metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.97D10, C10.982+ (1+4), C20⋊Q8⋊12C2, D10⋊D4⋊4C2, C20⋊7D4⋊30C2, C4⋊2D20⋊13C2, C4⋊C4.311D10, C4.D20⋊7C2, C42⋊D5⋊4C2, D20⋊8C4⋊13C2, C4.97(C4○D20), C42⋊C2⋊16D5, (C4×C20).27C22, (C2×C10).76C24, C20.199(C4○D4), (C2×C20).697C23, C22⋊C4.100D10, Dic5.5D4⋊4C2, (C2×D20).26C22, (C22×C4).197D10, C2.10(D4⋊8D10), C23.87(C22×D5), Dic5.34(C4○D4), C4⋊Dic5.196C22, (C2×Dic5).29C23, (C22×D5).24C23, C22.105(C23×D5), C23.D5.98C22, D10⋊C4.83C22, (C22×C20).233C22, (C22×C10).146C23, C5⋊1(C22.49C24), (C4×Dic5).218C22, (C2×Dic10).26C22, C10.D4.106C22, (C4×C5⋊D4)⋊13C2, C2.15(D5×C4○D4), C4⋊C4⋊7D5⋊12C2, C2.35(C2×C4○D20), C10.32(C2×C4○D4), (C2×C4×D5).245C22, (C5×C42⋊C2)⋊18C2, (C5×C4⋊C4).312C22, (C2×C4).278(C22×D5), (C2×C5⋊D4).113C22, (C5×C22⋊C4).115C22, SmallGroup(320,1204)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 902 in 236 conjugacy classes, 97 normal (43 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×11], C22, C22 [×12], C5, C2×C4 [×2], C2×C4 [×4], C2×C4 [×13], D4 [×8], Q8 [×2], C23, C23 [×3], D5 [×3], C10 [×3], C10, C42 [×2], C42 [×3], C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4 [×2], C4⋊C4 [×4], C22×C4, C22×C4 [×3], C2×D4 [×6], C2×Q8 [×2], Dic5 [×2], Dic5 [×4], C20 [×2], C20 [×5], D10 [×9], C2×C10, C2×C10 [×3], C42⋊C2, C42⋊C2 [×3], C4×D4 [×2], C4⋊D4 [×4], C4.4D4 [×4], C4⋊Q8, Dic10 [×2], C4×D5 [×6], D20 [×4], C2×Dic5 [×3], C2×Dic5 [×2], C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×4], C2×C20 [×2], C22×D5, C22×D5 [×2], C22×C10, C22.49C24, C4×Dic5, C4×Dic5 [×2], C10.D4, C10.D4 [×2], C4⋊Dic5, D10⋊C4, D10⋊C4 [×8], C23.D5, C4×C20 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4 [×2], C2×Dic10 [×2], C2×C4×D5, C2×C4×D5 [×2], C2×D20, C2×D20 [×2], C2×C5⋊D4, C2×C5⋊D4 [×2], C22×C20, C42⋊D5 [×2], C4.D20 [×2], D10⋊D4 [×2], Dic5.5D4 [×2], C20⋊Q8, C4⋊C4⋊7D5, D20⋊8C4, C4⋊2D20, C4×C5⋊D4, C20⋊7D4, C5×C42⋊C2, C42.97D10
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], C22.49C24, C4○D20 [×2], C23×D5, C2×C4○D20, D5×C4○D4, D4⋊8D10, C42.97D10
Generators and relations
G = < a,b,c,d | a4=b4=1, c10=d2=b2, ab=ba, cac-1=ab2, ad=da, bc=cb, dbd-1=a2b, dcd-1=c9 >
►On 160 points
Generators in S160
(1 153 49 89)(2 144 50 100)(3 155 51 91)(4 146 52 82)(5 157 53 93)(6 148 54 84)(7 159 55 95)(8 150 56 86)(9 141 57 97)(10 152 58 88)(11 143 59 99)(12 154 60 90)(13 145 41 81)(14 156 42 92)(15 147 43 83)(16 158 44 94)(17 149 45 85)(18 160 46 96)(19 151 47 87)(20 142 48 98)(21 129 109 76)(22 140 110 67)(23 131 111 78)(24 122 112 69)(25 133 113 80)(26 124 114 71)(27 135 115 62)(28 126 116 73)(29 137 117 64)(30 128 118 75)(31 139 119 66)(32 130 120 77)(33 121 101 68)(34 132 102 79)(35 123 103 70)(36 134 104 61)(37 125 105 72)(38 136 106 63)(39 127 107 74)(40 138 108 65)
(1 26 11 36)(2 27 12 37)(3 28 13 38)(4 29 14 39)(5 30 15 40)(6 31 16 21)(7 32 17 22)(8 33 18 23)(9 34 19 24)(10 35 20 25)(41 106 51 116)(42 107 52 117)(43 108 53 118)(44 109 54 119)(45 110 55 120)(46 111 56 101)(47 112 57 102)(48 113 58 103)(49 114 59 104)(50 115 60 105)(61 89 71 99)(62 90 72 100)(63 91 73 81)(64 92 74 82)(65 93 75 83)(66 94 76 84)(67 95 77 85)(68 96 78 86)(69 97 79 87)(70 98 80 88)(121 160 131 150)(122 141 132 151)(123 142 133 152)(124 143 134 153)(125 144 135 154)(126 145 136 155)(127 146 137 156)(128 147 138 157)(129 148 139 158)(130 149 140 159)
(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 65 11 75)(2 74 12 64)(3 63 13 73)(4 72 14 62)(5 61 15 71)(6 70 16 80)(7 79 17 69)(8 68 18 78)(9 77 19 67)(10 66 20 76)(21 152 31 142)(22 141 32 151)(23 150 33 160)(24 159 34 149)(25 148 35 158)(26 157 36 147)(27 146 37 156)(28 155 38 145)(29 144 39 154)(30 153 40 143)(41 126 51 136)(42 135 52 125)(43 124 53 134)(44 133 54 123)(45 122 55 132)(46 131 56 121)(47 140 57 130)(48 129 58 139)(49 138 59 128)(50 127 60 137)(81 116 91 106)(82 105 92 115)(83 114 93 104)(84 103 94 113)(85 112 95 102)(86 101 96 111)(87 110 97 120)(88 119 98 109)(89 108 99 118)(90 117 100 107)
G:=sub<Sym(160)| (1,153,49,89)(2,144,50,100)(3,155,51,91)(4,146,52,82)(5,157,53,93)(6,148,54,84)(7,159,55,95)(8,150,56,86)(9,141,57,97)(10,152,58,88)(11,143,59,99)(12,154,60,90)(13,145,41,81)(14,156,42,92)(15,147,43,83)(16,158,44,94)(17,149,45,85)(18,160,46,96)(19,151,47,87)(20,142,48,98)(21,129,109,76)(22,140,110,67)(23,131,111,78)(24,122,112,69)(25,133,113,80)(26,124,114,71)(27,135,115,62)(28,126,116,73)(29,137,117,64)(30,128,118,75)(31,139,119,66)(32,130,120,77)(33,121,101,68)(34,132,102,79)(35,123,103,70)(36,134,104,61)(37,125,105,72)(38,136,106,63)(39,127,107,74)(40,138,108,65), (1,26,11,36)(2,27,12,37)(3,28,13,38)(4,29,14,39)(5,30,15,40)(6,31,16,21)(7,32,17,22)(8,33,18,23)(9,34,19,24)(10,35,20,25)(41,106,51,116)(42,107,52,117)(43,108,53,118)(44,109,54,119)(45,110,55,120)(46,111,56,101)(47,112,57,102)(48,113,58,103)(49,114,59,104)(50,115,60,105)(61,89,71,99)(62,90,72,100)(63,91,73,81)(64,92,74,82)(65,93,75,83)(66,94,76,84)(67,95,77,85)(68,96,78,86)(69,97,79,87)(70,98,80,88)(121,160,131,150)(122,141,132,151)(123,142,133,152)(124,143,134,153)(125,144,135,154)(126,145,136,155)(127,146,137,156)(128,147,138,157)(129,148,139,158)(130,149,140,159), (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), (1,65,11,75)(2,74,12,64)(3,63,13,73)(4,72,14,62)(5,61,15,71)(6,70,16,80)(7,79,17,69)(8,68,18,78)(9,77,19,67)(10,66,20,76)(21,152,31,142)(22,141,32,151)(23,150,33,160)(24,159,34,149)(25,148,35,158)(26,157,36,147)(27,146,37,156)(28,155,38,145)(29,144,39,154)(30,153,40,143)(41,126,51,136)(42,135,52,125)(43,124,53,134)(44,133,54,123)(45,122,55,132)(46,131,56,121)(47,140,57,130)(48,129,58,139)(49,138,59,128)(50,127,60,137)(81,116,91,106)(82,105,92,115)(83,114,93,104)(84,103,94,113)(85,112,95,102)(86,101,96,111)(87,110,97,120)(88,119,98,109)(89,108,99,118)(90,117,100,107)>;
G:=Group( (1,153,49,89)(2,144,50,100)(3,155,51,91)(4,146,52,82)(5,157,53,93)(6,148,54,84)(7,159,55,95)(8,150,56,86)(9,141,57,97)(10,152,58,88)(11,143,59,99)(12,154,60,90)(13,145,41,81)(14,156,42,92)(15,147,43,83)(16,158,44,94)(17,149,45,85)(18,160,46,96)(19,151,47,87)(20,142,48,98)(21,129,109,76)(22,140,110,67)(23,131,111,78)(24,122,112,69)(25,133,113,80)(26,124,114,71)(27,135,115,62)(28,126,116,73)(29,137,117,64)(30,128,118,75)(31,139,119,66)(32,130,120,77)(33,121,101,68)(34,132,102,79)(35,123,103,70)(36,134,104,61)(37,125,105,72)(38,136,106,63)(39,127,107,74)(40,138,108,65), (1,26,11,36)(2,27,12,37)(3,28,13,38)(4,29,14,39)(5,30,15,40)(6,31,16,21)(7,32,17,22)(8,33,18,23)(9,34,19,24)(10,35,20,25)(41,106,51,116)(42,107,52,117)(43,108,53,118)(44,109,54,119)(45,110,55,120)(46,111,56,101)(47,112,57,102)(48,113,58,103)(49,114,59,104)(50,115,60,105)(61,89,71,99)(62,90,72,100)(63,91,73,81)(64,92,74,82)(65,93,75,83)(66,94,76,84)(67,95,77,85)(68,96,78,86)(69,97,79,87)(70,98,80,88)(121,160,131,150)(122,141,132,151)(123,142,133,152)(124,143,134,153)(125,144,135,154)(126,145,136,155)(127,146,137,156)(128,147,138,157)(129,148,139,158)(130,149,140,159), (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), (1,65,11,75)(2,74,12,64)(3,63,13,73)(4,72,14,62)(5,61,15,71)(6,70,16,80)(7,79,17,69)(8,68,18,78)(9,77,19,67)(10,66,20,76)(21,152,31,142)(22,141,32,151)(23,150,33,160)(24,159,34,149)(25,148,35,158)(26,157,36,147)(27,146,37,156)(28,155,38,145)(29,144,39,154)(30,153,40,143)(41,126,51,136)(42,135,52,125)(43,124,53,134)(44,133,54,123)(45,122,55,132)(46,131,56,121)(47,140,57,130)(48,129,58,139)(49,138,59,128)(50,127,60,137)(81,116,91,106)(82,105,92,115)(83,114,93,104)(84,103,94,113)(85,112,95,102)(86,101,96,111)(87,110,97,120)(88,119,98,109)(89,108,99,118)(90,117,100,107) );
G=PermutationGroup([(1,153,49,89),(2,144,50,100),(3,155,51,91),(4,146,52,82),(5,157,53,93),(6,148,54,84),(7,159,55,95),(8,150,56,86),(9,141,57,97),(10,152,58,88),(11,143,59,99),(12,154,60,90),(13,145,41,81),(14,156,42,92),(15,147,43,83),(16,158,44,94),(17,149,45,85),(18,160,46,96),(19,151,47,87),(20,142,48,98),(21,129,109,76),(22,140,110,67),(23,131,111,78),(24,122,112,69),(25,133,113,80),(26,124,114,71),(27,135,115,62),(28,126,116,73),(29,137,117,64),(30,128,118,75),(31,139,119,66),(32,130,120,77),(33,121,101,68),(34,132,102,79),(35,123,103,70),(36,134,104,61),(37,125,105,72),(38,136,106,63),(39,127,107,74),(40,138,108,65)], [(1,26,11,36),(2,27,12,37),(3,28,13,38),(4,29,14,39),(5,30,15,40),(6,31,16,21),(7,32,17,22),(8,33,18,23),(9,34,19,24),(10,35,20,25),(41,106,51,116),(42,107,52,117),(43,108,53,118),(44,109,54,119),(45,110,55,120),(46,111,56,101),(47,112,57,102),(48,113,58,103),(49,114,59,104),(50,115,60,105),(61,89,71,99),(62,90,72,100),(63,91,73,81),(64,92,74,82),(65,93,75,83),(66,94,76,84),(67,95,77,85),(68,96,78,86),(69,97,79,87),(70,98,80,88),(121,160,131,150),(122,141,132,151),(123,142,133,152),(124,143,134,153),(125,144,135,154),(126,145,136,155),(127,146,137,156),(128,147,138,157),(129,148,139,158),(130,149,140,159)], [(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)], [(1,65,11,75),(2,74,12,64),(3,63,13,73),(4,72,14,62),(5,61,15,71),(6,70,16,80),(7,79,17,69),(8,68,18,78),(9,77,19,67),(10,66,20,76),(21,152,31,142),(22,141,32,151),(23,150,33,160),(24,159,34,149),(25,148,35,158),(26,157,36,147),(27,146,37,156),(28,155,38,145),(29,144,39,154),(30,153,40,143),(41,126,51,136),(42,135,52,125),(43,124,53,134),(44,133,54,123),(45,122,55,132),(46,131,56,121),(47,140,57,130),(48,129,58,139),(49,138,59,128),(50,127,60,137),(81,116,91,106),(82,105,92,115),(83,114,93,104),(84,103,94,113),(85,112,95,102),(86,101,96,111),(87,110,97,120),(88,119,98,109),(89,108,99,118),(90,117,100,107)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 39 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 6 | 6 | 0 | 0 | 0 | 0 |
| 35 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 7 |
| 0 | 0 | 0 | 0 | 18 | 23 |
| 35 | 35 | 0 | 0 | 0 | 0 |
| 40 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,0,0,0,39,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[6,35,0,0,0,0,6,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,18,18,0,0,0,0,7,23],[35,40,0,0,0,0,35,6,0,0,0,0,0,0,0,9,0,0,0,0,32,0,0,0,0,0,0,0,9,0,0,0,0,0,0,9] >;
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20AB |
| 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 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | C4○D4 | D10 | D10 | D10 | D10 | C4○D20 | 2+ (1+4) | D5×C4○D4 | D4⋊8D10 |
| kernel | C42.97D10 | C42⋊D5 | C4.D20 | D10⋊D4 | Dic5.5D4 | C20⋊Q8 | C4⋊C4⋊7D5 | D20⋊8C4 | C4⋊2D20 | C4×C5⋊D4 | C20⋊7D4 | C5×C42⋊C2 | C42⋊C2 | Dic5 | C20 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C4 | C10 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 4 | 2 | 16 | 1 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{97}D_{10} % in TeX
G:=Group("C4^2.97D10"); // GroupNames label
G:=SmallGroup(320,1204);
// by ID
G=gap.SmallGroup(320,1204);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,387,100,675,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=d^2=b^2,a*b=b*a,c*a*c^-1=a*b^2,a*d=d*a,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^9>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀