metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.163D10, C10.1402+ (1+4), (C4×D20)⋊16C2, C4⋊2D20⋊37C2, C4⋊C4.214D10, C42⋊D5⋊8C2, C42⋊2C2⋊6D5, D10⋊D4⋊45C2, D20⋊8C4⋊42C2, (C4×C20).35C22, (C2×C20).96C23, C22⋊C4.81D10, D10.56(C4○D4), Dic5⋊4D4⋊37C2, (C2×C10).253C24, D10.13D4⋊42C2, C2.65(D4⋊8D10), C23.59(C22×D5), Dic5.49(C4○D4), Dic5.Q8⋊39C2, (C2×D20).175C22, C22.D20⋊30C2, C4⋊Dic5.318C22, (C22×C10).67C23, C22.274(C23×D5), D10⋊C4.46C22, (C2×Dic5).276C23, (C4×Dic5).160C22, (C22×D5).112C23, C5⋊11(C22.47C24), C10.D4.147C22, (C22×Dic5).153C22, (D5×C4⋊C4)⋊43C2, C4⋊C4⋊D5⋊43C2, C2.100(D5×C4○D4), (C5×C42⋊2C2)⋊8C2, C10.211(C2×C4○D4), (C2×C4×D5).144C22, (C2×C4).89(C22×D5), (C5×C4⋊C4).205C22, (C2×C5⋊D4).73C22, (C5×C22⋊C4).78C22, SmallGroup(320,1381)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 918 in 238 conjugacy classes, 95 normal (91 characteristic)
C1, C2 [×3], C2 [×5], C4 [×12], C22, C22 [×13], C5, C2×C4 [×6], C2×C4 [×13], D4 [×10], C23, C23 [×3], D5 [×4], C10 [×3], C10, C42, C42 [×2], C22⋊C4 [×3], C22⋊C4 [×7], C4⋊C4 [×3], C4⋊C4 [×7], C22×C4 [×6], C2×D4 [×6], Dic5 [×2], Dic5 [×4], C20 [×6], D10 [×2], D10 [×8], C2×C10, C2×C10 [×3], C2×C4⋊C4, C42⋊C2, C4×D4 [×4], C4⋊D4 [×4], C22.D4 [×2], C42.C2, C42⋊2C2, C42⋊2C2, C4×D5 [×7], D20 [×5], C2×Dic5 [×5], C2×Dic5, C5⋊D4 [×5], C2×C20 [×6], C22×D5 [×3], C22×C10, C22.47C24, C4×Dic5 [×2], C10.D4 [×5], C4⋊Dic5 [×2], D10⋊C4 [×7], C4×C20, C5×C22⋊C4 [×3], C5×C4⋊C4 [×3], C2×C4×D5 [×5], C2×D20 [×3], C22×Dic5, C2×C5⋊D4 [×3], C42⋊D5, C4×D20, Dic5⋊4D4 [×2], D10⋊D4 [×3], C22.D20, Dic5.Q8, D5×C4⋊C4, D20⋊8C4, D10.13D4, C4⋊2D20, C4⋊C4⋊D5, C5×C42⋊2C2, C42.163D10
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×4], C24, D10 [×7], C2×C4○D4 [×2], 2+ (1+4), C22×D5 [×7], C22.47C24, C23×D5, D5×C4○D4 [×2], D4⋊8D10, C42.163D10
Generators and relations
G = < a,b,c,d | a4=b4=1, c10=d2=a2, ab=ba, cac-1=dad-1=a-1b2, cbc-1=a2b, dbd-1=a2b-1, dcd-1=c9 >
►On 160 points
Generators in S160
(1 160 11 150)(2 53 12 43)(3 142 13 152)(4 55 14 45)(5 144 15 154)(6 57 16 47)(7 146 17 156)(8 59 18 49)(9 148 19 158)(10 41 20 51)(21 69 31 79)(22 106 32 116)(23 71 33 61)(24 108 34 118)(25 73 35 63)(26 110 36 120)(27 75 37 65)(28 112 38 102)(29 77 39 67)(30 114 40 104)(42 89 52 99)(44 91 54 81)(46 93 56 83)(48 95 58 85)(50 97 60 87)(62 125 72 135)(64 127 74 137)(66 129 76 139)(68 131 78 121)(70 133 80 123)(82 153 92 143)(84 155 94 145)(86 157 96 147)(88 159 98 149)(90 141 100 151)(101 128 111 138)(103 130 113 140)(105 132 115 122)(107 134 117 124)(109 136 119 126)
(1 135 99 24)(2 126 100 35)(3 137 81 26)(4 128 82 37)(5 139 83 28)(6 130 84 39)(7 121 85 30)(8 132 86 21)(9 123 87 32)(10 134 88 23)(11 125 89 34)(12 136 90 25)(13 127 91 36)(14 138 92 27)(15 129 93 38)(16 140 94 29)(17 131 95 40)(18 122 96 31)(19 133 97 22)(20 124 98 33)(41 117 159 71)(42 108 160 62)(43 119 141 73)(44 110 142 64)(45 101 143 75)(46 112 144 66)(47 103 145 77)(48 114 146 68)(49 105 147 79)(50 116 148 70)(51 107 149 61)(52 118 150 72)(53 109 151 63)(54 120 152 74)(55 111 153 65)(56 102 154 76)(57 113 155 67)(58 104 156 78)(59 115 157 69)(60 106 158 80)
(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 28 11 38)(2 37 12 27)(3 26 13 36)(4 35 14 25)(5 24 15 34)(6 33 16 23)(7 22 17 32)(8 31 18 21)(9 40 19 30)(10 29 20 39)(41 113 51 103)(42 102 52 112)(43 111 53 101)(44 120 54 110)(45 109 55 119)(46 118 56 108)(47 107 57 117)(48 116 58 106)(49 105 59 115)(50 114 60 104)(61 155 71 145)(62 144 72 154)(63 153 73 143)(64 142 74 152)(65 151 75 141)(66 160 76 150)(67 149 77 159)(68 158 78 148)(69 147 79 157)(70 156 80 146)(81 137 91 127)(82 126 92 136)(83 135 93 125)(84 124 94 134)(85 133 95 123)(86 122 96 132)(87 131 97 121)(88 140 98 130)(89 129 99 139)(90 138 100 128)
G:=sub<Sym(160)| (1,160,11,150)(2,53,12,43)(3,142,13,152)(4,55,14,45)(5,144,15,154)(6,57,16,47)(7,146,17,156)(8,59,18,49)(9,148,19,158)(10,41,20,51)(21,69,31,79)(22,106,32,116)(23,71,33,61)(24,108,34,118)(25,73,35,63)(26,110,36,120)(27,75,37,65)(28,112,38,102)(29,77,39,67)(30,114,40,104)(42,89,52,99)(44,91,54,81)(46,93,56,83)(48,95,58,85)(50,97,60,87)(62,125,72,135)(64,127,74,137)(66,129,76,139)(68,131,78,121)(70,133,80,123)(82,153,92,143)(84,155,94,145)(86,157,96,147)(88,159,98,149)(90,141,100,151)(101,128,111,138)(103,130,113,140)(105,132,115,122)(107,134,117,124)(109,136,119,126), (1,135,99,24)(2,126,100,35)(3,137,81,26)(4,128,82,37)(5,139,83,28)(6,130,84,39)(7,121,85,30)(8,132,86,21)(9,123,87,32)(10,134,88,23)(11,125,89,34)(12,136,90,25)(13,127,91,36)(14,138,92,27)(15,129,93,38)(16,140,94,29)(17,131,95,40)(18,122,96,31)(19,133,97,22)(20,124,98,33)(41,117,159,71)(42,108,160,62)(43,119,141,73)(44,110,142,64)(45,101,143,75)(46,112,144,66)(47,103,145,77)(48,114,146,68)(49,105,147,79)(50,116,148,70)(51,107,149,61)(52,118,150,72)(53,109,151,63)(54,120,152,74)(55,111,153,65)(56,102,154,76)(57,113,155,67)(58,104,156,78)(59,115,157,69)(60,106,158,80), (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), (1,28,11,38)(2,37,12,27)(3,26,13,36)(4,35,14,25)(5,24,15,34)(6,33,16,23)(7,22,17,32)(8,31,18,21)(9,40,19,30)(10,29,20,39)(41,113,51,103)(42,102,52,112)(43,111,53,101)(44,120,54,110)(45,109,55,119)(46,118,56,108)(47,107,57,117)(48,116,58,106)(49,105,59,115)(50,114,60,104)(61,155,71,145)(62,144,72,154)(63,153,73,143)(64,142,74,152)(65,151,75,141)(66,160,76,150)(67,149,77,159)(68,158,78,148)(69,147,79,157)(70,156,80,146)(81,137,91,127)(82,126,92,136)(83,135,93,125)(84,124,94,134)(85,133,95,123)(86,122,96,132)(87,131,97,121)(88,140,98,130)(89,129,99,139)(90,138,100,128)>;
G:=Group( (1,160,11,150)(2,53,12,43)(3,142,13,152)(4,55,14,45)(5,144,15,154)(6,57,16,47)(7,146,17,156)(8,59,18,49)(9,148,19,158)(10,41,20,51)(21,69,31,79)(22,106,32,116)(23,71,33,61)(24,108,34,118)(25,73,35,63)(26,110,36,120)(27,75,37,65)(28,112,38,102)(29,77,39,67)(30,114,40,104)(42,89,52,99)(44,91,54,81)(46,93,56,83)(48,95,58,85)(50,97,60,87)(62,125,72,135)(64,127,74,137)(66,129,76,139)(68,131,78,121)(70,133,80,123)(82,153,92,143)(84,155,94,145)(86,157,96,147)(88,159,98,149)(90,141,100,151)(101,128,111,138)(103,130,113,140)(105,132,115,122)(107,134,117,124)(109,136,119,126), (1,135,99,24)(2,126,100,35)(3,137,81,26)(4,128,82,37)(5,139,83,28)(6,130,84,39)(7,121,85,30)(8,132,86,21)(9,123,87,32)(10,134,88,23)(11,125,89,34)(12,136,90,25)(13,127,91,36)(14,138,92,27)(15,129,93,38)(16,140,94,29)(17,131,95,40)(18,122,96,31)(19,133,97,22)(20,124,98,33)(41,117,159,71)(42,108,160,62)(43,119,141,73)(44,110,142,64)(45,101,143,75)(46,112,144,66)(47,103,145,77)(48,114,146,68)(49,105,147,79)(50,116,148,70)(51,107,149,61)(52,118,150,72)(53,109,151,63)(54,120,152,74)(55,111,153,65)(56,102,154,76)(57,113,155,67)(58,104,156,78)(59,115,157,69)(60,106,158,80), (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), (1,28,11,38)(2,37,12,27)(3,26,13,36)(4,35,14,25)(5,24,15,34)(6,33,16,23)(7,22,17,32)(8,31,18,21)(9,40,19,30)(10,29,20,39)(41,113,51,103)(42,102,52,112)(43,111,53,101)(44,120,54,110)(45,109,55,119)(46,118,56,108)(47,107,57,117)(48,116,58,106)(49,105,59,115)(50,114,60,104)(61,155,71,145)(62,144,72,154)(63,153,73,143)(64,142,74,152)(65,151,75,141)(66,160,76,150)(67,149,77,159)(68,158,78,148)(69,147,79,157)(70,156,80,146)(81,137,91,127)(82,126,92,136)(83,135,93,125)(84,124,94,134)(85,133,95,123)(86,122,96,132)(87,131,97,121)(88,140,98,130)(89,129,99,139)(90,138,100,128) );
G=PermutationGroup([(1,160,11,150),(2,53,12,43),(3,142,13,152),(4,55,14,45),(5,144,15,154),(6,57,16,47),(7,146,17,156),(8,59,18,49),(9,148,19,158),(10,41,20,51),(21,69,31,79),(22,106,32,116),(23,71,33,61),(24,108,34,118),(25,73,35,63),(26,110,36,120),(27,75,37,65),(28,112,38,102),(29,77,39,67),(30,114,40,104),(42,89,52,99),(44,91,54,81),(46,93,56,83),(48,95,58,85),(50,97,60,87),(62,125,72,135),(64,127,74,137),(66,129,76,139),(68,131,78,121),(70,133,80,123),(82,153,92,143),(84,155,94,145),(86,157,96,147),(88,159,98,149),(90,141,100,151),(101,128,111,138),(103,130,113,140),(105,132,115,122),(107,134,117,124),(109,136,119,126)], [(1,135,99,24),(2,126,100,35),(3,137,81,26),(4,128,82,37),(5,139,83,28),(6,130,84,39),(7,121,85,30),(8,132,86,21),(9,123,87,32),(10,134,88,23),(11,125,89,34),(12,136,90,25),(13,127,91,36),(14,138,92,27),(15,129,93,38),(16,140,94,29),(17,131,95,40),(18,122,96,31),(19,133,97,22),(20,124,98,33),(41,117,159,71),(42,108,160,62),(43,119,141,73),(44,110,142,64),(45,101,143,75),(46,112,144,66),(47,103,145,77),(48,114,146,68),(49,105,147,79),(50,116,148,70),(51,107,149,61),(52,118,150,72),(53,109,151,63),(54,120,152,74),(55,111,153,65),(56,102,154,76),(57,113,155,67),(58,104,156,78),(59,115,157,69),(60,106,158,80)], [(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)], [(1,28,11,38),(2,37,12,27),(3,26,13,36),(4,35,14,25),(5,24,15,34),(6,33,16,23),(7,22,17,32),(8,31,18,21),(9,40,19,30),(10,29,20,39),(41,113,51,103),(42,102,52,112),(43,111,53,101),(44,120,54,110),(45,109,55,119),(46,118,56,108),(47,107,57,117),(48,116,58,106),(49,105,59,115),(50,114,60,104),(61,155,71,145),(62,144,72,154),(63,153,73,143),(64,142,74,152),(65,151,75,141),(66,160,76,150),(67,149,77,159),(68,158,78,148),(69,147,79,157),(70,156,80,146),(81,137,91,127),(82,126,92,136),(83,135,93,125),(84,124,94,134),(85,133,95,123),(86,122,96,132),(87,131,97,121),(88,140,98,130),(89,129,99,139),(90,138,100,128)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 18 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 21 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 2 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 34 | 34 | 0 | 0 | 0 | 0 |
| 7 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 5 | 0 | 0 |
| 0 | 0 | 16 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 39 |
| 0 | 0 | 0 | 0 | 1 | 40 |
| 7 | 7 | 0 | 0 | 0 | 0 |
| 40 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 2 |
| 0 | 0 | 0 | 0 | 40 | 1 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,32,0,0,0,0,0,18,9],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,21,0,0,0,0,0,9,0,0,0,0,0,0,40,0,0,0,0,0,2,1],[34,7,0,0,0,0,34,1,0,0,0,0,0,0,40,16,0,0,0,0,5,1,0,0,0,0,0,0,1,1,0,0,0,0,39,40],[7,40,0,0,0,0,7,34,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,40,40,0,0,0,0,2,1] >;
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 4O | 4P | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 20A | ··· | 20L | 20M | ··· | 20R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 10 | 10 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | C4○D4 | D10 | D10 | D10 | 2+ (1+4) | D5×C4○D4 | D4⋊8D10 |
| kernel | C42.163D10 | C42⋊D5 | C4×D20 | Dic5⋊4D4 | D10⋊D4 | C22.D20 | Dic5.Q8 | D5×C4⋊C4 | D20⋊8C4 | D10.13D4 | C4⋊2D20 | C4⋊C4⋊D5 | C5×C42⋊2C2 | C42⋊2C2 | Dic5 | D10 | C42 | C22⋊C4 | C4⋊C4 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 2 | 6 | 6 | 1 | 8 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{163}D_{10} % in TeX
G:=Group("C4^2.163D10"); // GroupNames label
G:=SmallGroup(320,1381);
// by ID
G=gap.SmallGroup(320,1381);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,219,184,1571,297,192,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1*b^2,c*b*c^-1=a^2*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^9>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀