39 | 20 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 13 | 13 |
| 0 | 0 | 0 | 0 | 0 | 0 | 9 | 28 |
G:=sub<GL(8,GF(41))| [2,28,2,28,0,0,0,0,13,39,13,39,0,0,0,0,37,26,39,13,0,0,0,0,15,4,28,2,0,0,0,0,0,0,0,0,39,13,0,0,0,0,0,0,28,2,0,0,0,0,0,0,0,0,39,13,0,0,0,0,0,0,28,2],[1,0,1,0,0,0,0,0,0,1,0,1,0,0,0,0,39,0,40,0,0,0,0,0,0,39,0,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[3,38,2,39,0,0,0,0,3,21,2,14,0,0,0,0,39,2,38,3,0,0,0,0,39,27,38,20,0,0,0,0,0,0,0,0,13,28,0,0,0,0,0,0,13,9,0,0,0,0,0,0,0,0,28,13,0,0,0,0,0,0,28,32],[3,21,2,14,0,0,0,0,3,38,2,39,0,0,0,0,39,27,38,20,0,0,0,0,39,2,38,3,0,0,0,0,0,0,0,0,13,9,0,0,0,0,0,0,13,28,0,0,0,0,0,0,0,0,13,9,0,0,0,0,0,0,13,28] >;
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | ··· | 4M | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 20A | ··· | 20L | 20M | ··· | 20R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 20 | 4 | ··· | 4 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | D10 | D10 | D10 | 2+ (1+4) | 2- (1+4) | D4⋊8D10 | D4.10D10 |
| kernel | C42.165D10 | C20⋊2Q8 | C42⋊2D5 | Dic5.14D4 | C23.D10 | D10.12D4 | Dic5.5D4 | C22.D20 | C20⋊Q8 | Dic5.Q8 | C4.Dic10 | D10⋊Q8 | D10⋊2Q8 | C4⋊C4⋊D5 | C5×C42⋊2C2 | C42⋊2C2 | C42 | C22⋊C4 | C4⋊C4 | C10 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 1 | 2 | 4 | 8 |
In GAP, Magma, Sage, TeX
C_4^2._{165}D_{10} % in TeX
G:=Group("C4^2.165D10"); // GroupNames label
G:=SmallGroup(320,1384);
// by ID
G=gap.SmallGroup(320,1384);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,219,1571,570,297,80,12550]);
// 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=a*b^2,d*a*d^-1=a^-1,c*b*c^-1=a^2*b^-1,d*b*d^-1=b^-1,d*c*d^-1=c^9>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀